Quadrant Model of Reality Book 7 Art and Philosophy

Flight formation aerobatics are flown by teams of up to sixteen aircraft, although most teams fly between four and ten aircraft. 16 is the squares of the quadrant model

The "Finger-four" formation (also known as the "four finger formation"), is a flight formation used by fighter aircraft. It consists of four aircraft, and four of these formations can be combined into a squadron formation. The formation consists of a flight of four aircraft, composed of a "lead element" and a "second element", each of two aircraft. When viewing the formation from above, the positions of the planes resemble the tips of the four fingers of a human right hand (without the thumb), giving the formation its name. Four Finger Formation.png Four Finger Squadron.PNG The lead element is made up of the flight leader at the very front of the formation and one wingman to his rear left. The second element is made up of an additional two planes, the element leader and his wingman. The element leader is to the right and rear of the flight leader, followed by the element wingman to his right and rear. Both the flight leader and element leader have offensive roles, in that they are the ones to open fire on enemy aircraft while the flight remains intact. Their wingmen have a defensive role — the flight wingman covers the rear of the second element and the element wingman covers the rear of the element lead. Four of these flights can be assembled to form a squadron formation which consists of two staggered lines of fighters, one in front of the other. Each flight is usually designated by a color (i.e. Red, Blue, Yellow, and Green). A squadron formation is 16 squares. That is the squares of the quadrant model.

The formation was developed by several air forces independently in the 1930s. The Finnish Air Force adopted it during 1934-1935.[1] [2] Luftwaffe pilots developed the formation independently in 1938 during the Spanish Civil War, and were the first to use it in combat. Most notable in its development and use in the Luftwaffe were Günther Lützow and Werner Mölders and their fellow airmen. In the German Luftwaffe the flight (German: Schwarm) was made up of two pairs (German: Rotte) of aircraft. Each Rotte was composed of a leader and a wingman. The aircraft in the Schwarm had greater vertical and horizontal separation, so they were free to scan in all directions for enemy aircraft rather than focusing on maintaining a close formation. This allowed the pilots to maintain greater situational awareness and reduce the chance of being spotted by the enemy due to the looser formation. The two Rotten could split up at any time and attack on their own. The Rottenführer (pair leader) would attack enemy aircraft, leaving his wingman to scan for threats and protect him while he engaged the enemy. The Finnish Air Force's approach was even more flexible by allowing the pilot who spotted the enemy to become the leader of the pair or even the whole flight for the duration of the attack as he had the best situational awareness at that moment in time. The Luftwaffe continued the use of this formation during the Battle of Britain, in which its effectiveness was shown to be considerably greater than the standard three-aircraft "Vic" close formation used by the Royal Air Force (RAF).[citation needed] The RAF and later the United States Army Air Forces (USAAF) and Soviet Air Forces adopted this formation and used it themselves against the Luftwaffe.[citation needed] The Finnish Air Force proved the effectiveness by achieving a 16:1 kill ratio with the finger-four during the 1939-1940 Winter War against the Soviet Air Force, which at the time used the conventional Vic formation and superior aircraft.[citation needed] The Soviet air force units in the Spanish Civil War adopted the formation against the Germans but reverted to the "V" on return to the Soviet Union. The flying ace Douglas Bader was the first RAF pilot to adopt the formation in 1940. The United States Army Air Corps and Naval Aviation began using a concept called "Fighting Pair" from 1940–41. Japan too adopted the finger-four formation during World War II.[3][4][5] Missing man formation Main article: Missing man formation The finger-four formation became less common after World War II. However, it is still used in the "Missing Man Formation" at pilots' funeral ceremonies. The formation performs a fly-by in level flight over the funeral, at which point the second element leader climbs vertically and departs the formation, symbolizing the departure of the person being honored.

At the outbreak of the Second World War the Vic was still in use by both bombers and fighter formations in most air forces; however the German air forces fighter units had changed to the more flexible and aggressive pair (Rotte) and four (Schwarm) combination. These comprised a pair (leader and wingman) and four (two pairs) in a “finger-four” arrangement

The missing man formation is an aerial salute performed as part of a flypast of aircraft at a funeral or memorial event, typically in memory of a fallen pilot.[1][2] The formation is often called the "missing man flyby" or "flypast".[3] Several variants of the formation are seen. The formation most commonly used in the United States is based on the “finger-four” aircraft combat formation composed of a pair of two-aircraft elements.[4] The aircraft fly in a V-shape with the flight leader at the point and his wingman on his left. The second element leader and his wingman fly to his right. The formation flies over the ceremony low enough to be clearly seen and the second element leader abruptly pulls up out of the formation while the rest of the formation continues in level flight until all aircraft are out of sight. In an older variant the formation is flown with the second element leader position conspicuously empty. In another variation, the flight approaches from the south, preferably near sundown, and one of the aircraft will suddenly split off to the west, flying into the sunset. In all cases, the aircraft performing the pull-up, split off, or missing from the formation, is honoring the person (or persons) who has died, and it represents their departure to the heavens. In the movie Hell Divers from 1932, the closing flyby shows a missing man formation. U.S. Navy F/A-18 jets fly a missing man formation at a memorial service for astronaut Neil Armstrong on 31 August 2012. In 1936, King George V received the first recorded flypast for a non-RAF funeral. The United States adopted the tradition in 1938 during the funeral for Major General Oscar Westover with over 50 aircraft and one blank file.[3] By the end of World War II, the missing man formation had evolved to include the pull-up. In April 1954, United States Air Force General Hoyt Vandenberg was buried at Arlington National Cemetery without the traditional horse-drawn artillery caisson. Instead, Vandenberg was honored by a flyover of jet aircraft with one plane missing from the formation. On November 26, 1999, four Air Force F-16s flew the missing man formation over Kyle Field to honor the 12 Aggies who died during the Aggie Bonfire collapse.[5] The Delaware Air National Guard flew the missing man formation over the Dover International Speedway on June 3, 2001 to honor NASCAR driver Dale Earnhardt Sr., who had perished in a wreck on the final lap of the 2001 Daytona 500 race on February 18. In December 2004, as a final tribute to Prince Bernhard of the Netherlands's former military role in the Royal Netherlands Air Force, three modern F-16 jet fighters and a World War II Spitfire performed a missing man formation during his funeral. The missing man formation was flown at a family memorial service in Indian Hill, Ohio on 31 August 2012 in honour of former American astronaut, US Navy pilot, and test pilot Neil Armstrong, the first man to walk on the Moon. In November 2014 the state memorial service for former Australian Prime Minister Gough Whitlam, who had served as a navigator in the Royal Australian Air Force during World War II, concluded with a missing man formation flight conducted by four RAAF F/A-18 Hornet fighters.[6] On 29 March 2015, the Republic of Singapore Air Force's Black Knights attempted to fly the missing man formation as an aerial salute to long-serving Prime Minister Lee Kuan Yew during his funeral procession from Parliament House to the University Cultural Centre of the National University of Singapore, but was unable to do so, due to poor weather conditions. [7] [8] Motorsport variant The missing man formation is also used in various types of motorsport to commemorate the death of a driver, rider, or official.[9] In case of a rolling start, during the pace laps before the race begins, the driver in the pole position drops back a row into the second row and the field paces with no vehicle in the lead position.[10] Similarly, the pole position on a starting grid can be left empty for a standing start. In drag racing, the variant of the missing man formation only is put into place if a driver has lost his/her life sometime before the race but after having qualified. Should that happen, the deceased driver's lane remains vacant for what would have been his/her quarterfinal race and the opposing driver, who must cross the finish line without being disqualified in order to proceed, will slowly drive their car (referred to as "idling") down the track as a sign of respect for the fallen opponent (a recent example occurring when Robert Hight did this in honor of Scott Kalitta, who was killed in a qualifying crash in 2008). Rolling Honor Guard variant The missing man formation is also used for the motorcycle Rolling Honor Guards. A common formation of motorcycles is five in front of the hearse: two motorcycles in tandem (#1 and #2, left and right, from the perspective of the hearse), two motorcycles directly in front of the hearse, in tandem (#5 and #6, left and right, as noted), and a solo rider in the resultant #4 position, and the missing motorcycle (in the #3 position) representing the fallen. This is performed for both the loss of a person who was a member of the motorcycle club/organization, or, may be provided as a sign of respect by groups such as the Patriot Guard Riders.

The thach weave when the planes cross, making a sort of quadrant with their movment. In the Battle of Midway it was done by four planes. The Thach Weave (also known as a Beam Defense Position) is an aerial combat tactic developed by naval aviator John S. Thach of the United States Navy soon after the United States' entry into World War II. Thach had heard, from a report published in the 22 September 1941 Fleet Air Tactical Unit Intelligence Bulletin, of the Japanese Mitsubishi Zero's extraordinary maneuverability and climb rate. Before even experiencing it for himself, he began to devise tactics meant to give the slower-turning American F4F Wildcat fighters a chance in combat. While based in San Diego, he would spend every evening thinking of different tactics that could overcome the Zero's maneuverability, and would then test them in flight the following day.[citation needed] Working at night with matchsticks on the table, he eventually came up with what he called "Beam Defense Position", but which soon became known as the "Thach Weave". It was executed either by two fighter aircraft side-by-side or by two pairs of fighters flying together. When an enemy aircraft chose one fighter as his target (the "bait" fighter; his wingman being the "hook"), the two wingmen turned in towards each other. After crossing paths, and once their separation was great enough, they would then repeat the exercise, again turning in towards each other, bringing the enemy plane into the hook's sights. A correctly executed Thach Weave (assuming the bait was taken and followed) left little chance of escape to even the most maneuverable opponent. The basic Thach Weave, executed by two wingmen. Thach called on Ensign Edward "Butch" O'Hare, who led the second section in Thach's division, to test the idea. Thach took off with three other Wildcats in the role of defenders, Butch O'Hare meanwhile led four Wildcats in the role of attackers. The defending aircraft had their throttles wired (to restrict their performance), while the attacking aircraft had their engine power unrestricted - this simulated an attack by superior fighter aircraft.[1] Trying a series of mock attacks, Butch found that in every instance Thach's fighters, despite their power handicap, had either ruined his attack or actually maneuvered into position to shoot back. After landing, Butch excitedly congratulated Thach: "Skipper, it really worked. I couldn't make any attack without seeing the nose of one of your airplanes pointed at me." Thach carried out the first test of the tactic in combat during the Battle of Midway in June 1942, when a squadron of Zeroes attacked his flight of four Wildcats. Thach's wingman, Ensign R. A. M. Dibb, was attacked by a Japanese pilot and turned towards Thach, who dove under his wingman and fired at the incoming enemy aircraft's belly until its engine ignited. The maneuver soon became standard among US Navy pilots and was adopted by USAAF pilots. Marines flying Wildcats from Henderson Field on Guadalcanal also adopted the Thach Weave. The tactic initially confounded the Japanese Zero pilots flying out of Rabaul. Saburō Sakai, the famous Japanese ace, relates their reaction to the Thach Weave when they encountered Guadalcanal Wildcats using it:[2] For the first time Lt. Commander Tadashi Nakajima encountered what was to become a famous double-team maneuver on the part of the enemy. Two Wildcats jumped on the commander's plane. He had no trouble in getting on the tail of an enemy fighter, but never had a chance to fire before the Grumman's team-mate roared at him from the side. Nakajima was raging when he got back to Rabaul; he had been forced to dive and run for safety. The maneuver proved so effective that American pilots also used it during the Vietnam War, and it remains an applicable tactic as of 2013.[3]

Mouse in the maze was another video game made in 1959. These prototype video games were extraordinarily simple. It basically involved a dot moving through a maze but the maze was extremely simple, merely a quadrant grid

Space War was also one of the first 1950 video game prototypes. The game could not be played on a regular computer but needed to be played on a University campus computer so it could never be marketed. It involved two players trying to fight each other in space around a sun. Player controls include clockwise and counterclockwise rotation, thrust, fire, and hyperspace. Initially these were controlled using the front-panel test switches, with four switches for each player, but these proved to wear out very quickly under normal gameplay, and the location of the switches left one player off to one side of the CRT display and visually disadvantaged as a result.[3] Most sites used custom control boxes wired into the same switches, although joysticks and other inputs were also used. Four optional features were controlled by sense switches on the console: no star (and thus no gravity) enable angular momentum disable background starfield the "Winds of Space"- a warping factor on trajectories that require the pilot to make careful adjustments every time they move.

Tennis For Two was an electronic game developed in 1958 on a Donner Model 30 analog computer, which simulates a game of tennis or ping pong on an oscilloscope. Created by American physicist William Higinbotham for visitors at the Brookhaven National Laboratory, it is important in the history of video games as one of the first electronic games to use a graphical display. Tennis for two was probably the second video game ever created. It was played on a screen that looked like a quadrant grid

In 1952, Alexander S. Douglas created OXO, a software program for the Electronic Delay Storage Automatic Calculator (EDSAC) computer, which simulates a game of tic-tac-toe. The EDSAC was the first computer to have memrory that could be read from or written to, and filled an entire room; it included three 35×16 dot matrix cathode ray tubes to graphically display the state of the computer's memory.[9][18] As a part of a thesis on human–computer interaction, Douglas used one of these screens to portray other information to the user; he chose to do so via displaying the current state of a game.[19] The player entered input using a rotary telephone controller, selecting which of the nine squares on the board they wished to move next. Their move would appear on the screen, and then the computer's move would follow.[20] The game was not available to the general public, and was only available to be played in the University of Cambridge's Mathematical Laboratory, by special permission, as the EDSAC could not be moved.[21] Like other early video games, after serving Douglas's purpose, the game was discarded.[9] Around the same time, Strachey expanded his draughts program for another mainframe computer, the Manchester Mark 1, culminating in a version for the Ferranti Mark 1 in 1952, which had a CRT display.[22] Like OXO, the display was mostly static, updating only when a move was made.[23] OXO and Strachey's draughts program are the earliest known games to display visuals on an electronic screen. Again, the genesis of video games was tic tac toe, checkers, and chess, all games made up of quadrants.

The first publicly demonstrated electronic game was created in 1950. Bertie the Brain was an arcade game of tic-tac-toe, built by Dr. Josef Kates for the 1950 Canadian National Exhibition. As I mentioned tic tac toe is made up of quadrants Around this time, non-visual games were being developed at various research computer laboratories; for example, Christopher Strachey developed a simulation of the game draughts, or checkers, for the Pilot ACE that he unsuccessfully attempted to run for the first time in July 1951 at the British National Physical Laboratory and completed in 1952; this is the first known computer game to be created for a general-purpose computer, rather than a machine specifically made for the game like Bertie. Checkers was the second video game created, also made of quadrants

The Legend of Zelda: A Link to the Past and Four Swords[a] is an action-adventure game co-developed by Nintendo and Capcom and published by Nintendo for the Game Boy Advance. It is the ninth installment in the The Legend of Zelda video game series. There are always four Link characters (differentiated by different colors: green, red, blue and purple) in play, regardless of the number of people playing; "extra" Links are attached to those directly controlled and positioned around the controlling character. Normally, the extra Links follow the player, but players can separate an individual Link and control independently, or put the four Links into formations. These techniques are required to solve puzzles and defeat enemies. Players are encouraged to work together to gather enough Force Gems to empower the Four Sword, and failing to do so by the time the boss is defeated or the dark barrier is reached results in having to go back to the beginning of the stage to collect more. However, once the requisite gems are collected, players are automatically transported to the dark barrier and therefore do not have to repeat the entire stage. The Links eventually save the shrine maidens, retrieve the Dark Mirror, destroy Shadow Link and Vaati, and face Ganon in an ultimate showdown. The Links defeats Ganon and seal him firmly in the Four Sword. Peace returns to Hyrule and the people celebrate as all traces of the evil that plagued Hyrule are vanquished.[3] Link then returns the Four Sword back to its resting pedestal and the Four Links become one again. The nature of the quadrant model is there are four quadrants but they are really all one. Zelda is one of the most popular video games of all time.

The Legend of Zelda: The Minish Cap (/ˈmɪnɪʃ/) (Japanese: ゼルダの伝説 ふしぎのぼうし Hepburn: Zeruda no Densetsu: Fushigi no Bōshi?, lit. The Legend of Zelda: The Mysterious Cap) is an action-adventure game and the twelfth entry in the The Legend of Zelda series. Developed by Capcom, with Nintendo overseeing the development process, it was released for the Game Boy Advance handheld game console in Japan and Europe in 2004 and in North America and Australia the following year.[1] In this story Link retrieves the four elemental artifacts and uses them to restore the Picori Blade to the Four Sword,[11] capable of defeating Vaati. The sign of zelda is the sierpinsky triangle which I described is four triangles, one within the other three, which is the quadrant pattern in its essence.

In the Legend of Zelda: Majora's Mask,The gameplay is centered on the perpetually repeating three-day cycle and the use of various masks, some of which allow Link to transform into different beings. Link learns to play several melodies on his ocarina, which have a variety of effects like controlling the flow of time or opening passages to four temples, which house challenges Link must overcome Link has three masks that can transform him into different forms. So he has four forms. His three transformations receive different reactions from non-player characters.[10] For instance, the Goron and Zora are allowed to exit Clock Town at will, whereas the Deku Scrub is not permitted to leave by reason of his childlike resemblance. Animals also interact differently with the four forms of Link. The Legend of Zelda: Majora's Mask is set in Termina, a land parallel to Hyrule,[18][19] the latter being the main setting of most games in the series. According to legend, Termina was split into four areas by four magical giants that live in four regions of the land. At the center of Termina lies Clock Town, which features a large clock tower that counts down the days before the Carnival of Time—a major festival where the people of Termina pray for good luck and harvests. Termina Field surrounds Clock Town; beyond lie a swamp, mountain range, bay, and canyon in each of the four cardinal directions. Link must then travel between the four cardinal regions of Termina: Woodfall, Snowhead, the Great Bay, and Ikana Canyon, for each region conceals one of the Four Giants who will be able, once reunited, to halt the moon's crashing. At the same time, each region has been struck with a terrible curse by the Skull Kid which plagues its inhabitants and seals away its giant. To lift the curse and free the giants, Link must enter a dungeon in each region and defeat its boss. After doing so, he obtains the power to summon the giant he has set free. With all four curses lifted, Link climbs on top of the Clock Tower at midnight on the third day to confront the Skull Kid again. There and then, he summons the Four Giants, who halt the moon's descent toward Termina by holding it up with their arms. Now seeing the Skull Kid as a useless puppet, Majora's Mask drops his grip on him and flies up to possess the moon instead. With Tatl at his side, Link follows the Majora's Mask inside the moon and defeats him once and for all, returning the moon to its proper place in the sky.[22] The Four Giants return to their sleep. Tatl and Tael reunite with the newly liberated Skull Kid. The Happy Mask Salesman takes Majora's Mask, stating it has been purified of its evil power. Link rides away on Epona while the people of Termina celebrate the Carnival of Time and the dawn of a new day. The game ends with a post-credits scene depicting Link and Epona back in the mysterious forest, resuming Link's search for his friend, as they ride off towards a mysterious light breaking through the thick forest. A drawing on a tree stump of Link, Tatl, Tael, the Skull Kid, and the Four Giants is shown after. Notice the repetition of fours.

In Zelda II: The Adventure of Link, Link begins the game with four Heart Containers and four Magic Containers and can acquire up to four more of each, permanently increasing his life points and magic points respectively.

As I mentioned, Zelda is one of the most popular video games of all time. I do not think it is a coincidence that the games portray the quadrant four a lot. In Oracle of Seasons, the environment changes with the four seasons. From spring, summer, winter, autumn. Gameplay is sometimes affected by the seasons; during the winter for example, a path opens up that cannot be accessed during any other season; or during spring, the flower can be used to access unreachable ledges. The central item of Oracle of Seasons is the Rod of Seasons. By standing on a stump and swinging the rod, Link can change the season and affect his surroundings.[20] For example, to cross a body of water, Link can change the season to winter and walk on the ice. Changing the season to summer causes vines to flourish, which Link can use to scale cliffs. When Link obtains the rod, he initially cannot use it.[21] In the course of the game, Link visits four towers that house the four spirits of the seasons; each tower Link visits allows him to switch to an additional season.[21]

In Zelda there are four Light Spirits (光の精霊 Hikari no Seirei?) throughout Hyrule. All of the light spirits are found in Twilight Princess. The first light spirit is Ordona (ラトアーヌ Ratoānu?), best described as an Ordon Goat. Ordona has a spiraling circular orb in between her antlers and first appears at Ordon Spring. She appears once Link has defeated the first Shadow Beast. The second light spirit, Faron (フィローネ Firōne?), is described as a monkey/ape. He is holding his golden orb with his tail eclipsed over his head. He appears in Faron Spring, and will fully appear when Link has completed the first collection of twilight bugs. The third light spirit, Eldin (オルディン Orudin?), is described as an eagle, with his orb between his feet. He appears in the lake near the shaman's house in Kakariko Village, and will fully appear once Link has completed the second collection of twilight bugs. The fourth and last light spirit, Lanayru (ラネール Ranēru?), is described as a serpent, with his orb inside his mouth. He appears in the cave at Lake Hylia. He fully appears when Link has completed the third and final collection of twilight bugs. But once Link is back to his human form, and after Lanayru tells him the story of the three goddesses and the three Fused Shadows, Zant appears. Zant will then embed the Shadow Crystal in Link's skull that will allow Link to transform into his human and wolf form. He will also expose Midna to Lanayru, causing her to become very ill, and sending Link on the quest for the Master Sword.

In Zelda the Deku (デクナッツ Dekunattsu?) are a race of plant-like creatures which are introduced in Ocarina of Time. They appear mostly in the overworld and dungeons. Deku are generally short and have leaves sprouting out from their heads. They often have red, glowing eyes, and their mouths are short, hollow tubes that can shoot "Deku Nuts". Their bodies consist entirely of wood and leaves, and they perish quickly if set on fire. They can fly by using large leaves to glide, and some can use the leaves on their head to fly for indefinite periods after taking off from a "Deku Flower." There are four types of Deku depicted in the series: Deku Scrubs, Mad Scrubs, Business Scrubs, and Royal Scrubs. Deku Scrubs are the most common type, which have green leaves. They often give information when caught. Mad Scrubs are violent, have red and yellow leaves, and do not talk. Business Scrubs are traders who offer to sell their wares and services. Royal Scrubs have larger heads, bigger eyes, smaller mouths, and they also have extra leaves covering their body. In The Legend of Zelda: Majora's Mask, Link can inhabit the body of an unknown Deku Scrub (who is heavily implied to be the son of the Deku King's butler) and can fly for a limited time with use of a Deku Flower and can shoot bubbles from its mouth (once he receives the magic meter from the Great Fairy). The Deku Scrub cannot go into deep water but hops on top of it five times and then sinks.

Tetra is princess Zelda In Zelda the princess's name is Tetra. Tetra means four. Tetra's name could be derived from the Ancient Greek word meaning "four," similar to Tetris. It could also be derived from an actual Tetra, a type of South American freshwater fish, adding to the sea-faring theme of the game. Tetra could also allude to tetrahedron, the physical shape of the Triforce. (the sierpinski triangle, the triangle with four triangles within it) In The Wind Waker Tetra is the leader of the pirates. When Link first meets Tetra, she is being kidnapped by the Helmaroc King. The monster bird is distracted by Tetra's crew and she is dropped into the Fairy Woods. Link comes to her rescue, but because of this distraction, his sister Aryll, mistaken for Tetra, is kidnapped instead.

Berzerk was one of the original video games made in 1980. Again there are four enemies in this game with the fourth being different than the other three. There are three colors of robots, and then there is the fourth evil otto, the nemesis of the humanoid protagonist. It is a very simple game where the player can move the four directions up and down and left and right kind of in a quadrant manner.

Dark yellow robots that do not fire

Red robots that can fire 1 bullet (500 points)

Dark cyan robots that can fire 2 bullets (1,500 points)

In this version of the game, after 5,000 points, Evil Otto doubles his speed, moving as fast as the player while robots remain in the maze, and twice as fast as the player after all the robots are destroyed.

In the sitcom My Name is Earl (Season 1, Episode 8), the character "Crabman" is portrayed, playing Berzerk and scoring high. He afterwards would take a polaroid photograph of the screen, pinning the highscore to his personal wall of fame.[20]

In the Futurama episode "Fear of a Bot Planet", the Anti-Human Patrol robots, along with the PA loudspeaker, use the sound samples of "Get the humanoid!" and "Intruder alert! Intruder alert!" from the original game.[21] The episode "Anthology of Interest II" features an actual robot from the game, and the spoken line of the robot references the style of the sound samples ("Fork 'em over! FORK 'EM OVER!").[22]

In The Simpsons episode "Homer Goes to College", Homer visits some nerds who mutter "Intruder alert" and "Stop the humanoid".[23]

In the NewsRadio episode "Rosebowl", news director Dave Nelson introduces an unpopular new employee evaluation system. In the fracas following the adoption of this new system, Dave is referred to as "Evil Otto" by the two news anchors, Bill McNeal and Catherine Duke.[24]

In the popular online game World of Warcraft, Gnomish Alarm-O-Bots call out "Intruder Alert!" when attacked in the same robotic voice as Evil Otto.

Donkey Kong Jr. (ドンキーコングJR. Donkī Kongu Junia?) is a 1982 arcade-style platform video game by Nintendo. It first appeared in arcades, and, over the course of the 1980s, was later released for a variety of platforms, most notably the Nintendo Entertainment System. The game's title is written out as Donkey Kong Junior in the North American arcade version and various ports to non-Nintendo systems. Its eponymous star, Donkey Kong Jr., also called simply Junior[3] or abbreviated as DK Jr.,[4] is trying to rescue his father Donkey Kong, who has been imprisoned. Donkey Kong's cage is guarded by Mario, in his only appearance as an antagonist in a Nintendo video game. This game is the sequel to the video game Donkey Kong, which featured Mario as the hero and Junior's father as the villain (while in this game, it's the other way around). Like its predecessor, Donkey Kong, Jr. is an arcade-style platform game. There are a total of four stages, each with a unique theme. DK Jr. can run left and right, jump, and grab vines/chains/ropes to climb higher on the screen. He can slide down faster by holding only one vine, or climb faster by holding two. Enemies include "Snapjaws," which resemble bear traps with eyes, bird-like creatures called "Nitpickers", and "Sparks" that roam across the wiring in one of Mario's hideouts. To pass the first three stages, DK Jr. must reach the key at the top. In the fourth stage, DK Jr. must push six keys into locks near the top of the stage to free Donkey Kong. After a brief cutscene, the player is taken back to the first stage at an increased difficulty. DK Jr. loses a life when he touches any enemy or projectile, falls too great a distance, or falls off the bottom of the screen. Additionally, he loses a life if the timer counts down to zero. The game ends when the player loses all of his or her lives The first three stages are different than the fourth. This is the nature of the quadrant model. The original games Mario and Donkey Kong, which remained the most popular video games throughout video game history, started reflecting completely the quadrant model pattern.

Lemmings was a later 1991 video game. But it was divided into four difficulty categories. So it was not one fo the original prototypes that really reflected the quadrant model. Lemmings is divided into a number of levels, grouped into four difficulty categories.[1] Each level begins with a trap door opening from above, releasing a steady line of lemmings who all follow each other.[2] Levels include a variety of obstacles that prevent lemmings from reaching the exit, such as large drops, booby traps and pools of lava. The four difficulty groups – "Fun", "Tricky", "Taxing" and "Mayhem" – are used to organize the levels to reflect their overall difficulty.[7] This rating reflects several factors, including the number of obstacles the player has to surpass, the limitation on the number of types of skills available to assign, the time limit, the minimum rate of lemming release, and the percentage of lemmings that must be saved.[5]

Minecraft is a very popular video game. The game primarily consists of four game modes: survival, creative, adventure, and spectator. It also has a changeable difficulty system of four levels; the easiest difficulty (peaceful) removes any hostile creatures that spawn.[34] Survival mode The Minecraft crafting screen, showing the crafting pattern of two stone axes In this mode, players have to gather natural resources (such as wood and stone) found in the environment in order to craft certain blocks and items.[22] Depending on the difficulty, monsters spawn in darker areas in a certain radius of the character, requiring the player to build a shelter at night.[22] The mode also features a health bar which is depleted by attacks from monsters, falls, drowning, falling into lava, suffocation, starvation, and other events. Players also have a hunger bar, which must be periodically refilled by eating food in-game, except in "Peaceful" difficulty, in which the hunger bar does not drain. If the hunger bar is depleted, automatic healing will stop and eventually health will deplete. Health replenishes when players have a nearly full hunger bar, and also regenerates regardless of fullness if players play on the easiest difficulty. There are a wide variety of items that players can craft in Minecraft.[35] Players can craft armor, which can help mitigate damage from attacks, while weapons such as swords can be crafted to kill enemies and other animals more easily. Players may acquire resources to craft tools, such as axes, shovels, or pickaxes, used to chop down trees, dig soil, and mine ores, respectively; tools made of iron perform their tasks more quickly than tools made of stone or wood and can be used more heavily before they break. Players may also trade goods with villager mobs through a bartering system involving trading emeralds for different goods.[36] Villagers often trade with emeralds, wheat or other materials.[25][36] The game has an inventory system, and players can carry a limited number of items. Upon dying, items in the players' inventories are dropped, and players re-spawn at the current spawn point, which is set by default where players begin the game, but can be reset if players sleep in a bed.[37] Dropped items can be recovered if players can reach them before they despawn. Players may acquire experience points by killing mobs and other players, mining, smelting ores, breeding animals, and cooking food. Experience can then be spent on enchanting tools, armor and weapons.[34] Enchanted items are generally more powerful, last longer, or have other special effects.[34] Players may also play in hardcore mode, this being a variant of survival mode that differs primarily in the game being locked to the hardest gameplay setting as well as featuring permadeath; upon players' death, their world is deleted.[38] Creative mode An example of a creation constructed in Minecraft In creative mode, players have access to all of the resources and items in the game through the inventory menu, and can place or remove them instantly.[39] Players, who are able to fly freely around the game world, do not take environmental or mob damage, and are not affected by hunger.[40][41] The game mode helps players focus on building and creating large projects.[39] Adventure mode Adventure mode was added to Minecraft in version 1.3; it was designed specifically so that players could experience user crafted custom maps and adventures.[42][43][44] Gameplay is similar to survival mode but introduces various player restrictions, which can be applied to the game world by the creator of the map. This is so that players can obtain the required items and experience adventures in the way that the mapmaker intended.[44] Another addition designed for custom maps is the command block; this block allows mapmakers to expand interactions with players through certain server commands.[45] Spectator mode Spectator mode allows players to fly around through blocks and watch game play without interacting. In this mode, the inventory becomes a menu that allows the player to teleport to players in the world. It is also possible to view from the point of view of another player or creature. Some things may look different from another creature's point of view.

The TurboGrafx-16 Entertainment SuperSystem, known in Japan and in France as the PC Engine (PCエンジン Pī Shī Enjin?), is a home video game console joint-developed by Hudson Soft and NEC, released in Japan on October 30, 1987, in the United States on August 29, 1989, and in France on November 22, 1989. It was the first console released in the 16-bit era, albeit still utilizing an 8-bit CPU. Originally intended to compete with the Nintendo Entertainment System (NES), it ended up competing against the Mega Drive/Genesis, and later on the Super Famicom/Super NES. The TurboGrafx-16 has an 8-bit CPU and a dual 16-bit GPU. 16 is the number of squares in the quadrant model.

Bomberman is example of another original video game that was played within a visual quadrant grid with four possible directions, up, down, left, or right. Although it was not one of the 1970s original games.

Although Lumines is not one of the original video games from which the others spawned, it still reflects the quadrant model image. It is one of the most popular arcade games and puzzle of all time with tetris, which also reflects the quadrant pattern Lumines is a block-dropping game that may seem at first to be similar to Columns and Tetris. A 2x2 square (an O tetromino/ quadrant) made of four smaller block pieces is dropped into the playing field, which may appear different as the player advances through levels or skins. The small blocks that comprise the larger blocks will be one of two different colors. The objective is to rotate and align the blocks in such a way as to create 2x2 squares of the same color, which may span multiple blocks and, indeed, share blocks. For example, if one should get a 2x3 area of matching blocks, the middle portion will "share" itself with both the left and right halves and create two 2x2 squares. After the "timeline", which is synchronized to the music, sweeps over the matching blocks, they disappear. When too many unmatched blocks pile up to the point where no more blocks may be dropped in the playing field, the game ends. When part of a falling block hits an obstruction, the unobstructed portion of the block will split off and continue to fall. More points are scored by creating the largest number of squares during one "timeline" sweep. Increasing score multipliers are earned by repeatedly clearing squares on consecutive timeline sweeps. Bonuses are also awarded by reducing all remaining tiles to one single color or for removing all non-active tiles from the screen altogether. Occasionally, a block falls with a special square of one of the two colors with a "jewel" in the center. This square, when cleared as part of a matched 2x2 square, will cause all individual blocks of the same color that are horizontally or vertically adjacent to the matched 2x2 square, or to an adjacent square, to be cleared without score. These can be used for both generating large bonuses, since generally several blocks of the other color will be formed once these are removed, as well as to help the player recover if the field becomes too cluttered. There are four basic modes in the game: Challenge, Time Attack, Puzzle, Vs., and Vs. CPU Mode. Challenge Mode cycles through skins in a fixed order of generally increasing difficulty, and is played until the blocks pile up to the top of the screen. The maximum score in Challenge Mode is 999,999 points. Time Attack games give the player a limited time to clear as many blocks as possible. Puzzle mode challenges the player to create pictures (such as a cat, dog, cross, etc.) by forming the picture with one color while surrounding it with the opposite color. Vs. CPU mode is a series of battles against A.I. opponents. A line splits the playing field in half, and deleting blocks or combinations of blocks shifts the line towards the opposing player, giving the opposing player less room on their side. The battle ends when blocks pile up all the way to the top of the screen for one player. Two players with PSPs can use their wireless connection to play in the same way.

Whac-A-Mole is a popular arcade redemption game invented in 1976 by Aaron Fechter of Creative Engineering, Inc.. In Japan, もぐら退治 (mogura taiji, "Mole Buster") is a popular arcade game invented in 1975 by Kazuo Yamada of TOGO, based on ten of the designer's pencil sketches from 1974, licensed to Bandai in 1977.[1] ) It can also be commonly found at Japanese festivals. Whac-A-Mole arcade games originally had four holes in a quadrant formation out of which a mole would emerge and you had to hit it on the had. Later more holes were added

Skee ball was another very popular game played at arcades. It involved rolling a ball on a ramp where it would fly into holes. Traditional skee ball machines like this one do not include the two additional "100 points" holes, located on the uppermost corners of the machine, on either side of the "50 points" hole. Traditional skee ball had four scoring options. You could score 10 points by rolling the ball into the large first loop. You could get 30 by the second which was smaller. You could get 40 points by getting it in the hole above that. You get 50 points by getting it in the fourth highest hole. The fourth hole was different from the previous three because the previous three holes were encircled by the 10 point hole. The fourth square is always transcendent.

Pinball was a very popular game and had at most four flippers. The flippers in pinball are one or more small mechanically or electromechanically controlled levers, roughly 3 to 7 cm in length, used for redirecting the ball up the playfield. They are the main control that the player has over the ball. Careful timing and positional control allows the player to intentionally direct the ball in a range of directions with various levels of velocity. With the flippers, the player attempts to move the ball to hit various types of scoring targets, and to keep the ball from disappearing off the bottom of the playfield. The very first pinball games appeared in the early 1930s and did not have flippers; after launch the ball simply proceeded down the playfield, directed by static nails (or "pins") to one of several scoring areas. (These pins gave the game its name.) In 1947, the first mechanical flippers appeared on Gottlieb's Humpty Dumpty[23] and by the early 1950s, the familiar two-flipper configuration, with the flippers at the bottom of the playfield above the center drain, had become standard. Some machines also added a third or fourth flipper midway up the playfield. The new flipper ushered in the "golden age" of pinball, where the fierce competition between the various pinball manufacturers led to constant innovation in the field. Various types of stationary and moving targets were added, spinning scoring reels replaced games featuring static scores lit from behind. Multiplayer scores were added soon after, and then bells and other noise-makers, all of which began to make pinball less a game and more of an experience. The flippers have loaned pinball its common name in many languages, where the game is known mainly as "flipper".

TV Basketball of 1974 was the first sprite game. It involved four players in one line in two dimensions playing basketball

The players usually count aloud to 3, or speak the name of the game (e.g. "Rock Paper Scissors!" or "Ro Sham Bo!"), each time either raising one hand in a fist and swinging it down on the count or holding it behind. On the fourth count (saying, "Shoot!" or "Sho!"), the players change their hands into one of three gestures, which they then "throw" by extending it towards their opponent. Variations include a version where players use only three counts before throwing their gesture (thus throwing on the count of "Scissors!" or "Bo!"), or a version where they shake their hands three times before "throwing." There are usually three choices, rock paper or scissors. But occasionally a fourth is added. The fourth is always transcendent. Sometimes a fifth is added. The fourth always points to the fifth.

Literature

We categorize sentences into four main types, depending on the number and type of clauses they contain: Simple Sentence It has one independent clause. We drove from Connecticut to Tennessee in one day. Compound Sentence It has one more than one independent clause We were exhausted, but we arrived in time for my father's birthday party. Complex Sentence It has one independent clause and at least one or more dependent clause Although he is now 79 years old, he still claims to be 65. Compound-complex Sentence It has one more than one independent clause and at least one dependent clause The types are elucidated by a auadrant woth two axes Dichotomy one is one dependent vlause or many dependent clauses. Dichotomy 2 is one independent claise or many independent clauses. This yields four types

In September 1982, Arcade Express reviewed the ColecoVision port and scored Donkey Kong 9 out of 10.[33] Computer and Video Games reviewed the ColecoVision port in its September 1984 issue and scored it 4 out of 4 in all four categories of Action, Graphics, Addiction and Theme.[34]

Donkey Kong (Japanese: ドンキーコング Hepburn: Donkī Kongu?) is an arcade game released by Nintendo in 1981. It is an early example of the platform game genre, as the gameplay focuses on maneuvering the main character across a series of platforms while dodging and jumping over obstacles. In the game, Mario (originally named Mr. Video but then changed to "Jumpman") must rescue adamsel in distress named Pauline (originally named Lady), from a giant ape named Donkey Kong. The hero and ape later became two of Nintendo's most popular and recognizable characters. Donkey Kong is one of the most important titles from the Golden Age of Video Arcade Games, and is one of the most popular arcade games of all time. The game is divided into four different single-screen stages. Each represents 25 meters of the structure Donkey Kong has climbed, one stage being 25 meters higher than the previous. The final stage occurs at 100 meters. Stage one involves Mario scaling a construction site made of crooked girders and ladders while jumping over or hammering barrels and oil barrels tossed by Donkey Kong. Stage two involves climbing a five-story structure of conveyor belts, each of which transports cement pans. The third stage involves the player riding elevators while avoiding bouncing springs. The final stage involves Mario removing eight rivets which support Donkey Kong. Removing the final rivet causes Donkey Kong to fall and the hero to be reunited with Pauline.[16] These four stages combine to form a level. Upon completion of the fourth stage, the level then increments, and the game repeats the stages with progressive difficulty. For example, Donkey Kong begins to hurl barrels faster and sometimes diagonally, and fireballs get speedier. The victory music alternates between levels 1 and 2. The 22nd level is colloquially known as the kill screen, due to an error in the game's programming that kills Mario after a few seconds, effectively ending the game. Miyamoto then thought of using sloped platforms, barrels and ladders. When he specified that the game would have multiple stages, the four-man programming team complained that he was essentially asking them to make the game repeatedly.[19]:38–39 Nevertheless, they followed Miyamoto's design, creating a total of approximately 20 kilobytes of content.[20]:530 Yukio Kaneoka composed a simplistic soundtrack to serve as background music for the levels and story events.[

Mario Bros. (マリオブラザーズ Mario Burazāzu?) is a platform game published and developed for arcades by Nintendo in 1983. It was created by Shigeru Miyamoto. The player gains points by defeating multiple enemies consecutively and can participate in a bonus round to gain more points. Enemies are defeated by kicking them over once they have been flipped on their back. This is accomplished by hitting the platform the enemy is on directly beneath them. If the player allows too much time to pass after doing this, the enemy will flip itself back over, changing in color and increasing speed. Each phase has a certain number of enemies, with the final enemy immediately changing color and increasing its speed. There are four enemies: Square 1: the Shellcreeper, which simply walks around; Square 2: the Sidestepper, which requires two hits to flip over; Square 3: the Fighter Fly, which moves by jumping and can only be flipped when it is touching a platform; and Square 4: the Slipice, which turns platforms into slippery ice. When bumped from below, the Slipice dies immediately instead of flipping over. The original versions of Mario Bros.—the arcade version and the Family Computer/Nintendo Entertainment System (FC/NES) version—were received positively by critics.

Donkey Kong spawned the sequels Donkey Kong Jr. and Donkey Kong 3, as well as the spin-off Mario Bros. A complete remake of the original arcade game on the Game Boy, named Donkey Kong or Donkey Kong '94 contains levels from both the original Donkey Kong and Donkey Kong Jr arcades. It starts with the same damsel-in-distress premise and four basic locations as the arcade game and then progresses to 97 additional puzzle-based levels. It is the first game to have built-in enhancement for the Super Game Boy accessory. The arcade version makes an appearance in Donkey Kong 64 in the Frantic Factory level.

In the history of computer and video games, the fourth generation (more commonly referred to as the 16-bit era) of games consoles began on October 30, 1987 with the Japanese release of Nippon Electric Company's (NEC) PC Engine (known as the TurboGrafx-16 in North America). Although NEC released the first fourth generation console, and was second to the SNES in Japan, this era's sales were mostly dominated by the rivalry between Nintendo and Sega's consoles in North America: the Super Nintendo Entertainment System (the Super Famicom in Japan) and the Mega Drive (named the Sega Genesis in North America due to trademark issues). Nintendo was able to capitalize on its previous success in the third generation and managed to win the largest worldwide market share in the fourth generation as well. Sega was extremely successful in this generation and began a new franchise, Sonic the Hedgehog, to compete with Nintendo's Mario series of games. Several other companies released consoles in this generation, but none of them were widely successful. Nevertheless, several other companies started to take notice of the maturing video game industry and began making plans to release consoles of their own in the future. This generation ended with the Super Nintendo Entertainment System's discontinuation in 1999 in North America, Australia and Europe.[citation needed] 16 is the squares in the quadrant model

In The Legend of Zelda: Spirit Tracks Zelda demands to know how to prevent the Demon King's return, to which Anjean replies that the Spirit Tracks have to be restored by retrieving and completing several ancient rail maps (one for each of the four realms of Hyrule) from the above floors of the Tower to restore the Spirit Tracks. Then, the Tower must be linked up by the Spirit Tracks to four different temples, one in each of the four known realms.

In street fighter there is the final four boss opponents. These bosses I think are known as the Four Grand Masters.

In Metroid 4 there are four space pirate bosses In Metroid Prime, unlike those in earlier games in the series, the beam weapons in Metroid Prime have no stacking ability, in which the traits of each beam merge. Instead, the player must cycle the four beam weapons; there are charge combos with radically different effects for each

In most video games throughout the history of video games since advanced consoles like nintendo 64, four players engage in play using a split screen. The split screen looks like a quadrant.

Pong (marketed as PONG) is one of the earliest arcade video games and the very first sports arcade video game. It is a tennis sports game featuring simple two-dimensional graphics.

Bushnell felt the best way to compete against imitators was to create better products, leading Atari to produce sequels in the years followings the original's release: Pong Doubles, Super Pong, Ultra Pong, Quadrapong, and Pin-Pong.[3] The sequels feature similar graphics, but include new gameplay elements; for example, Pong Doubles allows four players to compete in pairs, while Quadrapong has them compete against each other in a four way field. There is no five way pong. The forth square is always different. The fifth is always questionable.

Space invaders was also the first game where players were given multiple lives,[64] had to repel hordes of enemies,[18] could take cover from enemy fire, and use destructible barriers,[65] in addition to being the first game to use a continuous background soundtrack, with four simple diatonic descending bass notes repeating in a loop, which was dynamic and changed pace during stages,[66] like a heartbeat sound that increases pace as enemies approached.[67] Whereas videogame music prior to Space Invaders was restricted to the extremities (i.e., a short introductory theme with game-over counterpart), the alien-inspired hit featured continuous music—the well-known four-note loop—throughout, uninterrupted by sound effects. "It was thus the first time that sound effects and music were superimposed to form a rich sonic landscape. Not only do players receive feedback related directly to their actions through sound effects; they also receive stimulus in a more subtle, non-interactive fashion through music

In each level of the original space invaders arcade game there were four types of enemies in each level. Each enemy type afforded a different amount of points. The fourth type afforded often a mystery amount of points (and it was a lot). The fourth is always transcendent.

Also in the original Space invaders video games there were four barriers that could protect your ship from enemy missiles.

Galaxy Wars is an arcade video game developed by Universal and manufactured by Taito in 1979. Gameplay[edit] Galaxy Wars Arcade game screen shot There are four parts to the game after getting through a certain number of levels. They are Good, Very Good, Wonderful and Fantastic. Depress the fire button for a missile. The missile speed increases when depressing the fire button continuously. Guide the missile from a stationary launch pad to the top of the screen to blow up the invading fleet of armed UFOs while dodging meteorites and bombs. Points are awarded for blowing up various ships and range from 50-550 depending on the ship. There is a bonus chance of 600 points for one pattern. After clearing a level or "pattern" as the back of the flyer calls it, the player was rewarded with messages like "Good!!" after 3 screens cleared, "Very Good!!" after 7 screens cleared, "Wonderful!!" after 10 screens cleared, and "Fantastic!!" after 15 screens cleared. Players who failed to score any points were told to "Give Up!!" A launcher appears every 3,000 additional points (5,000 if the adjustment is made in the controlling dip switches in the arcade cabinet). The game has a 1up and 2up player score and High Score tallied at the top of the screen. The arcade cabinet has one joystick to move the launcher left to right and guide the missiles.

Galaxians was another video game with the same type of schematic where there would be four types of enemies, and each level of enemy would give you a higher amount of points if killed. The fourth type would give a huge amount of points. The fourth is always transcendent.

Asteroids was another one of the original late 70's video games. It too had four iterations of enemies. There were large asteroids, that broke into medium asteroids, that broke into small asteroids. And then there was the space ship. The transcendent fourth. These video games began the video game revolution, and it is from them all of the other video games emerged.

Metroid: Other M, which takes place between Super Metroid and Fusion, provides more information about Samus's backstory and her emotional connection to both the Metroid hatchling and her former commander, Adam Malkovich, as well as her relation to all four Mother Brain designs, namely Zebes' Mother Brains, Aurora Unit 313 and MB.[16] Mother Brain (Japanese: マザーブレイン?) is a fictional character created by Nintendo for the Metroid series. She is one of the most prominent antagonists within the series, serving as the main antagonist of Metroid and Super Metroid.

Sea Wolf was one of the first video games. It featured a 1,2,3,4 on the screen. The light would cycle through 1,2,3,4 and then the gun would reload.

Tank uses a black and white Motorola television for its display.[3] The control panel consists of four military-style joysticks, two per player, with a fire button mounted on top of the right joystick of each pair

In 2010 a series of four classic Atari game CD-ROMs (Centipede, Lunar Lander, Super Breakout and Asteroids) were given away in kids' meals, and were also available for purchase separately

Atari's breakout was another one of the original video games out of which all video games emerged. Breakout is an arcade game developed and published by Atari, Inc.[2] It was conceptualized by Nolan Bushnell and Steve Bristow, influenced by the 1972 Atari arcade game Pong, and built by Steve Wozniak aided by Steve Jobs. The premise of the game was to break through four colors of blocks. The colors were yellow, red, blue and green. The red blocks make the balls go faster than the yellow blocks. The blue are faster than the red. The green are the fastest and make the ball go extremely fast. The fourth square is always transcendent.

Nibbler was a later game than the first 1970s games, but it was still an elementary early video game. It involved a snake moving through a maze that was based around a quadrant grid.

Snake was another popular elemental video game put on mobile phones. It involves a snake moving a possible four directions up down right or left. Its coordinates were centered around a quadrant orientation. Blockade was the prototype 1970s video game it grew out of. Blockade had four directional buttons.

007 was a very popular game in Nintendo 64. GoldenEye 007 is a first-person shooter video game developed by Rare and based on the 1995 James Bond film GoldenEye. It was exclusively released for the Nintendo 64 video game console in August 1997. The game features a single-player campaign in which players assume the role of British Secret Intelligence Service agent James Bond as he fights to prevent a criminal syndicate from using a satellite weapon against London to cause a global financial meltdown. The game also includes a split-screen multiplayer mode in which two, three, or four players can compete in different types of deathmatch games. The screen was usually split into four screens, forming a quadrant image. Diddy Kong racing was another game that was usually split into four screens as four friends would race each other

Gran Trak 10 is a single-player racing arcade game released by Atari in 1974. In the game, the player races against the clock with the intent of accumulating as many points as possible. Primitive diode-based ROM was used to store the sprites for the car, score and game timer, and the race track. The game's controls — steering wheel, four-position gear shifter, and accelerator and brake foot pedals — were also all firsts for arcade games. Steve Wozniak famously played Gran Trak 10 during the four-day development of a prototype for another Atari game, Breakout.[1]

Battle the Midway was an early video game that involved you using cross hairs of your periscope to shoot ships. The cross hairs looked like a quadrant. The game was basically a quadrant aiming at ships

Virtua Racing or V.R. for short, is a Formula One racing arcade game, developed by Sega AM2 and released in 1992. It was revolutionary because when selecting a car, the player can choose different transmission types.[3] VR introduced the "V.R. View System" by allowing the player to choose one of four views to play the game. This feature was then used in most other Sega arcade racing games (and is mentioned as a feature in the attract mode of games such as Daytona USA). It was later ported to home consoles, starting with the Mega Drive/Genesis in 1994.

Kwirk, known in Japan as Puzzle Boy (パズルボーイ?), is an action/transport puzzle video game first developed and published by Atlus in Japan on November 24, 1989 for the original Game Boy. In each room, Kwirk must navigate around and interact with various four obstacles in order to progress. Obstacles: Brick Walls – Cannot be moved nor walked through. Brick walls must be maneuvered around and blocks must be pushed around them. Turnstiles – Blocks set on an axis that turn 90 degrees when pushed by a character. They come in single, double, triple, and quadruple variations. They cannot turn if something is blocking their radius of movement. Blocks – Basic blocks of various sizes. They can be pushed by characters and may block paths necessary for a character to reach the stairs. Blocks can also fill holes to allow characters to walk past. Holes – Can't be walked over. Instead, blocks can be used to fill holes or characters must maneuver around the holes. At certain points in the game "Going Up?," one or all of Kwirk’s Veggie Friends will appear to help. They don’t have special abilities, but instead play exactly like Kwirk to allow maneuvers that weren't possible with only one character. The player switches between characters by pressing the select button and all of the four Veggie Friends must be brought to the stairs to clear the floor. The four Veggie Friends: Curly Carrot Eddie Eggplant Pete the Pepper Sass the Squash

Tomato Adventure (トマトアドベンチャー Tomato Adobenchā?) is a role-playing video game (RPG) developed by AlphaDream and published by Nintendo in Japan for the Game Boy Advance on January 25, 2002.

During battles against enemies, the player must fight using toy-like weapons called "Gimmicks",[2] which require the player to play a mini game correctly in order to land direct hits on the enemies, depending on which Gimmick the player uses. While using Gimmicks correctly, the player will earn stars for the extreme attacks, but if the player increases the difficulty of the Gimmicks, the player will increase the attack points of the Gimmicks and earn more stars, while the mini games would be more difficult than before. When one or two gears light up above the Gimmick meter, the player now has the choice to use one of two extreme attacks, if the player has a partner joined in. If the player fails in any mini game, the Gimmick meter will drop down to zero, and the player will have to start it all over. There are four different types of Gimmicks.

There are four types of Gimmicks:

Time - This type makes you finish a mini game that requires timing, like pressing a button whenever something comes to a certain part.

Speed - This type makes you finish a mini game that requires you to finish a task correctly before time runs out.

Excite (or Doki-Doki) - This type makes you finish miscellaneous mini games.

Input - This type makes you finish a mini game that either requires you to repeatedly hit the proper button(s) or to insert the information in a certain order or amount of times.

Star Fox is one of the most popular games of all time. The first Star Fox game had four levels in which there were four characters, Fox McCloud, Peppy Hare, Slippy Toad, and Falco Lombardi trying to save the galaxy. The first Star Fox (スターフォックス Sutā Fokkusu?), released as Starwing in Europe (to avoid confusion with an association named "StarVox" in Germany[1]), is the first game in the Star Fox series of video games, released on February 21, 1993 in Japan, on March 26, 1993 in North America, and on June 3, 1993 in Europe for the Super Famicom/Super Nintendo Entertainment System. It was the second three-dimensional Nintendo-developed game (behind 1992's X, also developed by Nintendo EAD together with Argonaut Software) but it is Nintendo's first game to use 3D polygon graphics. It accomplished this by being the first ever game to use the Super FX graphics acceleration coprocessor powered GSU-1. The complex display of three-dimensional models with polygons was still new and uncommon in console video games, and the game was much-hyped as a result. There are four players. In each level, Fox McCloud, is accompanied by three computer-controlled wingmen: Peppy Hare, Slippy Toad, and Falco Lombardi. Fox McCloud, the leader of the team, is accompanied by his teammates Falco Lombardi, Peppy Hare, and Slippy Toad- a sort of Fantastic Four elite fighting team

Star Fox Adventures was a later Star Fox game for Game Cube features both the established four main characters of the series—Fox, Falco Lombardi, Slippy Toad, and Peppy Hare, although Falco does not appear until near the game's end—and a host of new characters. Major additions are a quiet, mysterious blue fox named Krystal and the small dinosaur Prince Tricky, Fox's helper during the game. The entire planet is populated with dinosaurs, like the tyrannical General Scales, and other prehistoric animals such as pterosaurs and mammoths.[6] The entire game takes place on the world of Dinosaur Planet (in later games called "Sauria") and a number of detached pieces of the planet that are suspended in orbit around it. Dinosaur Planet is ruled by the EarthWalker tribe, which resemble Triceratops, and the rival CloudRunner tribe, similar to pterosaurs and birds. The SharpClaw tribe, which are the major antagonists in Adventures, are humanoid theropods.[6]

Super Smash brothers is a popular video game where four people can fight each other at one time. A common component of newer video games is they allow four people to play each other or with each other at one time

In Star Fox 64 Andross launches an attack across the Lylat system. The Star Fox team, now consisting of four members Fox, Peppy, Falco Lombardi and Slippy Toad have to defeat Andross. While traveling through several planets, including the Lylat system's star, Solar, and the asteroid field Meteo, the team battles with several of Andross' henchmen, including the rival mercenaries, Star Wolf. Star Fox was one of the most popular games in Nintendo 64.

In Star Fox 64 The Arwing is the primary craft used by the Star Fox team. The Arwing can use its boost meter to perform four special moves to avoid collisions and get the drop on pursuers: boost, brake, the U-turn, and the aforementioned somersault.

Fox 64 features multiplayer support for up to four players simultaneously.At first users can only play using the Arwing spaceship, but by earning certain medals in Story Mode, players can unlock the Landmaster tank, as well as the option to fight on foot as one of the four members of Star Fox equipped with a bazooka. Multiplayer is the only place where players can use a Landmaster with upgraded lasers.

Quadrature is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of mathematical analysis. Mathematicians of Ancient Greece, according to the Pythagorean doctrine, understood calculation of area as the process of constructing geometrically a square having the same area (squaring). That is why the process was named quadrature. For example, a quadrature of the circle, Lune of Hippocrates, The Quadrature of the Parabola. This construction must be performed only by means of compass and straightedge. Antique method to find the Geometric mean For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side (the Geometric mean of a and b). For this purpose it is possible to use the following fact: if we draw the circle with the sum of a and b as the diameter, then the height BH (from a point of their connection to crossing with a circle) equals their geometric mean. The similar geometrical construction solves a problem of a quadrature for a parallelogram and a triangle. The area of a segment of a parabola Problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge had been proved in the 19th century to be impossible. Nevertheless, for some figures (for example Lune of Hippocrates) a quadrature can be performed. The quadratures of a sphere surface and a parabola segment done by Archimedes became the highest achievement of the antique analysis. The area of the surface of a sphere is equal to quadruple the area of a great circle of this sphere. The area of a segment of the parabola cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment. For the proof of the results Archimedes used the Method of exhaustion of Eudoxus. In medieval Europe the quadrature meant calculation of area by any method. More often the Method of indivisibles was used; it was less rigorous, but more simple and powerful. With its help Galileo Galilei and Gilles de Roberval found the area of a cycloid arch, Grégoire de Saint-Vincent investigated the area under a hyperbola (Opus Geometricum, 1647), and Alphonse Antonio de Sarasa, de Saint-Vincent's pupil and commentator noted the relation of this area to logarithms. John Wallis algebrised this method: he wrote in his Arithmetica Infinitorum (1656) series that we now call the definite integral, and he calculated their values. Isaac Barrow and James Gregory made further progress: quadratures for some algebraic curves and spirals. Christiaan Huygens successfully performed a quadrature of some Solids of revolution. The quadrature of the hyperbola by Saint-Vincent and de Sarasa provided a new function, the natural logarithm, of critical importance. With the invention of integral calculus came a universal method for area calculation. In response, the term quadrature has become traditional, and instead the modern phrase "computation of a univariate definite integral" is more common.

The solution of the problem of squaring the circle by compass and straightedge demands construction of the number , and the impossibility of this undertaking follows from the fact that pi is a transcendental (non-algebraic and therefore non-constructible) number. If the problem of the quadrature of the circle is solved using only compass and straightedge, then an algebraic value of pi would be found, which is impossible. Johann Heinrich Lambert conjectured that pi was transcendental in 1768 in the same paper in which he proved its irrationality, even before the existence of transcendental numbers was proven. It was not until 1882 that Ferdinand von Lindemann proved its transcendence. The transcendence of pi implies the impossibility of exactly "circling" the square, as well as of squaring the circle. It is possible to construct a square with an area arbitrarily close to that of a given circle. If a rational number is used as an approximation of pi, then squaring the circle becomes possible, depending on the values chosen. However, this is only an approximation and does not meet the constraints of the ancient rules for solving the problem. Several mathematicians have demonstrated workable procedures based on a variety of approximations. Bending the rules by allowing an infinite number of compass-and-straightedge operations or by performing the operations on certain non-Euclidean spaces also makes squaring the circle possible. For example, although the circle cannot be squared in Euclidean space, it can be in Gauss–Bolyai–Lobachevsky space. Indeed, even the preceding phrase is overoptimistic.[7][8] There are no squares as such in the hyperbolic plane, although there are regular quadrilaterals, meaning quadrilaterals with all sides congruent and all angles congruent (but these angles are strictly smaller than right angles). There exist, in the hyperbolic plane, (countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area. However, there is no method for starting with a regular quadrilateral and constructing the circle of equal area, and there is no method for starting with a circle and constructing a regular quadrilateral of equal area (even when the circle has small enough radius such that a regular quadrilateral of equal area exists).

In mathematics, quadrature is a historical term which means determining area. Quadrature problems served as one of the main sources of problems in the development of calculus, and introduce important topics in mathematical analysis.

Mathematicians of ancient Greece, according to the Pythagorean doctrine, understood determination of area of a figure as the process of geometrically constructing a square having the same area (squaring), thus the name quadrature for this process. The Greek geometers were not always successful (see quadrature of the circle), but they did carry out quadratures of some figures whose sides were not simply line segments, such as the lunes of Hippocrates and the quadrature of the parabola. By Greek tradition, these constructions had to be performed using only a compass and straightedge.

For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side (the geometric mean of a and b). For this purpose it is possible to use the following: if one draws the circle with diameter made from joining line segments of lengths a and b, then the height (BH in the diagram) of the line segment drawn perpendicular to the diameter, from the point of their connection to the point where it crosses the circle, equals the geometric mean of a and b. A similar geometrical construction solves the problems of quadrature of a parallelogram and of a triangle.

The area of a segment of a parabola is 4/3 that of the area of a certain inscribed triangle.

Problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge was proved in the 19th century to be impossible. Nevertheless, for some figures (for example a lune of Hippocrates) a quadrature can be performed. The quadratures of the surface of a sphere and a parabola segment discovered by Archimedes became the highest achievement of analysis in antiquity.

The area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere.

The area of a segment of a parabola determined by a straight line cutting it is 4/3 the area of a triangle inscribed in this segment.

For the proof of these results, Archimedes used the method of exhaustion[1]:113 of Eudoxus.

In medieval Europe, quadrature meant the calculation of area by any method. Most often the method of indivisibles was used; it was less rigorous than the geometric constructions of the Greeks, but it was simpler and more powerful. With its help, Galileo Galilei and Gilles de Roberval found the area of a cycloid arch, Grégoire de Saint-Vincent investigated the area under a hyperbola (Opus Geometricum, 1647),[1]:491 and Alphonse Antonio de Sarasa, de Saint-Vincent's pupil and commentator, noted the relation of this area to logarithms.[1]:492[2]

John Wallis algebrised this method; he wrote in his Arithmetica Infinitorum (1656) some series which are equivalent to what is now called the definite integral, and he calculated their values. Isaac Barrow and James Gregory made further progress: quadratures for some algebraic curves and spirals. Christiaan Huygens successfully performed a quadrature of the surface area of some solids of revolution.

The quadrature of the hyperbola by Saint-Vincent and de Sarasa provided a new function, the natural logarithm, of critical importance. With the invention of integral calculus came a universal method for area calculation. In response, the term quadrature has become traditional (some[who?] would say archaic), and instead the modern phrase finding the area is more commonly used for what is technically the computation of a univariate definite integral

In mathematics, a quadratrix (from the Latin word quadrator, squarer) is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhausen, which are both related to the circle.

The quadratrix of Dinostratus (also called the quadratrix of Hippias) was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it affected a mechanical solution of squaring the circle. Pappus, in his Collections, treats its history, and gives two methods by which it can be generated. Let a helix be drawn on a right circular cylinder; a screw surface is then obtained by drawing lines from every point of this spiral perpendicular to its axis. The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix. A right cylinder having for its base an Archimedean spiral is intersected by a right circular cone which has the generating line of the cylinder passing through the initial point of the spiral for its axis. From every point of the curve of intersection, perpendiculars are drawn to the axis. Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix. Quadratrix of Dinostratus (in red) Another construction is as follows. DAB is a quadrant in which the line DA and the arc DB are divided into the same number of equal parts. Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to AB and through the corresponding points on the radius DA. The locus of these intersections is the quadratrix. Quadratrix of Dinostratus with a central portion flanked by infinite branches Letting A be the origin of the Cartesian coordinate system, D be the point (a,0), a units from the origin along the x axis, and B be the point (0,a), a units from the origin along the y axis, the curve itself can be expressed by the equation[1] Because the cotangent function is invariant under negation of its argument, and has a simple pole at each multiple of π, the quadratrix has reflection symmetry across the y axis, and similarly has a pole for each value of x of the form x = 2na, for integer values of n, except at x = 0 where the pole in the cotangent is canceled by the factor of x in the formula for the quadratrix. These poles partition the curve into a central portion flanked by infinite branches. The point where the curve crosses the y axis has y = 2a/π; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a rational multiple of 1/π, leading to a solution of the classical problem of squaring the circle. Since this is impossible with compass and straightedge, the quadratrix in turn cannot be constructed with compass and straightedge. An accurate construction of the quadratrix would also allow the solution of two other classical problems known to be impossible with compass and straightedge, doubling the cube and trisecting an angle.

The quadratrix of Tschirnhausen[2] is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before. The mutual intersections of the lines drawn from the points of division of the arc parallel to DA, and the lines drawn parallel to AB through the points of division of DA, are points on the quadratrix. The cartesian equation is y=a cos 2a. The curve is periodic, and cuts the axis of x at the points x= (2n - I)a, n being an integer; the maximum values of y are =a. Its properties are similar to those of the quadratrix of Dinostratus.

The quadratrix was discovered by Hippias of Elis in 430 BC. It may have been used by him for trisecting an angle and squaring the circle. The curve may be used for dividing an angle into any number of equal parts. Later it was studied by Dinostratus in 350 BC who used the curve to square the circle. Hippias of Elis was a statesman and philosopher who travelled from place to place taking money for his services. Plato describes him as a vain man being both arrogant and boastful. He had a wide but superficial knowledge. His only contribution to mathematics seems to be the quadratrix.

The quadratrix or trisectrix of Hippias (also quadratrix of Dinostratos) is a curve, which is created by a uniform motion. It is one of the oldest examples for a kinematic curve, that is a curve created through motion. Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem (hence trisectrix). Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle (hence quadratrix).

The quadratrix is mentioned in the works of Proklos (412–485), Pappos (3rd and 4th centuries) and Iamblichus (c. 240–325). Proklos names Hippias as the inventor of a curve called quadratrix and describes somewhere else how Hippias has applied the curve on the trisection problem. Pappos only mentions how a curve named quadratrix was used by Dinostratos, Nicomedes and others to square the circle. He neither mentions Hippias nor attributes the invention of the quadratrix to a particular person. Iamblichus just writes in a single line, that a curve called quadratrix was used bei Nicomedes to square the circle.[10][11][12] Though based on Proklos' name for the curve it is conceivable that Hippias himself used it for squaring the circle or some other curvilinear figure, most math historians assume that Hippias invented the curve but used it only for the trisection of angles. Its use for squaring the circle only occurred decades later and was due to mathematicians like Dinostratos and Nicomedes. This interpretation of the historical sources goes back to the German mathematician and historian Moritz Cantor.

Dinostratus (Greek: Δεινόστρατος, c. 390 BCE – c. 320 BCE) was a Greek mathematician and geometer, and the brother of Menaechmus. He is known for using the quadratrix to solve the problem of squaring the circle. Dinostratus' chief contribution to mathematics was his solution to the problem of squaring the circle. To solve this problem, Dinostratus made use of the trisectrix of Hippias, for which he proved a special property (Dinostratus' theorem) that allowed him the squaring of the circle. Due to his work the trisectrix later became known as the quadratrix of Dinostratus as well.[1] Although Dinostratus solved the problem of squaring the circle, he did not do so using ruler and compass alone, and so it was clear to the Greeks that his solution violated the foundational principles of their mathematics.[1] Over two thousand years later it would be proved impossible to square a circle using straight edge and compass alone.

Vector monitors were also used by some late-1970s to mid-1980s arcade games such as Star Wars and Asteroids.[1] Atari used the term Quadrascan to describe the technology when used in their video game arcades. The QUADRASCAN converter modulates the elevation and azimuth error signals in phase quadrature and adds them to the sum channel.

In electrical engineering, a sinusoid with angle modulation can be decomposed into, or synthesized from, two amplitude-modulated sinusoids that are offset in phase by one-quarter cycle (π/2 radians). All three functions have the same frequency. The amplitude modulated sinusoids are known as in-phase and quadrature components.[1] Some authors find it more convenient to refer to only the amplitude modulation (baseband) itself by those terms. when φ happens to be such that the in-phase component is zero, the current and voltage sinusoids are said to be in quadrature, which means they are orthogonal to each other. In that case, no electrical power is consumed. Rather it is temporarily stored by the device and given back, once every seconds. Note that the term in quadrature only implies that two sinusoids are orthogonal, not that they are components of another sinusoid.

Quadrature amplitude modulation (QAM) is both an analog and a digital modulation scheme. It conveys two analog message signals, or two digital bit streams, by changing (modulating) the amplitudes of two carrier waves, using the amplitude-shift keying (ASK) digital modulation scheme or amplitude modulation (AM) analog modulation scheme. The two carrier waves, usually sinusoids, are out of phase with each other by 90° and are thus called quadrature carriers or quadrature components — hence the name of the scheme.

In signal processing, a quadrature filter is the analytic representation of the impulse response of a real-valued filter: If the quadrature filter is applied to a signal , the result is which implies that is the analytic representation of . Since is an analytic signal, it is either zero or complex-valued. In practice, therefore, is often implemented as two real-valued filters, which correspond to the real and imaginary parts of the filter, respectively. An ideal quadrature filter cannot have a finite support, but by choosing the function carefully, it is possible to design quadrature filters which are localized such that they can be approximated reasonably well by means of functions of finite support.

In PSK, the constellation points chosen are usually positioned with uniform angular spacing around a circle. This gives maximum phase-separation between adjacent points and thus the best immunity to corruption. They are positioned on a circle so that they can all be transmitted with the same energy. In this way, the moduli of the complex numbers they represent will be the same and thus so will the amplitudes needed for the cosine and sine waves. Two common examples are "binary phase-shift keying" (BPSK) which uses two phases, and "quadrature phase-shift keying" (QPSK) which uses four phases, although any number of phases may be used. Since the data to be conveyed are usually binary, the PSK scheme is usually designed with the number of constellation points being a power of 2.

Constellation diagram for QPSK with Gray coding. Each adjacent symbol only differs by one bit. Sometimes this is known as quadriphase PSK, 4-PSK, or 4-QAM. (Although the root concepts of QPSK and 4-QAM are different, the resulting modulated radio waves are exactly the same.) QPSK uses four points on the constellation diagram, equispaced around a circle. With four phases, QPSK can encode two bits per symbol, shown in the diagram with Gray coding to minimize the bit error rate (BER) — sometimes misperceived as twice the BER of BPSK. The mathematical analysis shows that QPSK can be used either to double the data rate compared with a BPSK system while maintaining the same bandwidth of the signal, or to maintain the data-rate of BPSK but halving the bandwidth needed. In this latter case, the BER of QPSK is exactly the same as the BER of BPSK - and deciding differently is a common confusion when considering or describing QPSK. The transmitted carrier can undergo numbers of phase changes. Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed (maximum) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK - at the same BER. The engineering penalty that is paid is that QPSK transmitters and receivers are more complicated than the ones for BPSK. However, with modern electronics technology, the penalty in cost is very moderate. As with BPSK, there are phase ambiguity problems at the receiving end, and differentially encoded QPSK is often used in practice. The implementation of QPSK is more general than that of BPSK and also indicates the implementation of higher-order PSK. Writing the symbols in the constellation diagram in terms of the sine and cosine waves used to transmit them: This yields the four phases π/4, 3π/4, 5π/4 and 7π/4 as needed. This results in a two-dimensional signal space with unit basis functions The first basis function is used as the in-phase component of the signal and the second as the quadrature component of the signal. Hence, the signal constellation consists of the signal-space 4 points The factors of 1/2 indicate that the total power is split equally between the two carriers. Comparing these basis functions with that for BPSK shows clearly how QPSK can be viewed as two independent BPSK signals. Note that the signal-space points for BPSK do not need to split the symbol (bit) energy over the two carriers in the scheme shown in the BPSK constellation diagram. QPSK systems can be implemented in a number of ways. An illustration of the major components of the transmitter and receiver structure are shown below. Although QPSK can be viewed as a quaternary modulation, it is easier to see it as two independently modulated quadrature carriers. With this interpretation, the even (or odd) bits are used to modulate the in-phase component of the carrier, while the odd (or even) bits are used to modulate the quadrature-phase component of the carrier. BPSK is used on both carriers and they can be independently demodulated. As a result, the probability of bit-error for QPSK is the same as for BPSK: However, in order to achieve the same bit-error probability as BPSK, QPSK uses twice the power (since two bits are transmitted simultaneously). The symbol error rate is given by: . If the signal-to-noise ratio is high (as is necessary for practical QPSK systems) the probability of symbol error may be approximated: The modulated signal is shown below for a short segment of a random binary data-stream. The two carrier waves are a cosine wave and a sine wave, as indicated by the signal-space analysis above. Here, the odd-numbered bits have been assigned to the in-phase component and the even-numbered bits to the quadrature component (taking the first bit as number 1). The total signal — the sum of the two components — is shown at the bottom. Jumps in phase can be seen as the PSK changes the phase on each component at the start of each bit-period. The topmost waveform alone matches the description given for BPSK above. Timing diagram for QPSK. The binary data stream is shown beneath the time axis. The two signal components with their bit assignments are shown at the top, and the total combined signal at the bottom. Note the abrupt changes in phase at some of the bit-period boundaries. The binary data that is conveyed by this waveform is: 1 1 0 0 0 1 1 0. The odd bits, highlighted here, contribute to the in-phase component: 1 1 0 0 0 1 1 0 The even bits, highlighted here, contribute to the quadrature-phase component: 1 1 0 0 0 1 1 0 Offset QPSK (OQPSK)[edit] Signal doesn't cross zero, because only one bit of the symbol is changed at a time Offset quadrature phase-shift keying (OQPSK) is a variant of phase-shift keying modulation using 4 different values of the phase to transmit. It is sometimes called Staggered quadrature phase-shift keying (SQPSK). Difference of the phase between QPSK and OQPSK Taking four values of the phase (two bits) at a time to construct a QPSK symbol can allow the phase of the signal to jump by as much as 180° at a time. When the signal is low-pass filtered (as is typical in a transmitter), these phase-shifts result in large amplitude fluctuations, an undesirable quality in communication systems. By offsetting the timing of the odd and even bits by one bit-period, or half a symbol-period, the in-phase and quadrature components will never change at the same time. In the constellation diagram shown on the right, it can be seen that this will limit the phase-shift to no more than 90° at a time. This yields much lower amplitude fluctuations than non-offset QPSK and is sometimes preferred in practice. The picture on the right shows the difference in the behavior of the phase between ordinary QPSK and OQPSK. It can be seen that in the first plot the phase can change by 180° at once, while in OQPSK the changes are never greater than 90°. The modulated signal is shown below for a short segment of a random binary data-stream. Note the half symbol-period offset between the two component waves. The sudden phase-shifts occur about twice as often as for QPSK (since the signals no longer change together), but they are less severe. In other words, the magnitude of jumps is smaller in OQPSK when compared to QPSK. π /4–QPSK[edit] Dual constellation diagram for π/4-QPSK. This shows the two separate constellations with identical Gray coding but rotated by 45° with respect to each other. This variant of QPSK uses two identical constellations which are rotated by 45° ( radians, hence the name) with respect to one another. Usually, either the even or odd symbols are used to select points from one of the constellations and the other symbols select points from the other constellation. This also reduces the phase-shifts from a maximum of 180°, but only to a maximum of 135° and so the amplitude fluctuations of –QPSK are between OQPSK and non-offset QPSK. One property this modulation scheme possesses is that if the modulated signal is represented in the complex domain, it does not have any paths through the origin. In other words, the signal does not pass through the origin. This lowers the dynamical range of fluctuations in the signal which is desirable when engineering communications signals. On the other hand, –QPSK lends itself to easy demodulation and has been adopted for use in, for example, TDMA cellular telephone systems. The modulated signal is shown below for a short segment of a random binary data-stream. The construction is the same as above for ordinary QPSK. Successive symbols are taken from the two constellations shown in the diagram. Thus, the first symbol (1 1) is taken from the 'blue' constellation and the second symbol (0 0) is taken from the 'green' constellation. Note that magnitudes of the two component waves change as they switch between constellations, but the total signal's magnitude remains constant (constant envelope). The phase-shifts are between those of the two previous timing-diagrams. SOQPSK[edit] The license-free shaped-offset QPSK (SOQPSK) is interoperable with Feher-patented QPSK (FQPSK), in the sense that an integrate-and-dump offset QPSK detector produces the same output no matter which kind of transmitter is used.[9] These modulations carefully shape the I and Q waveforms such that they change very smoothly, and the signal stays constant-amplitude even during signal transitions. (Rather than traveling instantly from one symbol to another, or even linearly, it travels smoothly around the constant-amplitude circle from one symbol to the next.) The standard description of SOQPSK-TG involves ternary symbols. DPQPSK[edit] Dual-polarization quadrature phase shift keying (DPQPSK) or dual-polarization QPSK - involves the polarization multiplexing of two different QPSK signals, thus improving the spectral efficiency by a factor of 2. This is a cost-effective alternative, to utilizing 16-PSK instead of QPSK to double the spectral efficiency. Any number of phases may be used to construct a PSK constellation but 8-PSK is usually the highest order PSK constellation deployed. With more than 8 phases, the error-rate becomes too high and there are better, though more complex, modulations available such as quadrature amplitude modulation (QAM). Although any number of phases may be used, the fact that the constellation must usually deal with binary data means that the number of symbols is usually a power of 2 to allow an integer number of bits per symbol

The Takeda clan (武田氏 Takeda-shi?) is a Japanese clan active from the late Heian Period (794 – 1185). The clan was historically based in Kai Province in present-day Yamanashi Prefecture.[1

Their crest was the four diamonds/ cross

The Pocket Cube (also known as the Mini Cube or the Ice Cube) is the 2×2×2 equivalent of a Rubik's Cube. The cube consists of 8 pieces, all corners.

Siyi was a derogatory Chinese name for various peoples bordering ancient China, namely, the Dongyi 東夷 "Eastern Barbarians", Nanman 南蠻 "Southern Barbarians", Xirong 西戎 "Western Barbarians", and Beidi 北狄 "Northern Barbarians". The Chinese mytho-geography and cosmography of the Zhou Dynasty (c. 1046–256 BCE) was based upon a round heaven and a square earth. Tianxia 天下 "[everywhere] under heaven; the world" encompassed Huaxia 華夏 "China" (also known as Hua, Xia, etc.) in the center surrounded by non-Chinese "barbarian" peoples. See the Hua–Yi distinction for details of this literally Sinocentric worldview. The Siyi construct, or a similar one, was a logical necessity for the ancient tianxia system. Liu Junping and Huang Deyuan (2006:532) describe the universal monarch with combined political, religious, and cultural authorities: "According to the Chinese in the old times, heaven and earth were matched with yin and yang, with the heaven (yang) superior and the earth (yin) inferior; and the Chinese as an entity was matched with the inferior ethnic groups surrounding it in its four directions so that the kings could be valued and the barbarians could be rejected." The authors (2006:535) propose that Chinese ideas about the "nation" and "state" of China evolved from the "casual use of such concepts as "tianxia", "hainei"( four corners within the sea) and "siyi" 四夷 (barbarians in four directions)." Located in the cardinal directions of tianxia were the sifang 四方 "Four Directions/Corners", situ 四土 "Four Lands/Regions", sihai 四海 "Four Seas", and Siyi 四夷 "Four Barbarians/Foreigners". The (c. 3rd century BCE) Erya (9, Wilkinson 2000: 710) defines sihai as " the place where the barbarians lived, hence by extension, the barbarians": "九夷, 八狄,七戎, 六蠻, 謂之四海" – "the nine Yi, eight Di, seven Rong, and six Man are called the four seas". These Siyi directionally comprised Yi 夷 to the east of China, Man 蠻 in the south, Rong 戎 in the west, and Di 狄 in the north. Unlike the English language with one general word barbarian meaning "uncultured or uncivilized peoples", Chinese had many specific exonyms for foreigners. Scholars such as Herrlee Glessner Creel (1970: 197) agree that Yi, Man, Rong, and Di were originally the Chinese names of particular ethnic groups or tribes. During the Spring and Autumn Period (771–476 BC), these four exonyms were expanded into (Pu 2005: 45) "general designations referring to the barbarian tribes".

In Athens, the population was divided into four social classes based on wealth.

In Athens, the population was divided into four social classes based on wealth.

Philosophy

Marr's tri-level hypothesis[edit] According to David Marr, information processing systems must be understood at three distinct yet complementary levels of analysis - an analysis at one level alone is not sufficient.[1][2] Computational[edit] The computational level of analysis identifies what the information processing system does (e.g.: what problems does it solve or overcome) and similarly, why does it do these things. Algorithmic/representational[edit] The algorithmic/representational level of analysis identifies how the information processing system performs its computations, specifically, what representations are used and what processes are employed to build and manipulate the representations. Physical/implementation[edit] The physical level of analysis identifies how the information processing system is physically realized (in the case of biological vision, what neural structures and neuronal activities implement the visual system). Poggio's learning level[edit] After thirty years of the book Vision (David Marr. 1982. W. H. Freeman and Company), Tomaso Poggio adds one higher level beyond the computational level, that is the learning. I am not sure that Marr would agree, but I am tempted to add learning as the very top level of understanding, above the computational level. [...] Only then may we be able to build intelligent machines that could learn to see—and think—without the need to be programmed to do it. — Tomaso Poggio, Vision (David Marr. 2010. The MIT Press), Afterword, P.367 The fourth is always different and transcendent.

Heyawake (Japanese: へやわけ, "divided rooms") is a binary-determination logic puzzle published by Nikoli. As of 2013, five books consisting entirely of Heyawake puzzles have been published by Nikoli. It first appeared in Puzzle Communication Nikoli #39 (September 1992). Heyawake is played on a rectangular grid of cells with no standard size; the grid is divided into variously sized rectangular "rooms" by bold lines following the edges of the cells. Some rooms may contain a single number, typically printed in their upper-left cell; as originally designed, every room was numbered, but this is rarely necessary for solving and is no longer followed. Some of the cells in the puzzle are to be painted black; the object of the puzzle is to determine for each cell if it must be painted or must be left blank (remaining white). In practice, it is often easier to mark known "blank" cells in some way—for example, by placing a dot in the center of the cell. The following rules determine which cells are which: Rule 1: Painted cells may never be orthogonally connected (they may not share a side, although they can touch diagonally). Rule 2: All white cells must be interconnected (form a single polyomino). Rule 3: A number indicates exactly how many painted cells there must be in that particular room. Rule 4: A room which has no number may contain any number of painted cells, or none. Rule 5: Where a straight (orthogonal) line of connected white cells is formed, it must not contain cells from more than two rooms—in other words, any such line of white cells which connects three or more rooms is forbidden. The game is made up of quadrants

Hitori (Japanese for: Alone or one person) (ひとりにしてくれ Hitori ni shite kure; literally "leave me alone") is a type of logic puzzle published by Nikoli. The game is made up of quadrants Hitori is played with a grid of squares or cells, and each cell contains a number. The objective is to eliminate numbers by filling in the squares such that remaining cells do not contain numbers that appear more than once in either a given row or column. Filled-in cells cannot be horizontally or vertically adjacent, although they can be diagonally adjacent. The remaining un-filled cells must form a single component connected horizontally and vertically.

Hotaru Beam is a binary-determination logic puzzle published by Nikoli. Hotaru Beam is played on a rectangular grid, usually of dashed lines, in which numbers in circles appear at some of the intersections on the grid. Additionally, each circle has a dot on one of the grid lines leading into the circle. The object is to draw solid lines along the grid lines to connect the circles in a single, contiguous network. The network may not branch or cross over itself, except at a circle. The number in a circle indicates how many turns the network will take when traveling in the direction indicated by the dot before connecting to another circle. For example, a 4 circle with a dot on the right hand-side will have a line coming out of the right-hand side which will turn four times before connecting to another circle. The game is made up of quadrants

Inshi no heya ( 因子の部屋 ; lit. "factoring rooms") is a type of logic puzzle published by Nikoli. Inshi no heya is played on a square grid, broken into "rooms" by heavier borders. One of every room's dimensions will be a single cell; the length or width of the room varying by room. Each room may run either horizontally or vertically, and has a small number appearing in its upper left corner. The puzzle starts with all the cells empty. The goal is to fill all the cells with nonzero single-digit numbers (1 through n, where n is the length of the grid's edge) such that: The numbers in each room, when multiplied together, equal the small number in the upper left corner of the room No number appears twice in a column or row The game is made up of quadrants

Keisuke is a logic puzzle published by Nikoli. Keisuke is played on a rectangular grid, in which some cells of the grid are shaded. Additionally, external to the grid, several numeric values are given, some denoted as horizontal, and some denoted as vertical. The puzzle functions as a simple numeric crossword puzzle. The object is to fill in the empty cells with single digits, such that the given numeric values appear on the grid in the orientation specified.

Kuromasu (Japanese:黒どこ kurodoko) is a binary-determination logic puzzle published by Nikoli. As of 2005, one book consisting entirely of Kuromasu puzzles has been published by Nikoli. Kuromasu is played on a rectangular grid. Some of these cells have numbers in them. Each cell may be either black or white. The object is to determine what type each cell is. The following rules determine which cells are which: Each number on the board represents the number of white cells that can be seen from that cell, including itself. A cell can be seen from another cell if they are in the same row or column, and there are no black cells between them in that row or column. Numbered cells may not be black. No two black cells may be horizontally or vertically adjacent. All the white cells must be connected horizontally or vertically. The game is made of quadrants

Light Up (Japanese:美術館 bijutsukan), also called Akari, is a binary-determination logic puzzle published by Nikoli. As of 2011, three books consisting entirely of Light Up puzzles have been published by Nikoli. Light Up is played on a rectangular grid of white and black cells. The player places light bulbs in white cells such that no two bulbs shine on each other, until the entire grid is litup. A bulb sends rays of light horizontally and vertically, illuminating its entire row and column unless its light is blocked by a black cell. A black cell may have a number on it from 0 to 4, indicating how many bulbs must be placed adjacent to its four sides; for example, a cell with a 4 must have four bulbs around it, one on each side, and a cell with a 0 cannot have a bulb next to any of its sides. An unnumbered black cell may have any number of light bulbs adjacent to it, or none. Bulbs placed diagonally adjacent to a numbered cell do not contribute to the bulb count. The game is made of quadrants A typical starting point in the solution of a Light Up puzzle is to find a black cell with a 4, or a cell with a smaller number that is blocked on one or more sides (for example, a 3 against a wall or a 2 in a corner) and therefore has only one configuration of surrounding bulbs. After this step, other numbered cells may be illuminated on one or more sides, narrowing down the possible bulb configurations around them, and in some cases making only one configuration possible. Another common technique is to look for a cell that is not yet lit, and determine if there is only one possible cell in which a bulb can be placed to light it up. When it is unclear where to place a bulb, one may also place dots in white cells that cannot have bulbs, such as around a 0 or in places where a bulb would create a contradiction. For example, a light bulb placed diagonally adjacent to a 3 will block two of its surrounding cells, making it impossible to have three bulbs around it; therefore, the diagonal cells around a 3 can never have lights in them and can be always dotted. Similarly, one may put dots in places where a bulb would "trap" another unlit cell, making it impossible to light it up without breaking the rules. More advanced techniques tend to focus on different combinations of clues. Two 3s that are one space apart, for example, with nothing between them or to the other two sides of the cell in between, must have a lightbulb in that space, and the two spaces next to the two threes, on the line joining them. If not, then one would have two lightbulbs illuminating each other. Also, from this deduction, the remaining four cells surrounding the threes must contain two lightbulbs. Note that as the four spaces are arranged in two rows with nothing in between, one must have one lightbulb to each row, so one can mark all other spaces in those rows as empty. Another fairly common pattern is a 1 diagonally adjacent to a 2, with one of the spaces next to the 2 but not adjacent to the 1 either empty or walled off. At most one lightbulb can be placed in the two cells common to the two clues, so the last lightbulb must go in the last space around the 2. Now, it is known that there is exactly one lightbulb in those cells, so the other cells next to the 1 must both be empty.

LITS, formerly known as Nuruomino (ヌルオミノ), is a binary determination puzzle published by Nikoli. LITS is played on a rectangular grid, typically 10×10; the grid is divided into polyominoes, none of which have fewer than four cells. The goal is to shade in a tetromino within each pre-printed polyomino in such a way that no two matching tetrominoes are orthogonally adjacent (with rotations and reflections counting as matching), and that the shaded cells form a valid nurikabe: they are all orthogonally contiguous (form a single polyomino) and contain no 2×2 square tetrominoes as subsets. The game is made of quadrants

Masyu (ましゅ Mashu?, IPA [maɕu͍]; translates as "evil influence")[1]) is a type of logic puzzle designed and published by Nikoli. The purpose of its creation was to present a puzzle that uses no numbers or letters and yet retains depth and aesthetics. The game is made of quadrants Masyu is played on a rectangular grid of squares, some of which contain circles; each circle is either "white" (empty) or "black" (filled). The goal is to draw a single continuous non-intersecting loop that properly passes through all circled cells. The loop must "enter" each cell it passes through from the center of one of its four sides and "exit" from a different side; all turns are therefore 90 degrees.[1] The two varieties of circle have differing requirements for how the loop must pass through them: White circles must be traveled straight through, but the loop must turn in the previous and/or next cell in its path; Black circles must be turned upon, but the loop must travel straight through the next and previous cells in its path.

A maze is a path or collection of paths, typically from an entrance to a goal. The word is used to refer both to branching tour puzzles through which the solver must find a route, and to simpler non-branching ("unicursal") patterns that lead unambiguously through a convoluted layout to a goal. (The term "labyrinth" is generally synonymous, but also can connote specifically a unicursal pattern.[1]) The pathways and walls in a maze are typically fixed, but puzzles in which the walls and paths can change during the game are also categorised as mazes or tour puzzles. A lot of mazes are made on quadrant grids Loops and traps maze: Follow the arrows from and back to the star Block maze: Fill in four blocks to make a road connecting the stars. No diagonals. Number maze: Begin and end at the star. Using the number in your space, jump that number of blocks in a straight line to a new space. No diagonals

Monchikoros are made of quadrants The binary determination puzzles LITS and Mochikoro, also published by Nikoli, are similar to Nurikabe and employ similar solution methods. The binary determination puzzle Atsumari is similar to Nurikabe but based upon a hexagonal tiling rather than a square tiling. Mochikoro is a variant of the Nurikabe puzzle : Each numbered cell belongs to a white area, the number indicates how many cells belong to the white area. Some white areas may not include a numbered cell. All white areas must be diagonally connected. The black cell must not cover an area of 2x2 cells or larger.

Numberlink is a type of logic puzzle involving finding paths to connect numbers in a grid. Numberlinks are made of quadrants The player has to pair up all the matching numbers on the grid with single continuous lines (or paths). The lines cannot branch off or cross over each other, and the numbers have to fall at the end of each line (i.e., not in the middle). It is considered that a problem is well-designed only if it has a unique solution[1] and all the cells in the grid are filled, although some Numberlink designers do not stipulate this.

Ripple Effect (Japanese:波及効果 Hakyuu Kouka) is a logic puzzle published by Nikoli. As of 2007, two books consisting entirely of Ripple Effect puzzles have been published by Nikoli. The second was published on October 4, 2007. The game is made up of quadrants Ripple Effect is played on a rectangular grid divided into polyominoes. The solver must place one positive integer into each cell of the grid - some of which may be given in advance - according to these rules: Every polyomino must contain the consecutive integers from 1 to the quantity of cells in that polyomino inclusive. If two identical numbers appear in the same row or column, at least that many cells with other numbers must separate them. For example, two cells both containing '1' may not be orthogonally adjacent, but must have at least one cell between them with a different number. Two cells marked '3' in the same row or column must have at least three cells with other numbers between them in that row or column, and so on.

A picture maze is a maze puzzle that forms a picture when solved Picture mazes tend to be produced on quadrant orthogonal lines Picture mazes require no special rules or learning. The rules are very simple: Draw a path from the entrance to exit of the puzzle, avoiding the dead ends. Fill the squares along the path to reveal the hidden picture. It is interesting to note that because of two rules, it is far simpler to reverse the process to solve this puzzle. Starting at a dead end, the path is filled until it reaches an intersection with three or more paths connecting to it. This process is repeated until all dead-end paths are filled, often showing the shortest path. On larger pieces, this is the only way to solve, as there are too many possible but incorrect paths, and chance of finding the correct path based on intuition, logic or simple luck is almost nonexistent.

Nurikabe (hiragana: ぬりかべ) is a binary determination puzzle named for Nurikabe, an invisible wall in Japanese folklore that blocks roads and delays foot travel. Nurikabe was apparently invented and named by Nikoli; other names (and attempts at localization) for the puzzle include Cell Structure and Islands in the Stream. The game is composed of quadrants The puzzle is played on a typically rectangular grid of cells, some of which contain numbers. Cells are initially of unknown color, but can only be black or white. Two same-color cells are considered "connected" if they are adjacent vertically or horizontally, but not diagonally. Connected white cells form "walls", while connected black cells form "a stream". The challenge is to paint each cell black or white, subject to the following rules: Each numbered cell is a wall cell, the number in it is the number of cells in that wall. Each wall must contain exactly one numbered cell. There must be only one stream, which is not allowed to contain "pools", i.e. 2x2 areas of black cells. Human solvers will typically dot the non-numbered cells they've determined to be certain to belong to a wall, while trying to solve the puzzle. Like most other pure-logic puzzles, a unique solution is expected, and a grid containing random numbers is highly unlikely to provide a uniquely solvable Nurikabe puzzle.

Slitherlink (also known as Fences, Takegaki, Loop the Loop, Loopy, Ouroboros, Suriza and Dotty Dilemma) is a logic puzzle developed by publisher Nikoli. The game is composed of points in quadrant formations Slitherlink is played on a rectangular lattice of dots. Some of the squares formed by the dots have numbers inside them. The objective is to connect horizontally and vertically adjacent dots so that the lines form a simple loop with no loose ends. In addition, the number inside a square represents how many of its four sides are segments in the loop. Other types of planar graphs can be used in lieu of the standard grid, with varying numbers of edges per vertex or vertices per polygon. These patterns include snowflake, Penrose, Laves and Altair tilings. These add complexity by varying the number of possible paths from an intersection, and/or the number of sides to each polygon; but similar rules apply to their solution.

Hidato (Hebrew: חידאתו, originating from the Hebrew word Hida = Riddle) is a logic puzzle game invented by Dr. Gyora Benedek, an Israeli mathematician. The goal of Hidato is to fill the grid with consecutive numbers that connect horizontally, vertically, or diagonally. Numbrix puzzles, created by Marilyn vos Savant, are similar to Hidato except that diagonal moves are not allowed. Jadium puzzles (formerly Snakepit puzzles), created by Jeff Marchant, are a more difficult version of Numbrix with fewer given numbers and have appeared on the Parade magazine web site regularly since 2014. The names Numbrix and Hidato are registered trademarks. Some publishers use different names for this puzzle such as Number Snake. In Hidato, a grid of cells is given. It is usually square-shaped, like Sudoku or Kakuro, but it can also include irregular shaped grids like hearts, skulls, and so forth. It can have inner holes (like a disc), but it has to be made of only one piece. The puzzle is made up of quadrants

The 36 Cube is a three-dimensional sudoku puzzle created by ThinkFun. The puzzle consists of a gray base that resembles a city skyline, plus 36 colored towers. The towers come in six different colors and six different heights. The goal of the puzzle is to place all the towers onto the base so as to form a level cube with each of the six colors appearing once, and only once, in each row and column. The 36 cube was invented by Dr. Derrick Niederman, a PhD. at MIT. He came up with the idea while writing a book on whole numbers, after unearthing an 18th-century mathematical hypothesis. This supposition, the 36 officer problem, requires placing six regiments of six differently ranked officers in a 6-x-6 square without having any rank or regiment in the same column. Such an arrangement would form a Graeco-Latin square. Euler conjectured there was no solution to this problem. Although Euler was correct, his conjecture was not settled until Gaston Tarry came up with an exhaustive proof in 1901. The cube is made up of quadrants

The logic puzzle game is played within quadrants

Futoshiki (不等式 futōshiki?), or More or Less, is a logic puzzle game from Japan. Its name means "inequality". It is also spelled hutosiki (using Kunrei-shiki romanization).

The puzzle is played on a square grid, such as 5 x 5. The objective is to place the numbers 1 to 5 (or whatever the dimensions are) such that each row, and column contains each of the digits 1 to 5. Some digits may be given at the start. In addition, inequality constraints are also initially specified between some of the squares, such that one must be higher or lower than its neighbour. These constraints must be honoured as the grid is filled out.

Kenken is played within logic squares KenKen and KenDoku are trademarked names for a style of arithmetic and logic puzzle invented in 2004 by Japanese math teacher Tetsuya Miyamoto,[1] who intended the puzzles to be an instruction-free method of training the brain.[2] The names Calcudoku and Mathdoku are sometimes used by those who don't have the rights to use the KenKen or KenDoku trademarks.[3][4] The name derives from the Japanese word for cleverness (賢 ken, kashiko(i)?).[1] As in sudoku, the goal of each puzzle is to fill a grid with digits –– 1 through 4 for a 4×4 grid, 1 through 5 for a 5×5, etc. –– so that no digit appears more than once in any row or any column (a Latin square). Grids range in size from 3×3 to 9×9. Additionally, KenKen grids are divided into heavily outlined groups of cells –– often called “cages” –– and the numbers in the cells of each cage must produce a certain “target” number when combined using a specified mathematical operation (either addition, subtraction, multiplication or division). For example, a linear three-cell cage specifying addition and a target number of 6 in a 4×4 puzzle must be satisfied with the digits 1, 2, and 3. Digits may be repeated within a cage, as long as they are not in the same row or column. No operation is relevant for a single-cell cage: placing the "target" in the cell is the only possibility (thus being a "free space"). The target number and operation appear in the upper left-hand corner of the cage. In the English-language KenKen books of Will Shortz, the issue of the non-associativity of division and subtraction is addressed by restricting clues based on either of those operations to cages of only two cells in which the numbers may appear in any order. Hence if the target is 1 and the operation is - (subtraction) and the number choices are 2 and 3, possible answers are 2,3 or 3,2. Some puzzle authors have not done this and have published puzzles that use more than two cells for these operations.

One of the most Popular logic puzzles and a generic example of "thinking outside the box is the none dot problem in which you connect nine dots with four lines. The four lines makes you think of the quadrant. Most people get trapped at three lines but by making three long lines and one short line you connect all the dots with four lines. The fourth is always different than the first three

The tower of hanoi is an example

Of the quadrant model phenomena that the first three are similar and the fourth is different and transcendent. Also the graphical nature of the tower of hanoi is the sierpunski triangle, and i discussed the nature of the sierpinski triangle is one trianlge divided into four, reflecting the quadrant four

With four pegs and beyond Edit

Although the three-peg version has a simple recursive solution as outlined above, the optimal solution for the Tower of Hanoi problem with four pegs (called Reve's puzzle), let alone more pegs, is still an open problem. This is a good example of how a simple, solvable problem can be made dramatically more difficult by slightly loosening one of the problem constraints.

The fact that the problem with four or more pegs is an open problem does not imply that no algorithm exists for finding (all of) the optimal solutions. Simply represent the game by an undirected graph, the nodes being distributions of disks and the edges being moves and use breadth first search to find one (or all) shortest path(s) moving a tower from one peg onto another one. However, even smartly implemented on the fastest computer now available, this algorithm provides no way of effectively computing solutions for large numbers of disks; the program would require more time and memory than available. Hence, even having an algorithm, it remains unknown how many moves an optimal solution requires and how many optimal solutions exist for 1000 disks and 10 pegs.

Though it is not known exactly how many moves must be made, there are some asymptotic results. There is also a "presumed-optimal solution" given by the Frame-Stewart algorithm, discovered independently by Frame and Stewart in 1941.[17] The related open Frame-Stewart conjecture claims that the Frame-Stewart algorithm always gives an optimal solution. The optimality of the Frame-Stewart algorithm has been computationally verified for 4 pegs with up to 30 disks.[18]

For other variants of the four-peg Tower of Hanoi problem, see Paul Stockmeyer's survey paper.

Wang tiles reflect the quadrant image In 1961, Wang conjectured that if a finite set of Wang tiles can tile the plane, then there exists also a periodic tiling, i.e., a tiling that is invariant under translations by vectors in a 2-dimensional lattice, like a wallpaper pattern. He also observed that this conjecture would imply the existence of an algorithm to decide whether a given finite set of Wang tiles can tile the plane.[1][2] The idea of constraining adjacent tiles to match each other occurs in the game of dominoes, so Wang tiles are also known as Wang dominoes.[3] The algorithmic problem of determining whether a tile set can tile the plane became known as the domino problem.[4] According to Wang's student, Robert Berger,[4] The Domino Problem deals with the class of all domino sets. It consists of deciding, for each domino set, whether or not it is solvable. We say that the Domino Problem is decidable or undecidable according to whether there exists or does not exist an algorithm which, given the specifications of an arbitrary domino set, will decide whether or not the set is solvable. In other words, the domino problem asks whether there is an effective procedure that correctly settles the problem for all given domino sets. In 1966, Wang's student Robert Berger solved the domino problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if and only if the Turing machine does not halt. The undecidability of the halting problem (the problem of testing whether a Turing machine eventually halts) then implies the undecidability of Wang's tiling problem.[4] Combining Berger's undecidability result with Wang's observation shows that there must exist a finite set of Wang tiles that tiles the plane, but only aperiodically. This is similar to a Penrose tiling, or the arrangement of atoms in a quasicrystal. Although Berger's original set contained 20,426 tiles, he conjectured that smaller sets would work, including subsets of his set, and in his unpublished Ph.D. thesis, he reduced the number of tiles to 104. In later years, increasingly smaller sets were found.[5][6][7] For example, the set of 13 tiles given in the image above is an aperiodic set published by Karel Culik II in 1996.[6] It can tile the plane, but not periodically. Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense. For example, Wang cubes are equal-sized cubes with colored faces and side colors can be matched on any polygonal tessellation. Culik and Kari have demonstrated aperiodic sets of Wang cubes.[8] Winfree et al. have demonstrated the feasibility of creating molecular "tiles" made from DNA (deoxyribonucleic acid) that can act as Wang tiles.[9] Mittal et al. have shown that these tiles can also be composed of peptide nucleic acid (PNA), a stable artificial mimic of DNA.[10] Wang tiles have recently become a popular tool for procedural synthesis of textures, heightfields, and other large and nonrepeating bidimensional data sets; a small set of precomputed or hand-made source tiles can be assembled very cheaply without too obvious repetitions and without periodicity. In this case, traditional aperiodic tilings would show their very regular structure; much less constrained sets that guarantee at least two tile choices for any two given side colors are common because tileability is easily ensured and each tile can be selected pseudorandomly.[11][12][13][14] Wang tiles have also been used in cellular automata theory decidability proofs.[15] In popular culture Edit The short story Wang's Carpets, later expanded to the novel Diaspora, by Greg Egan, postulates a universe, complete with resident organisms and intelligent beings, embodied as Wang tiles implemented by patterns of complex molecules.[16]

TetraVex is a puzzle computer game, available for Windows and Linux systems. TetraVex is an edge-matching puzzle. The player is presented with a grid (by default, 3x3) and nine square tiles, each with a number on each edge. The objective of the game is to place the tiles in the grid in the proper position as fast as possible. Two tiles can only be placed next to each other if the numbers on adjacent faces match. The square tiles are divided in four reflecting the quadrant image

Connect Four (also known as Captain's Mistress, Four Up, Plot Four, Find Four, Fourplay, Four in a Row and Four in a Line) is a two-player connection game in which the players first choose a color and then take turns dropping colored discs from the top into a seven-column, six-row vertically suspended grid. The pieces fall straight down, occupying the next available space within the column. The objective of the game is to connect four of one's own discs of the same color next to each other vertically, horizontally, or diagonally before your opponent. Connect Four is a strongly solved game. The first player can always win by playing the right moves. The game was first sold under the famous Connect Four trademark by Milton Bradley in February 1974.

Connect 4x4 (spoken as Connect Four by Four) is a three-dimensional-thinking strategy game first released in 2009 by Milton Bradley. The goal of the game is identical to that of its similarly named predecessor, Connect Four. Players take turns placing game pieces in the grid-like, vertically suspended playing field until one player has four of his or her color lined up horizontally, vertically, or diagonally. Unlike its predecessor, Connect 4x4 uses a double grid, two different types of game pieces, and can be played by up to four people at once.

Nonograms are logic puzzles made of quadrants

Nonograms, also known as Hanjie, Picross or Griddlers, are picture logic puzzles in which cells in a grid must be colored or left blank according to numbers at the side of the grid to reveal a hidden picture. In this puzzle type, the numbers are a form of discrete tomography that measures how many unbroken lines of filled-in squares there are in any given row or column. For example, a clue of "4 8 3" would mean there are sets of four, eight, and three filled squares, in that order, with at least one blank square between successive groups.

These puzzles are often black and white, describing a binary image, but they can also be colored. If colored, the number clues are also colored to indicate the color of the squares. Two differently colored numbers may have a space in between them. For example, a black four followed by a red two could mean four black boxes, some empty spaces, and two red boxes, or it could simply mean four black boxes followed immediately by two red ones.

Nonograms have no theoretical limits on size,

and are not restricted to square layouts.

HMAS Quadrant (G11/D11/F01), named for the navigational instrument,[2] was a Q-class destroyer operated by the Royal Navy as HMS Quadrant (G67/D17) during World War II, and the Royal Australian Navy (RAN) from 1945 to 1957. The ship was built during the early 1940s as one of the War Emergency Programme destroyers, and entered service in 1942. During World War II, Quadrant served as a convoy escort in the Arctic, South Atlantic, and Indian Oceans, and operated with the British Eastern and British Pacific Fleets. At the war's end, the ship was decommissioned and transferred to the RAN, which operated her for two years before placing her in reserve. In 1950, the ship was docked for conversion into an anti-submarine frigate. Quadrant was recommissioned in 1953, and operated with the RAN until 1957, when she was paid off. The ship was sold for breaking in 1963. 4 × single 4.7-inch QF Mark XI** guns 1 × quadruple 2-pounder "pom-pom" 6 × single 20 mm Oerlikon guns 2 × quadruple torpedo tube sets for 21-inch torpedoes 4 × depth charge throwers, up to 70 depth charges Quadrant was built to the wartime Q class design; the third flotilla of War Emergency Programme destroyers. These ships had a displacement of 1,750 tons at standard load, and 2,388 tons at full load.[2] The destroyer was 358 feet 3 inches (109.19 m) in length overall, 339 feet 6 inches (103.48 m) long between perpendiculars, and had a beam of 35 feet 8 inches (10.87 m).[2] Propulsion was provided by two Admiralty 3-drum boilers connected to Parsons geared turbines; these provided 40,000 shaft horsepower to the destroyer's two propellers.[3] Quadrant could reach speeds of 31.5 knots (58.3 km/h; 36.2 mph).[2] The ship's company consisted of 220 officers and sailors.[3] Quadrant 's armament (at the end of World War II) consisted of four single 4.7-inch QF Mark XI** guns, a quadruple 2-pounder "pom-pom", six single 20 mm Oerlikon anti-aircraft guns, and two quadruple torpedo tube sets for 21-inch torpedoes.[2] The ship was also fitted with four depth charge throwers, with up to 70 depth charges carried.[2] During World War II, Quadrant served with the British Eastern and British Pacific Fleets.[4] Quadrant was engaged in convoy escort duties in the Arctic, South Atlantic, and Indian Oceans. She took part in the North African landings, aircraft carrier strikes against Surabaya and bombardment of the Nicobar Islands. She served with the British Pacific Fleet in 1945 where she took part in operations against Formosa (Taiwan), Okinawa, and the Japanese home islands.[4]

The Quadrant:MK is the national centre for Network Rail, opened in June 2012.[1] The complex, consisting of four linked buildings, is designed to accommodate 3,000 staff.[1] The complex, designed by architects GMW, is designed to provide 37,000 square metres (400,000 sq ft) of space.[2] The buildings are located on the site of the former England National Hockey Stadium, adjacent to Milton Keynes Central railway station.

Quadrant (Antarctica), one of four wedge-shaped divisions of Antarctica delimited by 90° lines of longitude converging at the South Pole

Quadrant Park[3] was a nightclub and in Liverpool, UK opened during the late 1980s to the early 1990s.[1] and one the most important in the UK at the time.[4] and was known to attract a number of international guest DJs. The main styles of music played were Italo house, rave and acid house.[5]

The Quadrant Shopping Centre is the principal under-cover shopping centre in Swansea, Wales. The centre opened in 1979.[1] Since the 1980s it has been home to the Swansea Devil, a controversial carved wooden statue of the Devil. The centre and surrounding areas are owned by the City and County of Swansea council.[2]

In cities with Cartesian-coordinate-based addressing systems, the streets that form the north-south and east-west dividing lines constitute the x and y axes of a Cartesian coordinate plane and thus divide the city into quadrants. The quadrants are typically identified in the street names, although the manner of doing so varies from city to city. For example, in one city, all streets in the northeast quadrant may have "NE" prefixed or suffixed to their street names, while in another, the intersection of North Calvert Street and East 27th Street can be only in the northeast quadrant

Swansea bus station, or Swansea city bus station,[1] is a bus station serving Swansea, Wales. It lies immediately to the west of the Quadrant Shopping Centre. The bus station also has a taxi rank to the south.

Quadrant is a 1977 (see 1977 in music) album by American jazz guitarist Joe Pass and vibraphonist Milt Jackson. It was re-issued in 1991 on CD by Original Jazz Classics.

Quadrant is an Australian literary and cultural journal. Quadrant reviews literature, as well as featuring essays on ideas and topics such as politics, history, universities, and the arts. It also publishes poetry and short stories.

The Quadrant Cycle Company was a company in Birmingham, England that was established in 1890 as a bicycle manufacturer. They advanced to make motorcycles from 1899 until their demise in 1928. They also made a tricar called Carette in 1899 and a small number of cars for about two years around 1906.

Quadrant was one of the earliest British motorcycle manufacturers, established in Birmingham in 1901. Famous for their big singles, Quadrant pioneered many innovations that proved important for motorcycle development but struggled after the First World War and the company was wound up in 1928.[

Gunter's quadrant is an instrument made of wood, brass or other substance, containing a kind of stereographic projection of the sphere on the plane of the equinoctial, the eye being supposed to be placed in one of the poles, so that the tropic, ecliptic, and horizon form the arcs of circles, but the hour circles are other curves, drawn by means of several altitudes of the sun for some particular latitude every year. This instrument is used to find the hour of the day, the sun's azimuth, etc., and other common problems of the sphere or globe, and also to take the altitude of an object in degrees. A rare Gunter quadrant, made by Henry Sutton and dated 1657, can be described as follows: It is a conveniently sized and high-performance instrument that has two pin-hole sights, and the plumb line is inserted at the vertex. The front side is designed as a Gunter quadrant and the rear side as a trigonometric quadrant. The side with the astrolabe has hour lines, a calendar, zodiacs, star positions, astrolabe projections, and a vertical dial. The side with the geometric quadrants features several trigonometric functions, rules, a shadow quadrant, and the chorden line.[9]

A treatise explaining the importance of the astrolabe by Nasir al-Din al-Tusi, Persian scientist. In this treatise he teaches how to create an astrolabe usijg 16 squares. He 16 squares is the quadrant model form

The sine quadrant (Arabic: Rub‘ul mujayyab, الربع المجيب) was a type of quadrant used by medieval Arabic astronomers. It is also known as a "sinecal quadrant" in the English-speaking world. The instrument could be used to measure celestial angles, to tell time, to find directions, or to determine the apparent positions of any celestial object for any time. The name is derived from the Arabic "rub‘‘‘" meaning a quarter and "mujayyab" meaning marked with sine.[1] It was described, according to King, by Muhammad ibn Mūsā al-Khwārizmī in 9th century Baghdad.

The instrument is a quarter of a circle made of wood or metal (usually brass) divided on its arc side into 90 equal parts or degrees. The 90 divisions are gathered in 18 groups of five degrees each and are generally numbered both ways from the ends of the arc. That is, one set of numbers begins at the left end of the arc and goes to 90 at the right end while the other set the zero is at the right and the 90 is at the left. This double numbering enables the instrument to measure either celestial altitude or zenith distance or both simultaneously. At the apex where the two graduated straight sides of the grid pattern meet in a right angle there is a pin hole with a cord in it and a small weight on the free end with a small bead that slides along the cord. The cord is called “Khait” and is used as a plumb line when measuring celestial altitudes. It is also used as the indicator of angles when doing calculations with the instrument. The sliding bead facilitates trigonometric calculations with the instrument. Traditionally the line from the beginning of the arc to the apex is called “Jaibs” and the line from the end of the arc to the apex is called “Jaib tamams”. Both jaibs and jaib tamams are divided into 60 equal units and the sixty parallel lines to the jaibs are called sitheeniys or” sixtys “ and the sixty parallel lines to the jaib tamams are “juyoobul mabsootah”. The reason for sixty divisions along the Jaibs and Jaib Tamams is that the instrument uses the Sexagesimal number system. That is it is graduated to the number base 60 and not to the base 10 or decimal system that we presently use. Time, angular measurement and geographical coordinate measurements are about the only hold overs from the Sumerian/Babylonian number system that are still in current use. Like the arc, the Jaibs and Jaib tamams have their sixty divisions gathered into groups of five that are numbered in both directions to and from the apex. The double numbering of the arc means that the “Jaibs” and “Jaib tamams” labels are relative to the measurement being taken or to the calculation being performed at the time and the terms are not attached to one or the other of the graduated scales on the instrument.

Edmund Gunter invented the cross bow quadrant, also called the mariner's bow, around 1623.[4] It gets its name from the similarity to the archer's crossbow. This instrument is interesting in that the arc is 120° but is only graduated as a 90° arc.[4] As such, the angular spacing of a degree on the arc is slightly greater than one degree. Examples of the instrument can be found with a 0° to 90° graduation or with two mirrored 0° to 45° segments centred on the midpoint of the arc.[4] The instrument has three vanes, a horizon vane (A in Figure 8) which has an opening in it to observe the horizon, a shadow vane (B) to cast a shadow on the horizon vane and a sighting vane (C) that the navigator uses to view the horizon and shadow at the horizon vane. This serves to ensure the instrument is level while simultaneously measuring the altitude of the sun. The altitude is the difference in the angular positions of the shadow and sighting vanes. With some versions of this instrument, the sun's declination for each day of the year was marked on the arc. This permitted the navigator to set the shadow vane to the date and the instrument would read the altitude directly.

An Elton's quadrant is a derivative of the Davis quadrant. It adds an index arm and artificial horizon to the instrument. It was invented by John Elton a sea captain who patented his design in 1728[1] and published details of the instrument in the Philosophical Transactions of the Royal Society in 1732.[2] Drawing of Elton's Quadrant. From Philosophical Transactions of the Royal Society, No. 423, Vol 37, 1731-1732. This instrument is derived from the Davis quadrant, adding an index arm with spirit levels to use as an artificial horizon. The instrument is signed J. Sisson fecit - Jonathan Sisson made the instrument as drawn. This instrument clearly reflects the shape and features of the Davis quadrant. The significant differences are the change in the upper arc to a simple triangular frame and the addition of an index arm. The triangular frame at the top spans 60° as did the arc on the backstaff. The main graduated arc subtends 30° as in the backstaff. The 30° arc is graduated in degrees and sixths of a degree, that is, at ten-minute intervals. The sighting vane of the backstaff is replaced with a sight (called an eye vane[2]) mounted on the end of the index arm. The index arm includes a nonius[3][2][4] to allow reading the large scale with ten divisions between the graduations on the scale. This provides the navigator with the ability to read the scale to the nearest minute of arc. The index arm has a spirit level to allow the navigator to ensure that the index is horizontal even when he cannot see the horizon. The instrument has a horizon vane like a Davis quadrant, but Elton refers to it as the shield or ray vane.[2] The shield is attached to the label.[5] The label is an arm that extends from the centre of the arc to the outside of the upper triangle and can be set to one of the three positions in the triangle (in the diagram, it appears to bisect the triangle as it is set to the centre or 30° position). At the upper end of the label is a Flamsteed glass or lens.[2] The three set positions allow the instrument to read 0° to 30°, 30° to 60° or 60° to 90°. The lens projects an image of the sun rather than a shadow of the sun on the shield. This provides an image even when the sky is hazy or lightly overcast. In addition, at the mid-span of the label there is a mounting point for a lantern to be used during nocturnal observations. There are two spirit levels on the shield. One, called the azimuth tube, ensures that the plane of the instrument is vertical. The other is perpendicular to the shield and will indicate when the plane of the shield is vertical and the label is horizontal.

Benjamin Cole quadrant Edit

Figure 7 – Cole quadrant from 1748.

A late addition to the collection of backstaves in the navigation world, this device was invented by Benjamin Cole in 1748.[4]

The instrument consists of a staff with a pivoting quadrant on one end. The quadrant has a shadow vane, which can optionally take a lens like the Davis quadrant's Flamsteed glass, at the upper end of the graduated scale (A in Figure 7). This casts a shadow or projects an image of the sun on the horizon vane (B). The observer views the horizon through a hole in the sight vane (D) and a slit in the horizon vane to ensure the instrument is level. The quadrant component is rotated until the horizon and the sun's image or shadow are aligned. The altitude can then be read from the quadrant's scale. In order to refine the reading, a circular vernier is mounted on the staff (D).

The fact that such an instrument was introduced in the middle of the 18th century shows that the quadrant was still a viable instrument even in the presence of the octant.

George Adams Sr. created a very similar backstaff at the same time. Adam's version ensured that the distance between the Flamsteed glass and horizon vane was the same as the distance from the vane to the sight vane.[8]

Thomas Hood invented this cross-staff in 1590.[4] It could be used for surveying, astronomy or other geometric problems. It consists of two components, a transom and a yard. The transom is the vertical component and is graduated from 0° at the top to 45° at the bottom. At the top of the transom, a vane is mounted to cast a shadow. The yard is horizontal and is graduated from 45° to 90°. The transom and yard are joined by a special fitting (the double socket in Figure 6) that permits independent adjustments of the transom vertically and the yard horizontally. It was possible to construct the instrument with the yard at the top of the transom rather than at the bottom.[7] Initially, the transom and yard are set so that the two are joined at their respective 45° settings. The instrument is held so that the yard is horizontal (the navigator can view the horizon along the yard to assist in this). The socket is loosened so that the transom is moved vertically until the shadow of the vane is cast at the yard's 90° setting. If the movement of just the transom can accomplish this, the altitude is given by the transom's graduations. If the sun is too high for this, the yard horizontal opening in the socket is loosened and the yard is moved to allow the shadow to land on the 90° mark. The yard then yields the altitude. It was a fairly accurate instrument, as the graduations were well spaced compared to a conventional cross-staff. However, it was a bit unwieldy and difficult to handle in wind.

There was a variation of the quadrant – the Back observation quadrant – that was used for measuring the sun's altitude by observing the shadow cast on a horizon vane

The cross-staff was normally a direct observation instrument. However, in later years it was modified for use with back observations

The plough was the name given to an unusual instrument that existed for a short time.[4] It was part cross-staff and part backstaff. In Figure 5, A is the transom that casts its shadow on the horizon vane at B. It functions in the same manner as the staff in Figure 1. C is the sighting vane. The navigator uses the sighting vane and the horizon vane to align the instrument horizontally. The sighting vane can be moved left to right along the staff. D is a transom just as one finds on a cross-staff. This transom has two vanes on it that can be moved closer or farther from the staff to emulate different-length transoms. The transom can be moved on the staff and used to measure angles.

The demi-cross was an instrument that was contemporary with the Davis quadrant. It was popular outside of England.[4] The vertical transom was like a half-transom on a cross-staff, hence the name demi-cross. It supported a shadow vane (A in Figure 4) that could be set to one of several heights (three according to May,[4] four according to de Hilster[6]). By setting the shadow vane height, the range of angles that could be measured was set. The transom could be slid along the staff and the angle read from one of the graduated scales on the staff. The sight vane (C) and horizon vane (B) were aligned visually with the horizon. With the shadow vane's shadow cast on the horizon vane and aligned with the horizon, the angle was determined. In practice, the instrument was accurate but more unwieldy than the Davis quadrant.[6] The plough

The Elton's quadrant derived from the Davis quadrant. It added an index arm with spirit levels to provide an artificial horizon.

Captain John Davis invented a version of the backstaff in 1594. Davis was a navigator who was quite familiar with the instruments of the day such as the mariner's astrolabe, the quadrant and the cross-staff. He recognized the inherent drawbacks of each and endeavoured to create a new instrument that could reduce those problems and increase the ease and accuracy of obtaining solar elevations.

One early version of the quadrant staff is shown in Figure 1.[3] It had an arc affixed to a staff so that it could slide along the staff (the shape is not critical, though the curved shape was chosen). The arc (A) was placed so that it would cast its shadow on the horizon vane (B). The navigator would look along the staff and observe the horizon through a slit in the horizon vane. By sliding the arc so that the shadow aligned with the horizon, the angle of the sun could be read on the graduated staff. This was a simple quadrant, but it was not as accurate as one might like. The accuracy in the instrument is dependent on the length of the staff, but a long staff made the instrument more unwieldy. The maximum altitude that could be measured with this instrument was 45°.

The next version of his quadrant is shown in Figure 2.[3] The arc on the top of the instrument in the previous version was replaced with a shadow vane placed on a transom. This transom could be moved along a graduated scale to indicate the angle of the shadow above the staff. Below the staff, a 30° arc was added. The horizon, seen through the horizon vane on the left, is aligned with the shadow. The sighting vane on the arc is moved until it aligns with the view of the horizon. The angle measured is the sum of the angle indicated by the position of the transom and the angle measured on the scale on the arc.

The instrument that is now identified with Davis is shown in Figure 3.[4] This form evolved by the mid-17th century.[4] The quadrant arc has been split into two parts. The smaller radius arc, with a span of 60°, was mounted above the staff. The longer radius arc, with a span of 30° was mounted below. Both arcs have a common centre. At the common centre, a slotted horizon vane was mounted (B). A moveable shadow vane was placed on the upper arc so that its shadow was cast on the horizon vane. A moveable sight vane was mounted on the lower arc (C).

It is easier for a person to place a vane at a specific location than to read the arc at an arbitrary position. This is due to Vernier acuity, the ability of a person to align two line segments accurately. Thus an arc with a small radius, marked with relatively few graduations, can be used to place the shadow vane accurately at a specific angle. On the other hand, moving the sight vane to the location where the line to the horizon meets the shadow requires a large arc. This is because the position may be at a fraction of a degree and a large arc allows one to read smaller graduations with greater accuracy. The large arc of the instrument, in later years, was marked with transversals to allow the arc to be read to greater accuracy than the main graduations allow.[5]

Thus Davis was able to optimize the construction of the quadrant to have both a small and a large arc, allowing the effective accuracy of a single arc quadrant of large radius without making the entire instrument so large. This form of the instrument became synonymous with the backstaff. It was one of the most widely used forms of the backstaff. Continental European navigators called it the English Quadrant.

A later modification of the Davis quadrant was to use a Flamsteed glass in place of the shadow vane; this was suggested by John Flamsteed.[4] This placed a lens on the vane that projected an image of the sun on the horizon vane instead of a shadow. It was useful under conditions where the sky was hazy or lightly overcast; the dim image of the sun was shown more brightly on the horizon vane where a shadow could not be seen.[5]

In heraldry, the Cross of Saint James, also called the Santiago cross or the cruz espada,[1] is a charge in the form of a cross. It combines a cross fitchy (the lower limb is pointed, as if to be driven into the ground) with either a cross fleury[2] (the arms end in fleurs-de-lys) or a cross moline[1] (the ends of the arms are forked and rounded). Most notably, a red Cross of Saint James with flourished arms, surmounted with an escallop,[2] was the emblem of the twelfth-century Spanish military Order of Santiago, named after Saint James the Greater. It is also used as a decorative element on the Tarta de Santiago, a traditional Galician sweet.

The term Jacob's staff, also cross-staff, a ballastella, a fore-staff, or a balestilha, is used to refer to several things. This can lead to considerable confusion unless one clarifies the purpose for which the object was named. In its most basic form, a Jacob's staff is a stick or pole with length markings; most staffs are much more complicated than that, and usually contain a number of measurement and stabilization features. The two most frequent uses are: in astronomy and navigation for a simple device to measure angles, later replaced by the more precise sextants; in surveying (and scientific fields that use surveying techniques, such as geology and ecology) for a vertical rod that penetrates or sits on the ground and supports a compass or other instrument. The simplest use of a Jacob's staff is to make qualitative judgements of the height and angle of an object relative to the user of the staff. The term Jacob's staff, also cross-staff, a ballastella, a fore-staff, or a balestilha, is used to refer to several things. This can lead to considerable confusion unless one clarifies the purpose for which the object was named. In its most basic form, a Jacob's staff is a stick or pole with length markings; most staffs are much more complicated than that, and usually contain a number of measurement and stabilization features. The two most frequent uses are: in astronomy and navigation for a simple device to measure angles, later replaced by the more precise sextants; in surveying (and scientific fields that use surveying techniques, such as geology and ecology) for a vertical rod that penetrates or sits on the ground and supports a compass or other instrument. The simplest use of a Jacob's staff is to make qualitative judgements of the height and angle of an object relative to the user of the staff. In the original form of the cross-staff, the pole or main staff was marked with graduations for length. The cross-piece (BC in the drawing to the right), also called the transom or transversal, slides up and down on the main staff. On older instruments, the ends of the transom were cut straight across. Newer instruments had brass fittings on the ends with holes in the brass for observation. In marine archaeology, these fittings are often the only components of a cross-staff that survive.[11] It was common to provide several transoms, each with a different range of angles it would measure. Three transoms were common. In later instruments, separate transoms were switched in favour of a single transom with pegs to indicate the ends. These pegs mounted in one of several pairs of holes symmetrically located on either side of the transom. This provided the same capability with fewer parts.[8] The transom on Frisius' version had a sliding vane on the transom as an end point.[8] Usage Edit The navigator places one end of the main staff against his cheek just below his eye. He sights the horizon at the end of the lower part of the transom (or through the hole in the brass fitting) (B), adjusting the cross arm on the main arm until the sun is at the other end of the transom (C). The altitude can then be determined by reading the position of the transom on the scale on the main staff. This value was converted to an angular measurement by looking up the value in a table. Cross-staff for navigation Edit The original version was not reported to be used at sea, until the Age of Discoveries. Its use was reported by João de Lisboa in his Treatise on the Nautical Needle of 1514.[12] Johannes Werner suggested the cross-staff be used at sea in 1514[8] and improved instruments were introduced for use in navigation. John Dee introduced it to England in the 1550s.[1] In the improved versions, the rod was graduated directly in degrees. This variant of the instrument is not correctly termed a Jacob's staff but is a cross-staff.[6] The cross-staff was difficult to use. In order to get consistent results, the observer had to position the end of the pole precisely against his cheek. He had to observe the horizon and a star in two different directions while not moving the instrument when he shifted his gaze from one to the other. In addition, observations of the sun required the navigator to look directly at the sun. This could be a painful exercise and made it difficult to obtain an accurate altitude for the sun. Mariners took to mounting smoked-glass to the ends of the transoms to reduce the glare of the sun.[8][13] As a navigational tool, this instrument was eventually replaced, first by the backstaff or quadrant, neither of which required the user to stare directly into the sun, and later by the octant and the sextant. Perhaps influenced by the backstaff, some navigators modified the cross-staff to operate more like the former. Vanes were added to the ends of the longest cross-piece and another to the end of the main staff. The instrument was reversed so that the shadow of the upper vane on the cross piece fell on the vane at the end of the staff. The navigator held the instrument so that he would view the horizon lined up with the lower vane and the vane at the end of the staff. By aligning the horizon with the shadow of the sun on the vane at the end of the staff, the elevation of the sun could be determined.[14] This actually increased the accuracy of the instrument, as the navigator no longer had to position the end of the staff precisely on his cheek. Another variant of the cross-staff was a spiegelboog, invented in 1660 by the Dutchman, Joost van Breen. Ultimately, the cross-staff could not compete with the backstaff in many countries. In terms of handling, the backstaff was found to be more easy to use.[15] However, it has been proven by several authors that in terms of accuracy, the cross-staff was superior to the backstaff.[16] Backstaves were no longer allowed on board Dutch East India Company vessels as per 1731, with octants not permitted until 1748.[16] Surveying In surveying the Jacob's staff, contemporaneously referred to as a jacob staff, is a single straight rod or staff made of nonferrous material, pointed and metal-clad at the bottom for penetrating the ground.[17] It also has a screw base and occasionally a ball joint on the mount, and is used for supporting a compass, transit, or other instrument.[18] The term cross-staff may also have a different meaning in the history of surveying. While the astronomical cross-staff was used in surveying for measuring angles, two other devices referred to as a cross-staff were also employed.[19] Cross-head, cross-sight, surveyor's cross or cross - a drum or box shaped device mounted on a pole. It had two sets of mutually perpendicular sights. This device was used by surveyors to measure offsets. Sophisticated versions had a compass and spirit levels on the top. The French versions were frequently eight-sided rather than round.[19] Optical square - an improved version of the cross-head, the optical square used two mirrors at 45° to each other. This permitted the surveyor to see along both axes of the instrument at once. Use of the Jacob's Staff as a support Edit In the past, many surveyor's instruments were used on a Jacob's staff. These include: Cross-head, cross-sight, surveyor's cross or cross Graphometer Circumferentor Holland circle Miner's dial Optical square Surveyor's Sextant Surveyor's target Abney level Some devices, such as the modern optical targets for laser-based surveying, are still in common use on a Jacob's staff.

Tycho Brahe's mural quadrant A mural instrument is an angle measuring device mounted on or built into a wall. For astronomical purposes, these walls were oriented so they lie precisely on the meridian. A mural instrument that measured angles from 0 to 90 degrees was called a mural quadrant. Many older mural quadrants have been constructed by marking directly on the wall surfaces. More recentinstruments were made with a frame that was constructed with precision and mounted permanently on the wall. The arc is marked with divisions, almost always in degrees and fractions of a degree. In the oldest instruments, an indicator is placed at the centre of the arc. An observer can move a device with a second indicator along the arc until the line of sight from the movable device's indicator through the indicator at the centre of the arc aligns with the astronomical object. The angle is then read, yielding the elevation or altitude of the object. In smaller instruments, an alidade could be used. More modern mural instruments would use a telescope with a reticle eyepiece to observe the object. Many mural quadrants were constructed, giving the observer the ability to measure a full 90° range of elevation. There were also mural sextants that read 60°. In order to measure the position of, for example, a star, the observer needs a sidereal clock in addition to the mural instrument. With the clock measuring time, a star of interest is observed with the instrument until it crosses an indicator showing that it is transiting the meridian. At this instant, the time on the clock is recorded as well as the angular elevation of the star. This yields the position in the coordinates of the instrument. If the instrument's arc is not marked relative to the celestial equator, then the elevation is corrected for the difference, resulting in the star's declination. If the sidereal clock is precisely synchronized with the stars, the time yields the right ascension directly.[1] Famous mural instruments

Large frame quadrants were used for astronomical measurements, notably determining the altitude of celestial objects. They could be permanent installations, such as mural quadrants. Smaller quadrants could be moved. Like the similar astronomical sextants, they could be used in a vertical plane or made adjustable for any plane. When set on a pedestal or other mount, they could be used to measure the angular distance between any two celestial objects. The details on their construction and use are essentially the same as those of the astronomical sextants; refer to that article for details. Navy: Used to gauge elevation on ships cannon, the quadrant had to be placed on each gun's trunnion in order to judge range, after the loading. The reading was taken at the top of the ship's roll, the gun adjusted,and checked, again at the top of the roll, and he went to the next gun, until all that were going to be fired were ready. The ship's Gunner was informed, who in turn informed the captain...You may fire when ready...at the next high roll, the cannon would be fired. In more modern applications, the quadrant is attached to the trunion ring or of a large naval gun to align it to benchmarks welded to the ship's deck. This is done to ensure firing of the gun hasn't "warped the deck." A flat surface on the mount gunhouse or turret is also checked against benchmarks, also, to ensure large bearings and/or bearing races haven't changed... to "calibrate" the gun.

Back observation quadrant Edit

In order to perform measurements of the altitude of the sun, a back observation quadrant was developed.[6]

With such a quadrant, the observer viewed the horizon from a sight vane (C in the figure on the right) through a slit in the horizon vane (B). This ensured the instrument was level. The observer moved the shadow vane (A) to a position on the graduated scale so as to cause its shadow to appear coincident with the level of the horizon on the horizon vane. This angle was the elevation of the sun

The geometric quadrant is a quarter-circle panel usually of wood or brass. Markings on the surface might be printed on paper and pasted to the wood or painted directly on the surface. Brass instruments had their markings scribed directly into the brass. For marine navigation, the earliest examples were found around 1460. They were not graduated in degrees but rather had the latitudes of the most common destinations directly scribed on the limb. When in use, the navigator would sail north or south until the quadrant indicated he was at the destination's latitude, turn in the direction of the destination and sail to the destination maintaining a course of constant latitude. After 1480, more of the instruments were made with limbs graduated in degrees.[6] Along one edge there were two sights forming an alidade. A plumb bob was suspended by a line from the centre of the arc at the top. In order to measure the altitude of a star, the observer would view the star through the sights and hold the quadrant so that the plane of the instrument was vertical. The plumb bob was allowed to hang vertical and the line indicated the reading on the arc's graduations. It was not uncommon for a second person to take the reading while the first concentrated on observing and holding the instrument in proper position. The accuracy of the instrument was limited by its size and by the effect the wind or observer's motion would have on the plumb bob. For navigators on the deck of a moving ship, these limitations could be difficult to overcome.

Islamic - King identified four types of quadrants that were produced by Muslim astronomers.[3] The sine quadrant (Arabic: Rubul Mujayyab) - also known as the "Sinecal Quadrant" – was used for solving trigonometric problems and taking astronomical observations. It was developed by al-Khwarizmi in 9th century Baghdad and prevalent until the nineteenth century. Its defining feature is a graph-paper like grid on one side that is divided into sixty equal intervals on each axis and is also bounded by a 90 degree graduated arc. A cord was attached to the apex of the quadrant with a bead, for calculation, and a plumb bob. They were also sometimes drawn on the back of astrolabes. The universal (shakkāzīya) quadrant – used for solving astronomical problems for any latitude: These quadrants had either one or two sets of shakkāzīya grids and were developed in the fourteenth century in Syria. Some astrolabes are also printed on the back with the universal quadrant like an astrolabe created by Ibn al-Sarrāj. The horary quadrant – used for finding the time with the sun: The horary quadrant could be used to find the time either in equal or unequal (length of the day divided by twelve) hours. Different sets of markings were created for either equal or unequal hours. For measuring the time in equal hours, the horary quadrant could only be used for one specific latitude while a quadrant for unequal hours could be used anywhere based on an approximate formula. One edge of the quadrant had to be aligned with the sun, and once aligned, a bead on the plumbline attached to the centre of the quadrant showed the time of the day. An example exists dated 1396, from European sources (Richard II of England).[4] The oldest horary quadrant was found during an excavation in 2013 in the Hanseatic town of Zutphen (Netherlands) and is dated ca. 1300 and is in the local Stedelijk Museum in Zutphen.).[5] The astrolabe/almucantar quadrant – a quadrant developed from the astrolabe: This quadrant was marked with one half of a typical astrolabe plate as astrolabe plates are symmetrical. A cord attached from the centre of the quadrant with a bead at the other end was moved to represent the position of a celestial body (sun or a star). The ecliptic and star positions were marked on the quadrant for the above. It is not known where and when the astrolabe quadrant was invented, existent astrolabe quadrants are either of Ottoman or Mamluk origin, while there have been discovered twelfth century Egyptian and fourteenth century Syrian treatises on the astrolabe quadrant. These quadrants proved to be very popular alternatives to astrolabes.

Altitude - The plain quadrant with plumb line, used to take the altitude of an object. Gunner's - A type of clinometer used by an artillerist to measure the elevation or depression angle of a gun barrel of a cannon or mortar, both to verify proper firing elevation, and to verify the correct alignment of the weapon-mounted fire control devices. Gunter's - A quadrant used for time determination. Invented by Edmund Gunter in 1623.

A quadrant is an instrument that is used to measure angles up to 90°. It was originally proposed by Ptolemy as a better kind of astrolabe.[1] Several different variations of the instrument were later produced by medieval Muslim astronomers. There are several types of quadrants: Mural quadrants used for measuring the altitudes of astronomical objects. Large frame-based instruments used for measuring angular distances between astronomical objects. Geometric quadrant used by surveyors and navigators. Davis quadrant a compact, framed instrument used by navigators for measuring the altitude of an astronomical object.

Quadrant in architecture refers to a curve in a wall or a vaulted ceiling. Generally considered to be an arc of 90 degrees - one quarter of a circle, or a half of the more commonly seen architectural feature - a crescent. The quadrant curve in architecture was a feature popularised by Palladio who used it often for the wings and colonnades which linked his classical style villas to their service wings and out-buildings. However, curved quadrant buildings should not be confused with the canted facades of Baroque architecture or the slightly curved buildings of the era such as the Quattro Canti in Palermo. The quadrant vault, a feature of Tudor architecture, is a curving interior - a continuous arc usually of brick, as seen in a tunnel - as opposed to a ribbed vault where a framework of ribs or arches supports the curves of the vault. A quadrant arch was often employed in Romanesque architecture to provide decorative support, as seen in the flying buttresses of Notre-Dame de Chartres built in the second half of the 12th century. During the 18th century, the quadrant once again became a popular design shape for the terraces of smart houses in fashionable spa towns such as Buxton. Henry Currey's "Quadrant", built to rival the architecture of Bath, is considered one of Buxton's finest buildings.

In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections. A cross section of three-dimensional space that is parallel to two of the axes is a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points of equal altitude.

A review of the existing empirical literature suggests the following as characteristics of each of the four phases of in- terest development. First, we define the phase. This is fol- lowed by description, information about the type of support that a person in this phase of interest typically needs, the way in which educational or instructional conditions might con- tribute to the development of interest, and the developmental progression related to this phase of development. Phase 1: Triggered Situational Interest Triggered situational interest refers to a psychological state of interest that results from short-term changes in affective and cognitive processing (Hidi & Baird, 1986, 1988; Mitch- ell, 1993): 1. Triggered situational interest can be sparked by envi- ronmental or text features such as incongruous, surprising in- formation; character identification or personal relevance; and intensity (Anderson, Shirey, Wilson, & Fielding, 1987; Gar- ner, Brown, Sanders, & Menke, 1992; Garner, Gillingham, & White, 1989; Hidi & Baird, 1986, 1988; Renninger & Hidi, 2002; Sadoski, 2001). 2. Triggered situational interest is typically, but not ex- clusively, externally supported (Bloom, 1985; Sloboda, 1990; Sosniak, 1990). 3. Instructional conditions or learning environments that include group work, puzzles, computers, and so on have been found to trigger situational interest (Cordova & Lepper, 1996; Hidi & Baird, 1988; Hidi, Weiss, Berndorff, & Nolan, 1998; Lepper & Cordova, 1992; Mitchell, 1993; Sloboda & Davidson, 1995). 4. Triggeredsituationalinterestmaybeaprecursortothe predisposition to reengage particular content over time as in more developed phases of interest (Renninger & Hidi, 2002; Renninger et al., 2004). Phase 2: Maintained Situational Interest Maintained situational interest refers to a psychological state of interest that is subsequent to a triggered state, involves fo- cused attention and persistence over an extended episode in time, and/or reoccurs and again persists: 1. Situational interest is held and sustained through meaningfulness of tasks and/or personal involvement (Harackiewicz et al., 2000; Mitchell, 1993). 2. A maintained situational interest is typically, but not exclusively, externally supported (Renninger & Hidi, 2002; Sansone & Morgan, 1992; Sansone et al., 1992; Schraw & Dennison, 1994; Wolters, 1998). 3. Instructional conditions or learning environments pro- vide meaningful and personally involving activities, such as project-based learning, cooperative group work, and one-on-one tutoring, can contribute to the maintenance of situational interest (Hidi et al., 1998; Hoffmann, 2002; Mitchell, 1993; Renninger et al., 2004; Schraw & Dennison, 1994; Sloboda & Davidson, 1995). 4. A maintained situational interest may or may not be a precursor to the development of a predisposition to reengage particular content over time as in more developed forms of interest (Harackiewicz et al., 2000; Hidi & Baird, 1988; Lipstein & Renninger, 2006; Mitchell, 1993). Phase 3: Emerging Individual Interest Emerging individual interest refers to a psychological state of interest as well as to the beginning phases of a relatively enduring predisposition to seek repeated reengagement with particular classes of content over time: 1. Emerging individual interest is characterized by posi- tive feelings, stored knowledge, and stored value (Bloom, 1985; Renninger, 1989, 1990, 2000; Renninger & Wozniak, 1985). Based on previous engagement, the student values the opportunity to reengage tasks related to his or her emerging individual interest and will opt to do these if given a choice (Bloom, 1985; Flowerday & Schraw, 2003; Katz, Kanat-Maymon, & Assor, 2003; Renninger & Hidi, 2002; Renninger & Shumar, 2002). The student begins to regularly generate his or her own “curiosity” questions about the con- tent of an emerging individual interest (Renninger, 1990; Renninger & Shumar, 2002). As an outcome of such curios- ity questions or self-set challenges, students may redefine and exceed task demands in their work with an emerging in- dividual interest (Lipstein & Renninger, 2006; Renninger, Boone, Luft, & Alejandre, in press; Renninger & Hidi, 2002; Renninger et al., 2004). The student is likely to be resource- ful when conditions do not immediately allow a question about content of emerging individual interest to be answered (Lipstein & Renninger, 2006; Renninger & Hidi, 2002; Renninger & Shumar, 2002). An emerging individual inter- est can enable a person to anticipate subsequent steps in pro- cessing work with content (Renninger & Hidi, 2002) and produce effort that feels effortless (Lipstein & Renninger, 2006; Renninger & Hidi, 2002; Renninger et al., 2004). 2. An emerging individual interest is typically but not ex- clusively self-generated (Cobb, 2004; Nolan, 2006; Renninger & Shumar, 2004; Sosniak, 1990). An emerging individual interest requires some external support, in the form of models or others such as peers, experts, and so on; such support can contribute to increased understanding (Heath & Roach, 1999; Krapp & Lewalter, 2001; Renninger, 2000; Renninger et al., in press) and be presented in the form of tasks or environments that challenge and provide opportu- nity (Nolan, 2006; Pressick-Kilborn & Walker, 2002; Renninger, 2000; Renninger & Shumar, 2002, 2004). A learner with emerging individual interest also may need en- couragement from others to persevere when confronted with difficulty (Bloom, 1985; Carey, Kleiner, Porch, Farris, & Burns, 2002; Renninger & Shumar, 2002; Sosniak, 1990). 3. Instructional conditions or the learning environment can enable the development of an emerging individual inter- est (Hannover, 1998; Hoffmann, 2002; Krapp & Lewalter, 2001; Lipstein & Renninger, 2006; Pressick-Kilborn & Walker, 2002; Renninger et al., 2004; Renninger & Shumar, 2002, 2004). 4. An emerging individual interest may or may not lead to well-developed individual interest (Bloom, 1985; Lipstein & Renninger, 2006; Sloboda, 1990). Phase 4: Well-Developed Individual Interest Well-developed individual interest refers to the psychological state of interest as well as to a relatively enduring predisposi- tion to reengage with particular classes of content over time: 1. A well-developed individual interest is characterized by positive feelings, and more stored knowledge and more stored value for particular content than for other activity in- cluding emerging individual interest (Renninger, 1989, 1990, 2000; Renninger et al., 2002; Renninger & Wozniak, 1985). Based on previous engagement, the student values the oppor- tunity to reengage tasks for which he or she has a well-devel- oped individual interest and will opt to pursue these if given a choice (Renninger, 1989, 1990; Renninger & Hidi, 2002; Renninger & Leckrone, 1991; Renninger et al., 2004). A well-developed individual interest may result in a student generating and seeking answers to curiosity questions (Lipstein & Renninger, 2006; Renninger & Hidi, 2002).5 The student is likely to be resourceful when conditions do not im- mediately allow a question concerning a well-developed in- dividual interest to be answered (Renninger & Hidi, 2002; Renninger & Shumar, 2002). A well-developed individual interest enables a person to anticipate subsequent steps in processing work with content (Renninger & Hidi, 2002). Well-developed individual interest produces effort that feels effortless (Lipstein & Renninger, 2006; Renninger & Hidi, 2002; Renninger et al., 2004). A well-developed individual interest enables a person to sustain long-term constructive and creative endeavors (Izard & Ackerman, 2000; Tomkins, 1962) and generates more types and deeper levels of strate- gies for work with tasks (Alexander & Murphy, 1998; Renninger, 1990; U. Schiefele & Krapp, 1996). A well-de- veloped individual interest leads a student to consider both the context and content of a task in the process of problem so- lution or passage comprehension (Renninger et al., 2002). Well-developed individual interest promotes self-regulation (Lipstein & Renninger, 2006; Renninger et al., 2004; Sansone & Smith, 2000). 2. Awell-developedindividualinterestistypicallybutnot exclusively self-generated (Bloom, 1985; Nolan, 2006; Pressick-Kilborn & Walker, 2002; Renninger et al., 2002; Renninger et al., 2004; Renninger & Shumar, 2004; Sloboda, 1990). Well-developed individual interest may also benefit from external support; support in the form of models or others such as peers, experts, and so on also can contribute to in- creased understanding (Csikszentmihalyi, Rathunde, & Whalen, 1993; Renninger, 2000). A learner with well-devel- oped individual interest will persevere to work, or address a question, even in the face of frustration (Fink, 1998; Prenzel, 1992; Renninger & Hidi, 2002; Renninger & Leckrone, 1991). 3. Instructional conditions or the learning environment can facilitate the development and deepening of well-devel- oped individual interest by providing opportunities that in- clude interaction and challenge that leads to knowledge building (Renninger & Hidi, 2002; Renninger & Shumar, 2002; Sloboda, 1990). Considered sequentially and from a developmental per- spective, the characteristics of each phase of interest may be considered mediators of subsequent development and the 5Curiosity questions refer to the type of verbal or nonverbal questioning that a learner generates in the process of organizing and accommodating new information (Renninger, 2000; see also Lindfors, 1987). INTEREST DEVELOPMENT 115  116 HIDI AND RENNINGER deepening of interest as well as outcomes of previous devel- opment. Most of the research on interest to date has been de- scriptive. Only a few studies have been conducted that have addressed the development of interest over time (e.g., Fink, 1998; Gisbert, 1998; Krapp & Fink, 1992; Krapp & Lewalter, 2001; Renninger & Leckrone, 1991), or provide evidence of causal relationships among phases of interest. Mitchell (1993), for example, demonstrated that although group work, puzzles, and computers would trigger adolescent students’ interest in math, only personal involvement and the meaningfulness of tasks maintain their interest over time. Harackiewicz et al. (2000) further demonstrated that factors that maintained college student interest were better predic- tors of continuing interest than were factors that only trig- gered their interest. More recently, Harackiewicz et al. (2002) replicated these findings and distinguished interest from students’ goals and performance. Further work that es- tablishes the predictive validity of the proposed four phases of interest development and examines the relations among them is the next needed step for interest research. Case Illustration The following case illustration provides an example of how the four phases of interest development appear to be linked: Julia is in her last term of college. While nervously waiting for a medical appointment, she picks up and flips through a magazine. Her attention is drawn to an article about a man who is an engineer and who re- cently gave up his partnership in a successful consult- ing practice to become a facilitator. A facilitator is a person who tries to help people or groups resolve con- flicts before they go to litigation. Julia likes the idea of working with people and wants to read more even though she has never heard of the occupation of facili- tator before now. Meanwhile, she is called to meet the doctor. She carefully marks the page she is reading and leaves the magazine on the table. Following her ap- pointment, she goes back to the table, finds the maga- zine, and sits down to finish reading the article. Julia’s case is an illustration of triggered and maintained situational interest. Her situational interest was triggered by encountering the presence of a new concept—facilitation. It was maintained through her desire to seek more information from the article and her ability to identify with the sense of possibility that facilitation could represent (Alexander, 2004; Hannover, 1998; Krapp, 2002b; Markus & Nurius, 1986). Julia wanted to learn more about facilitation. Although previ- ously she did not consider the possibility of becoming a facil- itator, Julia concluded that maybe she could do this type of job after reading a portion of the article, or that maybe she could recommend the job to others. Presumably, Julia recog- nized that she had qualities that are critical to the type of ne- gotiations in which facilitators engage and that she might en- joy this type of work. It should be noted that although in the example of Julia, the topic of facilitation is self-relevant, this aspect of a topic or content is not a necessary condition for in- terest development. Instead, Julia might have found an article about dinosaurs, become interested in the topic, and through continued engagement, developed an individual interest for dinosaurs. The article she found about facilitation, and the sense of possibility it suggested, maintained Julia’s situational interest. Once she returns to reading the article, Julia makes notes and decides to follow up on what she has read. She makes plans to go to the library, search the Internet, and talk to her advisor about her options. Even though she had little, if any, knowledge about facilita- tion prior to reading the article, she did have knowledge of related information such as different types of jobs that in- volve helping people, including helping others settle their differences. It is likely that this type of related information (Pressik-Kilbourn & Walker, 2002; Renninger & Shumar, 2002) combined with the concreteness of the content (Sadoski, 2001) initially triggered her interest and rendered the article accessible, despite Julia’s lack of background knowledge on the topic. It is also likely that it was this combination of related prior knowledge and text character- istics that enabled her to sustain her situational interest for the content of the article (Harackiewicz et al., 2000; Mitch- ell, 1993). Certainly, as Julia gained more knowledge and repeatedly sought out opportunities to reengage with ideas about facili- tation, her sense of possibility was confirmed. She continued to hold positive feelings, and her valuing of the facilitator role increased. Julia’s efforts to find more information about this type of job and her identification with its possibility sug- gest that through repeated engagement or magnification, as referred to by Silvia (2001), an individual interest for facilita- tion was emerging. Given repeated engagement over time, it is likely that Julia’s interest for facilitation could evolve into a well-de- veloped interest. A well-developed interest would be char- acterized by her continued effort to learn about facilitation, positive feelings about facilitation, and increased valuing for the concept relative to the other content with which she is involved (Renninger, 2000). In fact, it could be expected that she would persevere to think about and pursue a career in facilitation despite confusing or negative experiences (Ainley, 2002; Prenzel, 1992; Renninger, 2000; Renninger & Hidi, 2002). The case of John provides a contrast to that of Julia. A few days later, John, another student who is also waiting to see a doctor, picks up the same magazine. He flips through the pages, stopping at the same article, and reads intently until he hears his name called. In contrast to Julia, however, John does not finish the arti- cle. Once John is called in to meet with the doctor, his reading is interrupted, and his triggered situational in- terest for the article ceases. John, like Julia, picked up the magazine and initially felt some curiosity about what a facilitator might do. Because he exerted no ef- fort to learn more about facilitation or to finish reading the article after meeting with the doctor, in his situation the triggered situational interest is not maintained. As these examples demonstrate, once situational interest has been elicited, it can last for short or long periods of time. If an interest is maintained over time through repeated engage- ments and a person begins to identify with the content in question (Hannover, 1998; Jetton & Alexander, 2001), he or she can be described as having an emerging individual inter- est. Julia’s interest, for example, was triggered, and she be- came resourceful about finding information and seeking sup- port for learning more about facilitation. However, as illustrated by the cases of Julia and John, only some situational interests develop into individual inter- ests, and only some individual interests become well devel- oped. Moreover, it is important to acknowledge that although situational interest represents the initial phases of the devel- opment of individual interest, there are multiple possibilities for the person with an existing individual interest to experi- ence related situational interests (Bergin, 1999; Renninger & Hidi, 2002; Renninger & Shumar, 2002). The likelihood that Julia would develop a well-devel- oped interest for facilitation and that John would not, is sig- naled by Julia’s return to the article following the interview and John’s departure (Renninger, 1989, 1990; Renninger & Leckrone, 1991). At that point in time, Julia could have simply left the office, as John did. However, the interest that was triggered by the new information (the article about facilitation) was maintained and she wanted to finish read- ing the article (Dewey, 1913; Hidi & Baird, 1986; Mitchell, 1993). As Julia gathered more information about facilita- tion, her positive affect was sustained, and her knowledge continued to develop. It is possible that she reflected on fa- cilitation in terms of its utility as a source of a job (Wigfield, 1994; Wigfield & Eccles, 1992, 2002), which may have further supported her positive feelings and added to its value (Eccles et al., 1983). It is also possible that her situational interest for facilitation was maintained by her identification with the details of what facilitators do (Hannover, 1998), her feelings of self-efficacy (Bandura, 1977; Zimmerman, 2000a, 2000b), and her sense of possi- bility (Markus & Nurius, 1986). Julia’s search for addi- tional information and her self-regulation of her own activ- ity suggest that she had begun to develop a formative relation with facilitation (Boekaerts & Niemivirta, 2000; Sansone & Smith, 2000). This type of relation characterizes individual interest. INTEREST DEVELOPMENT 117 What Would Disprove the Four-Phase Model of Interest

Four Peaks (Yavapai: Wi:khoba[3]) is a prominent landmark on the eastern skyline of Phoenix. Part of the Mazatzal Mountains, it is located in the Tonto National Forest 40 miles (64 km) east-northeast of Phoenix, in the 61,074-acre (247.16 km2) Four Peaks Wilderness.[4] On rare occasions, Four Peaks offers much of the Phoenix metro area a view of snow-covered peaks, and is the highest point in Maricopa County. Four Peaks contains an amethyst mine that produces top-grade amethyst. The name Four Peaks is a reference to the four distinct peaks of a north–south ridge forming the massif's summit. The northernmost peak is named Brown's Peak and is the tallest of the four at 7,659 feet (2,334 m).[1] The remaining summits are unnamed, and from north to south are 7,644 feet (2,330 m),[5] 7,574 feet (2,309 m)[6] and 7,526 feet (2,294 m)[7] in altitude

Number 4 (四; accounting 肆; pinyin sì) is considered an unlucky number in Chinese because it is nearly homophonous to the word "death" (死 pinyin sǐ). Due to that, many numbered product lines skip the "4": e.g., Nokia cell phones (before the Lumia 640, there is no series containing a 4 in the name),[8] Palm[citation needed] PDAs, Canon PowerShot G's series (after G3 goes G5), etc. In East Asia, some buildings do not have a 4th floor. (Compare with the Western practice of some buildings not having a 13th floor because 13 is considered unlucky.) In Hong Kong, some high-rise residential buildings omit all floor numbers with "4", e.g., 4, 14, 24, 34 and all 40–49 floors, in addition to not having a 13th floor.[7] As a result, a building whose highest floor is number 50 may actually have only 35 physical floors. Singaporean public transport operator SBS Transit has omitted the number plates for some of its buses whose numbers end with '4' due to this, so if a bus is registered as SBS***3*, SBS***4* will be omitted and the next bus to be registered will be SBS***5*. Note that this only applies to certain buses and not others and that the final asterisk is a checksum letter and not a number. Another Singaporean public transport operator SMRT has omitted the '4' as the first digit of the serial number of the train cars as well as the SMRT Buses NightRider services. In the quadrant model the fourth square is death. The fourth square is transcendent and is death. I discussed in another book how knowledge and death are related. Knowledge is the fourth quadrant. The reason why th eChinese consider 13 unlucky is 13 is the first square of the fourth quadrant, which is the death square.

The Gang of Four (simplified Chinese: 四人帮; traditional Chinese: 四人幫; pinyin: Sìrén bāng) was a political faction composed of four Chinese Communist Party officials. They came to prominence during the Cultural Revolution (1966–76) and were later charged with a series of treasonous crimes. The gang's leading figure was Mao Zedong's last wife Jiang Qing. The other members were Zhang Chunqiao, Yao Wenyuan, and Wang Hongwen. The Gang of Four controlled the power organs of the Communist Party of China through the later stages of the Cultural Revolution, although it remains unclear which major decisions were made by Mao Zedong and carried out by the Gang, and which were the result of the Gang of Four's own planning. The Gang of Four, together with disgraced general Lin Biao, were labeled the two major "counter-revolutionary forces" of the Cultural Revolution and officially blamed by the Chinese government for the worst excesses of the societal chaos that ensued during the ten years of turmoil. Their downfall on October 6, 1976, a mere month after Mao's death, brought about major celebrations on the streets of Beijing and marked the end of a turbulent political era in China. I discussed that the reason why Asians are communist is because in the quadrant model communism is the first square political orientation and Asians are the first square race.

In the struggle between Hua Guofeng's and Deng Xiaoping's followers, a new term emerged,[citation needed] pointing to Hua's four closest collaborators, Wang Dongxing, Wu De, Ji Dengkui and Chen Xilian.[citation needed] In 1980, they were charged with "grave errors" in the struggle against the Gang of Four and demoted from the Political Bureau to mere Central Committee membership. "New Gang of Four" In the Xi Jinping era, some commentators and political observers have dubbed the loose political grouping of former security chief Zhou Yongkang, former Central Military Commission Vice-Chairman Xu Caihou, former Chongqing party chief Bo Xilai, and former General Office chief Ling Jihua as the "New Gang of Four".[6] All four were investigated for corruption-related offences between 2012 and 2014. Apart from sharing the name of the historical Gang of Four, the two "Gangs" had little in common, as whether the new "Gang" truly had a coherent set of shared political interests was not clear.

The Anti-Party Group (Russian: Антипартийная группа, tr. Antipartiynaya gruppa) was a group within the leadership of the Communist Party of the Soviet Union that unsuccessfully attempted to depose Nikita Khrushchev as First Secretary of the Party in June 1957. The group, named by that epithet by Khrushchev, was led by former Premiers Georgy Malenkov and Vyacheslav Molotov. The group rejected both Khrushchev's liberalisation of Soviet society and his denunciation of Joseph Stalin. During the stormy meeting of the Central Committee, Zhukov - a man of immense prestige given his role in the war and his reputation of fearlessness even in the face of Stalin's anger - delivered a bitter denunciation of the plotters, accusing them of having blood on their hands over Stalin's atrocities. He went further still saying that he had the military power to crush them even if they did win the vote and implied he would be able to have them all killed, but the triumphant Khrushchev rejected any such move. Malenkov, Molotov, Kaganovich and Shepilov - the only four names made public - were vilified in the press and deposed from their positions in party and government. They were given relatively unimportant positions: Molotov was sent as ambassador to Mongolia Malenkov became director of a hydroelectric plant in Kazakhstan Kaganovich became director of a small potassium factory in the Urals Shepilov became head of the Economics Institute of the local Academy of Sciences of Kyrgyzstan In 1961, in the wake of further de-Stalinisation, they were expelled from the Communist Party altogether and all lived mostly quiet lives from then on. Shepilov was allowed to rejoin the party by Khrushchev's successor Leonid Brezhnev in 1976 but remained on the sidelines.

Tetraphobia (from Greek τετράς - tetras, "four"[1] and φόβος - phobos, "fear"[2]) is the practice of avoiding instances of the number 4. It is a superstition most common in East Asian and Southeast Asian regions such as China, Taiwan, Singapore, Malaysia, Japan, Korea and Vietnam.

The Chinese word for four (四, pinyin: sì, jyutping: sei3), sounds quite similar to the word for death (死, pinyin: sǐ, jyutping: sei2), in many varieties of Chinese. Similarly, the Sino-Japanese, Sino-Korean, and Sino-Vietnamese words for four, shi (し, Japanese), and sa (사, Korean), sound similar or identical to death in each language (see Korean numerals, Japanese numerals, Vietnamese numerals). Special care may be taken to avoid occurrences or reminders of the number 4 during everyday life, especially during festive holidays, or when a family member is ill. So much so that just mentioning the number 4 around a sick relative is strongly avoided. For instance, you never want to give four of something and there is even a saying that "you don’t do things in fours…". Elevators in Asia and Asian neighborhoods will often be missing the 4th floor or any floor whose number contains the digit "4" (as 14, 24, etc.). Military aircraft and ships will also avoid the number 4 (such as the South Korean and Taiwanese navies) due to its extreme negative connotations with death. April 4 is also considered an exceptionally unlucky day (much like Friday the 13th in the West). Similarly, 14, 24, 42, etc. are also to be avoided due to the presence of the digit 4 in these numbers. In these countries, these floor numbers are often skipped in buildings, ranging from hotels to offices to apartments, as well as hospitals. Table number 4, 14, 24, 42, etc. are also often left out in wedding dinners or other social gatherings in these countries. In many residential complexes, building block 4, 14, 24 etc. are either omitted or replaced with block 3A, 13A, and 23A. Hospitals are of grave concern and the number 4 is regularly avoided altogether. Tetraphobia can dictate property prices. Neighborhoods have removed four from their street names and become more profitable as a result. In the same way, buildings with multiple fours can suffer price cuts of up to $30,000-$50,000. Four is also avoided in phone numbers, security numbers, business cards, addresses, ID numbers, and other numbers and are considered severe as they are personally attached to the person. Giving such numbers to Asian persons is considered extremely offensive and even grounds for law enforcement involvement or legal retaliation due to it being easily seen as a death threat and has been used as such by gangs, organised crime groups, and murderers. Tetraphobia far surpasses triskaidekaphobia (Western superstitions around the number 13). It even permeates the business world in these regions of Asia.[4]

Chinese is a tonal language with a comparatively small inventory of permitted syllables, resulting in an exceptionally large number of homophone words. Many of the numbers are homophones or near-homophones of other words and have therefore acquired superstitious meanings. The Chinese avoid phone numbers and addresses with fours, especially when they’re combined with another number that changes the meaning. Example: “94” could be interpreted as being dead for a long time. The Chinese government does not display tetraphobia by having military designations for People's Liberation Army with the number 4, for example, Dongfeng-4 ICBM, Type 094 Nuclear Submarine, Type 054A Frigate, etc. However some speculate that it does for aircraft (just as the United States generally skips the number 13 for their aircraft), seeing that it begins aircraft and engine destination with 5.[5] But the Taiwanese and the South Korean navies do not use the number 4 when assigning Pennant numbers to their ships. In Taiwan, the number 4 is banned in license plates and can only be used once in ID numbers (although even once, it is strongly avoided whatever possible). In Hong Kong, some apartments such as Vision City[6] and The Arch[7] skip all the floors from 40 to 49, which is the entire 40's. Immediately above the 39th floor is the 50th floor, leading many who are not aware of tetraphobia to believe that some floors are missing. Tetraphobia is not the main reason, but rather as an excuse to have apartments with 'higher' floors, thus increasing price, because higher floors in Hong Kong apartments are usually more expensive (see 39 Conduit Road). In Cantonese-speaking regions in China, 14 and 24 are considered more unlucky than the individual 4, since 14 sounds like "will certainly die" (實死), and 24 like "easy to die" (易死). While in Mandarin-speaking regions in China, 14 and 74 are considered more unlucky than the individual 4, since 14 sounds like "wants to die" (要死) and 74 like "will certainly die" or "will die in anger" (氣死). Where East Asian and Western cultures blend, such as in Hong Kong, it is possible in some buildings that both the thirteenth floor and the fourteenth floor are skipped, causing the twelfth floor to precede the fifteenth floor, along with all the other 4s. Thus a building whose top floor is numbered 100 would in fact have just seventy-nine floors. When Beijing lost its bid to stage the 2000 Olympic Games, it was speculated that the reason China did not pursue a bid for the following 2004 Games was due to the unpopularity of the number 4 in China. Instead, the city waited another four years, and would eventually host the 2008 Olympic Games, the number eight being a lucky number in Chinese culture.

Because of the significant population of Chinese and influence of Chinese culture in Southeast Asia, 4 is also considered to be unlucky. In buildings of Malaysia and Singapore, where Chinese are significant in population with 25% of Malaysians and 75% of Singaporeans being Chinese, the floor number 4 is occasionally skipped. Singaporean public transport operator SBS Transit has omitted the number plates for some of its buses whose numbers end with '4' due to this, so if a bus is registered as SBS***3*, SBS***4* will be omitted and the next bus to be registered will be SBS***5*[citation needed]. Note that this only applies to certain buses and not others and that the final asterisk is a checksum letter and not a number. Another Singaporean public transport operator SMRT has omitted the '4' as the first digit of the serial number of the train cars as well as the SMRT Buses NightRider services[citation needed]. Like Hong Kong, buildings of Singapore also skip the number 13 as Singapore is also a place where Eastern and Western cultures blend[citation needed]. The Grand Indonesia shopping centre in Jakarta replaced their 4th level with 3A. In Vietnam, the Sino-Vietnamese words for "four" (tứ) is used more in formal context than in everyday life and its spoken sound is clearly different from word for "death" (tử). The Chữ nôm word "bốn" equivalent to word "tứ" is often used, therefore the number 4 is rarely avoided. Even so, in the past Vietnamese people often named their children "tư" or "tứ", which means "the fourth child born in family".

In South Korea, tetraphobia is less extreme, but the floor number 4 is almost always skipped in hospitals and similar public buildings. In other buildings, the fourth floor is sometimes labelled "F" (Four) instead of "4" in elevators. Apartment numbers containing multiple occurrences of the number 4 (such as 404) are likely to be avoided to an extent that the value of the property is adversely affected. The national railroad, Korail, left out the locomotive number 4444 when numbering a locomotive class from 4401 upwards.

In Japan, many apartment houses and parking lots skip 4. Many hotels skip the 13th floor, similar to some western hotels. There is also much wordplay involved such as 24 can become nishi, aka double death (ニ死) 42 can become shini, aka “death” or “to death” (死に) 43 can become shisan which sounds like shizan, aka stillbirth (死産) 45 can be shigo, or “after death” (死後). 9 is also skipped, especially hospitals, due to the sound "ku" being associated with the word "to suffer" (「苦しむ」 "kurushimu"?). 49 is considered to be an especially unlucky number as it is evocative of the phrase "To suffer until death." (「死ぬまで苦しむ。」 "Shinu made kurushimu."?)

The software platform Symbian, used by Finnish telecommunications firm Nokia in their Series 60 platform, avoids releases beginning with 4, as it did when it was EPOC and owned by Psion (there was no Psion Series 4, and there was no 4th edition of S60). This was done "as a polite gesture to Asian customers".[10][11] Similarly, Nokia did not release any products under the 4xxx series, although some of Nokia's other products do contain the number 4, such as the Series 40 platform, and the Nokia 3410.

When area code 306 was nearing exhaustion in 2011, the Canadian Radio-television and Telecommunications Commission originally proposed that the new area code be 474.[12] However, representatives from SaskTel requested that the new area code be 639 instead, to avoid the negative connotations of 4 in Asian cultures. 639 was subsequently approved as the new area code.

Starting from Q4 2008, Samsung Telecommunications faced tetraphobia in its new 5-character model numbering scheme and no longer uses model codes containing the number 4, as previously it did (SGH-A400, C140, D410, D840, E740, F480, X450, X640, SGH-T499Y..

Triskaidekaphobia (pronunciation: /ˌtrɪskaɪˌdɛkəˈfoʊbiə, ˌtrɪskə-/, tris-kye-dek-ə-foh-bee-ə or tris-kə-dek-ə-foh-bee-ə; from Greek tris meaning "three", kai meaning "and", deka meaning "10" and phobos meaning "fear" or "morbid fear") is fear of the number 13 and avoidance to use it; it is a superstition and related to the specific fear of the 13th person at the Last Supper being Judas, who betrayed Jesus Christ and ultimately hanged himself. It is also a reason for the fear of Friday the 13th, called paraskevidekatriaphobia (from Παρασκευή Paraskevi, Greek for Friday) or friggatriskaidekaphobia (after Frigg, the Norse goddess after whom Friday is named in English). The term was first used by Isador Coriat in Abnormal Psychology 13 is the first square of the fourth quadrant. The fourth quadrant is the death quadrant/transcendent quadrant.

The fragrance wheel is shaped as a quadrant

Perfume is a billion dollar industry and has been very important throughout human history. Perfume oils usually contain tens to hundreds of ingredients and these are typically organized in a perfume for the specific role they will play. These ingredients can be roughly grouped into four groups: Primary scents (Heart): Can consist of one or a few main ingredients for a certain concept, such as "rose". Alternatively, multiple ingredients can be used together to create an "abstract" primary scent that does not bear a resemblance to a natural ingredient. For instance, jasmine and rose scents are commonly blends for abstract floral fragrances. Cola flavourant is a good example of an abstract primary scent. Modifiers: These ingredients alter the primary scent to give the perfume a certain desired character: for instance, fruit esters may be included in a floral primary to create a fruity floral; calone and citrus scents can be added to create a "fresher" floral. The cherry scent in cherry cola can be considered a modifier. Blenders: A large group of ingredients that smooth out the transitions of a perfume between different "layers" or bases. These themselves can be used as a major component of the primary scent. Common blending ingredients include linalool and hydroxycitronellal. Fixatives: Used to support the primary scent by bolstering it. Many resins, wood scents, and amber bases are used as fixatives. The top, middle, and base notes of a fragrance may have separate primary scents and supporting ingredients. The perfume's fragrance oils are then blended with ethyl alcohol and water, aged in tanks for several weeks and filtered through processing equipment to, respectively, allow the perfume ingredients in the mixture to stabilize and to remove any sediment and particles before the solution can be filled into the perfume bottles

Modern chypre perfumes have various connotations such as floral, fruity, green, woody-aromatic, leathery, and animalic notes, but can easily be recognized by their "warm" and "mossy-woody" base which contrasts the fresh citrus top, and a certain bitterness in the dry-down from the oak moss and patchouli. The accord consists of: Citrus: singular or blends of Bergamot, Orange, Lemon or Neroli Oak moss: mossy and woody Patchouli: camphoraceous and woody Musk: sweet, powdery, and animalic. Usually synthetic in modern times

The Fragrance wheel is a relatively new classification method that is widely used in retail and in the fragrance industry. The method was created in 1983 by Michael Edwards, a consultant in the perfume industry, who designed his own scheme of fragrance classification. The new scheme was created in order to simplify fragrance classification and naming scheme, as well as to show the relationships between each of the individual classes.[24] The five standard families consist of Floral, Oriental, Woody, Aromatic Fougère, and Fresh, with the first four families borrowing from the classic terminology and the last consisting of newer bright and clean smelling citrus and oceanic fragrances that have arrived in the past generation due to improvements in fragrance technology. Each of the families are in turn divided into subgroups and arranged around a wheel. In this classification scheme, Chanel No.5, which is traditionally classified as an aldehydic floral, would be located under the Soft Floral sub-group, and amber scents would be placed within the Oriental group. As a class, chypre perfumes are more difficult to place since they would be located under parts of the Oriental and Woody families. For instance, Guerlain's Mitsouko is placed under Mossy Woods, but Hermès Rouge, a chypre with more floral character, would be placed under Floral Oriental.

The Fragrance wheel is a relatively new classification method that is widely used in retail and in the fragrance industry. The method was created in 1983 by Michael Edwards, a consultant in the perfume industry, who designed his own scheme of fragrance classification after being inspired by a fragrance seminar by Firmenich. The new scheme was created in order to simplify fragrance classification and naming, as well as to show the relationships between each individual classes. The five standard families consist of Floral, Oriental, Woody, Fougère, and Fresh, with the former four families being more "classic" while the latter consists of newer, bright and clean smelling citrus and oceanic fragrances that have arrived due to improvements in fragrance technology. With the exception of the Fougère family, each of the families are in turn divided into three sub-groups and arranged around a wheel: 1. Floral Floral Soft Floral Floral Oriental 2. Oriental Soft Oriental Oriental Woody Oriental 3. Woody Wood Mossy Woods Dry Woods 4. Fresh Citrus Green Water 5. Fougère The Fougère family is placed at the center of this wheel since they are large family of scents that usually contain fragrance elements from each of the other four families. In this classification scheme, Chanel No.5, which is traditionally classified as a "Floral Aldehyde" would be located under Soft Floral sub-group, and "Amber" scents would be placed within the Oriental group. As a class, Chypres is more difficult to place since they would located under parts of the Oriental and Woody families. For instance, Guerlain Mitsouko, which is classically identified as a chypre will be placed under Mossy Woods, but Hermès Rouge, a chypre with more floral character, would be placed under Floral Oriental. According to Osmoz, there are eight major families: Chypre, Citrus, Floral and Oriental (feminine), and Aromatic, Citrus, Oriental and Woody (masculine). Each one of those olfactive families is then split into several subfamilies.

cross multiplication In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. Given an equation like: {\frac {a}{b}}={\frac {c}{d}} (where b and d are not zero), one can cross-multiply to get: ad=bc\qquad \mathrm {or} \qquad a={\frac {bc}{d}}. In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles.

Matrix multiplication is the foundation of advanced math and physics. It involves multiplying quadrants of numbers.

In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be multiplied according to elementary arithmetic. On the other hand, matrices are arrays of numbers, so there is no unique way to define "the" multiplication of matrices. As such, in general the term "matrix multiplication" refers to a number of different ways to multiply matrices. The key features of any matrix multiplication include: the number of rows and columns the original matrices have (called the "size", "order" or "dimension"), and specifying how the entries of the matrices generate the new matrix.

Like vectors, matrices of any size can be multiplied by scalars, which amounts to multiplying every entry of the matrix by the same number. Similar to the entrywise definition of adding or subtracting matrices, multiplication of two matrices of the same size can be defined by multiplying the corresponding entries, and this is known as the Hadamard product. Another definition is the Kronecker product of two matrices, to obtain a block matrix.

One can form many other definitions. However, the most useful definition can be motivated by linear equations and linear transformations on vectors, which have numerous applications in applied mathematics, physics, and engineering. This definition is often called the matrix product.[1][2] In words, if A is an n × m matrix and B is an m × p matrix, their matrix product AB is an n × p matrix, in which the m entries across the rows of A are multiplied with the m entries down the columns of B (the precise definition is below).

This definition is not commutative, although it still retains the associative property and is distributive over entrywise addition of matrices. The identity element of the matrix product is the identity matrix (analogous to multiplying numbers by 1), and a square matrix may have an inverse matrix (analogous to the multiplicative inverse of a number). A consequence of the matrix product is determinant multiplicativity. The matrix product is an important operation in linear transformations, matrix groups, and the theory of group representations and irreps.

Computing matrix products is both a central operation in many numerical algorithms and potentially time consuming, making it one of the most well-studied problems in numerical computing. Various algorithms have been devised for computing C = AB, especially for large matrices.

This article will use the following notational conventions: matrices are represented by capital letters in bold, e.g. A, vectors in lowercase bold, e.g. a, and entries of vectors and matrices are italic (since they are scalars), e.g. A and a. Index notation is often the clearest way to express definitions, and is used as standard in the literature. The i, j entry of matrix A is indicated by (A)ij or Aij, whereas a numerical label (not matrix entries) on a collection of matrices is subscripted only, e.g. A1, A2, etc

The cross product In mathematics and vector calculus, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space (R3) and is denoted by the symbol ×. Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with dot product (projection product). If two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero. More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The cross product is anticommutative (i.e. a × b = −b × a) and is distributive over addition (i.e. a × (b + c) = a × b + a × c). The space R3 together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space, but unlike the dot product, it also depends on a choice of orientation or "handedness". The product can be generalized in various ways; it can be made independent of orientation by changing the result to pseudovector, or in arbitrary dimensions the exterior product of vectors can be used with a bivector or two-form result. Also, using the orientation and metric structure just as for the traditional 3-dimensional cross product, one can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.[1] If one adds the further requirement that the product be uniquely defined, then only the 3-dimensional cross product qualifies. (See § Generalizations, below, for other dimensions.)

In 1911, at Jacques' home in Puteaux, the brothers hosted a regular discussion group with Cubist artists including Picabia, Robert Delaunay, Fernand Léger, Roger de La Fresnaye, Albert Gleizes, Jean Metzinger, Juan Gris, and Alexander Archipenko. Poets and writers also participated. The group came to be known as the Puteaux Group, or the Section d'Or. Uninterested in the Cubists' seriousness or in their focus on visual matters, Duchamp did not join in discussions of Cubist theory, and gained a reputation of being shy. However, that same year he painted in a Cubist style, and added an impression of motion by using repetitive imagery. During this period Duchamp's fascination with transition, change, movement and distance became manifest, and like many artists of the time, he was intrigued with the concept of depicting the fourth dimension in art.[13] His painting Sad Young Man on a Train embodies this concern: First, there's the idea of the movement of the train, and then that of the sad young man who is in a corridor and who is moving about; thus there are two parallel movements corresponding to each other. Then, there is the distortion of the young man—I had called this elementary parallelism. It was a formal decomposition; that is, linear elements following each other like parallels and distorting the object. The object is completely stretched out, as if elastic. The lines follow each other in parallels, while changing subtly to form the movement, or the form of the young man in question. I also used this procedure in the Nude Descending a Staircase.

The most prominent example of Duchamp's association with Dada was his submission of Fountain, a urinal, to the Society of Independent Artists exhibit in 1917. Artworks in the Independent Artists shows were not selected by jury, and all pieces submitted were displayed. However, the show committee insisted that Fountain was not art, and rejected it from the show. This caused an uproar amongst the Dadaists, and led Duchamp to resign from the board of the Independent Artists. Duchamps urinal called has four holes in a line in it where the urine drains. A lot of people question if Duchamp was poking fun at art and asking the question, what is art, and irreverently making a urinal artwork subverting art. The urinal artwork is one of the most famous artworks in history. It is a very taboo item, usually held in private, but here it is on display as a work of art. But it does have the four holes. Art is the third quadrant field of inquiry and thus is supposed to make people think and question. The third quadrant is destructive. Duchamp white urinal is kind of a play on the continuity of White throughout White history, where the urinal has elegant contours and White features like ancient White sculptures from Greece (in reality they were not White but were painted but became White after the paint eroded)

Originally the rubiks cube was a two by two cube (the quadrant) Rubik's Cube is a 3-D combination puzzle invented in 1974[1][2] by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube,[3] the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980[4] via businessman Tibor Laczi and Seven Towns founder Tom Kremer,[5] and won the German Game of the Year special award for Best Puzzle that year. As of January 2009, 350 million cubes had been sold worldwide[6][7] making it the world's top-selling puzzle game.[8][9] It is widely considered to be the world's best-selling toy.[10] In a classic Rubik's Cube, each of the six faces is covered by nine stickers, each of one of six solid colours: white, red, blue, orange, green, and yellow. In currently sold models, white is opposite yellow, blue is opposite green, and orange is opposite red, and the red, white and blue are arranged in that order in a clockwise arrangement.[11] On early cubes, the position of the colours varied from cube to cube.[12] An internal pivot mechanism enables each face to turn independently, thus mixing up the colours. For the puzzle to be solved, each face must be returned to have only one colour. Similar puzzles have now been produced with various numbers of sides, dimensions, and stickers, not all of them by Rubik. In March 1970, Larry Nichols invented a 2×2×2 "Puzzle with Pieces Rotatable in Groups" and filed a Canadian patent application for it. Nichols's cube was held together with magnets. Nichols was granted U.S. Patent 3,655,201 on April 11, 1972, two years before Rubik invented his Cube. On April 9, 1970, Frank Fox applied to patent his "Spherical 3×3×3". He received his UK patent (1344259) on January 16, 1974.[13]

The Rosicrucian Cosmo-Conception Chapter X The Earth Period The Globes of the Earth Period are located in the four densest states of matter--the Region of Concrete Thought, the Desire World, the Etheric, and the Chemical Regions (See Diagram 8). The densest Globe (Globe D) is our present Earth. When we speak of "the densest Worlds" or "the densest states of matter," the term must be taken in a relative sense. Otherwise it would imply a limitation in the absolute, and that is absurd. Dense and attenuated, up and down, east and west, are applicable only relatively to our own status and position. As there are higher, finer Worlds than those touched by our life wave, so there are also denser states of matter which are the field of evolution for other classes of beings. Nor must it be thought that these denser worlds are elsewhere in space; they are interpenetrated by our worlds in a manner similar to that in which the higher Worlds interpenetrate this Earth. The fancied solidity of the Earth and the forms we see are no bar to the passage of a denser body any more than our solid dense walls bar the passage of a human being clothed in his desire body. Neither is solidity synonymous with density, as may be illustrated by aluminum, a solid which is less dense than the fluidic mercury; nevertheless the latter, in spite of its density, will evaporate or exude through many solids. This being the fourth Period, we have at present four elements. In the Saturn Period there was but one element, Fire--i.e., there was warmth, or heat, which is incipient fire. In the second, or Sun Period, there were to elements, Fire and Air. In the third, or Moon Period, there were three elements, Water being added; and in the fourth, or Earth Period, was added the fourth element, Earth. Thus it will be seen that a new element was added for each Period. In the Jupiter Period an element of a spiritual nature will be added, which will unite with the speech so that words will invariable carry with them understanding--not misunderstanding, as is frequently the case now. For instance, when one says "house," he may mean a cottage, while the hearer may get the idea of a tenement flat building. To this environment of the four elements, as specified above, the different classes mentioned in diagram 10 were brought over by the Hierarchies in charge of them. We remember that in the Moon Period these classes formed three kingdoms--animal, animal-plant and plant-mineral. Here on Earth, however, the conditions are such that there can be no large half-way classes. There must be four distinctly different kingdoms. In this crystallized phase of existence the lines between them must be more sharply drawn than was the case in former Periods, where one kingdom gradually merged into the next. Therefore some of the classes mentioned in diagram 10 advanced one-half step, while others went back a half a step. Some of the mineral-plants advanced completely into the plant kingdom and became the verdure of the fields. Others went down and became the purely mineral soil in which the plants grew. Of the plant-animals some advanced into the animal kingdom, ahead of time, and those species have yet the colorless plant-blood and some, like star-fishes, have even the five points like the petals of flowers. All of class 2 whose desire bodies could be divided into two parts (as was the case with all of class 1) were fitted to become human vehicles and were therefore advanced into the human group. We must carefully remember that in the above paragraphs we are dealing with Form, not with the Life which dwells in the Form. The instrument is graded to suit the life that is to dwell in it. Those of class 2, in whose vehicles the above mentioned division could be made were raised to the human kingdom, but were given the indwelling spirit at a point in time later than class 1. Hence, they are not now so far evolved as class 1, and are therefore the lower races of mankind. Those whose desire bodies were incapable of division were put into the same division as classes 3a and 3b. They are our present anthropoids. They may yet overtake our evolution if they reach a sufficient degree of advancement before the critical point already mentioned, which will come in the middle of the fifth Revolution. If they do not overtake us by that time, they will have lost touch with our evolution. It was said that man had built his threefold body by the help of others higher than he, but in the previous Period there was no coordinating power; the threefold spirit, the Ego, was separate and apart from its vehicles. Now the time had come to unit the spirit and the body. Where the desire body separated, the higher part become somewhat master over the lower part and over the dense and vital bodies. It formed a sort of animal-soul with which the spirit could unite by means of the link of mind. Where there was no division of the desire body, the vehicle was given over to desires and passions without any check, and could therefore not be used as a vehicle within which the spirit could dwell. So it was put under the control of a group-spirit which ruled if from without. It became an animal body, and that kind has now degenerated into the body of the anthropoid. Where there was a division of the desire body, the dense body gradually assumed a vertical position, thus taking the spine out of the horizontal currents of the Desire World in which the group-spirit acts upon the animal through the horizontal spine. The Ego could then enter, work in and express itself through the vertical spine and build the vertical larynx and brain for its adequate expression in the dense body. A horizontal larynx is also under the domination of the group-spirit. While it is true that some animals, as the starling, raven, parrot, etc., previously mentioned, are able, because of the possession of a vertical larynx, to utter words, they cannot use them understandingly. The use of words to express thought is the highest human privilege and can be exercised only by a reasoning, thinking entity like man. If the student will keep this in mind, it will be easier to follow the different steps which lead up to this result. The Saturn Revolution of the Earth Period This is the Revolution during which, in each Period. the dense body is reconstructed. This time it was given the ability to form a brain and become a vehicle for the germ of mind which was to be added later. This addition constituted the final reconstruction of the dense body, rendering it capable of attaining the highest degree of efficiency possible to such a vehicle. Unspeakable Wisdom has been employed in its construction. It is a marvel. It can never be sufficiently impressed upon the mind of the student what immeasurable facilities for the gaining of knowledge are contained in this instrument, and what a great boon it is to man; how much he should prize it and how thankful he should be to have it. Some examples of the perfection of construction and intelligent adaptability displayed in this instrument have previously been given, but in order to further impress this great truth upon the mind of the student, it might not be out of place to illustrate more fully this Wisdom, also the work of the Ego in the blood. It is generally know, in a vague kind of way, that the gastric juice acts upon the food to promote assimilation; but only a very few people, outside of the medical profession, are aware that there are many different gastric juices, each appropriate to the treatment of a certain kind of food. The researches of Pavlov, however, have established the fact beyond doubt, that there is one kind of juice for the digestion of meat, another for milk, another for acid fruit, etc. That fact, by the way, is the reason why all foods do not mix well. Milk, for instance, requires a gastric juice that is widely different from almost any other kind except that required for the digestion of starchy foods, and is not readily digested with any food other than cereals. This alone would show marvelous wisdom; that the Ego working subconsciously is able to select the different juices which are appropriate to the different kinds of food taken into the stomach, making each of just the right strength and quantity to digest the food. What makes the matter still more wonderful, however, is the fact that the gastric juice is poured into the stomach in advance of the food. We do not consciously direct the process of mixing this fluid. The great majority of people know nothing of metabolism or any other phrase of chemistry. So it is not enough to say that, as we taste what is coming, we direct the process by means of signals through the nervous system. When this fact of the selection of juices was first proven, scientists were sorely puzzled trying to learn how the right kind of juice was selected and caused to enter the stomach before the food. They thought the signal was given along the nervous system. But it was demonstrated beyond doubt that the proper juice was poured into to the stomach even though the nervous system was blocked. At last Starling and Bayliss, in a series of experiments of brilliant ingenuity, proved that infinitesimal parts of the food are taken up by the blood as soon as the food enters the mouth, go in advance to the digestive glands and cause a flow of the proper juice. This again, is only the physical side of the phenomena. To understand the whole wonderful connection, we must turn to occult science. That alone explains why the signal is carried by the blood. The blood is one of the highest expressions of the vital body. The Ego guides and controls its dense instrument by means of the blood, therefore the blood is also the means used to act on the nervous system. During some of the time that digestion is going on, it acts partially through the nervous system, but (especially at the commencement of the digestive process) it acts directly upon the stomach. When, during scientific experiments, the nerves were blocked, the direct way through the blood was still open and the Ego derived the necessary information in that way. It will also be seen that the blood is driven to wherever the Ego unfolds the greatest activity at any time. If a situation requires sudden though and action, the blood is promptly driven to the head. If a heavy meal is to be digested the greater portion of the blood leaves the head, centering around the digestive organs. The Ego concentrates its efforts on ridding the body of the useless food. Therefore a man cannot think well after a heavy meal. He is sleepy because so much blood has left the brain that the residue is insufficient to carry on the functions necessary to full waking consciousness, besides, nearly all the vital fluid or solar energy specialized by the spleen is absorbed by the blood rushing through that organ after a meal in greater volume than between meals. Thus the rest of the system is also deprived of the vital fluid in a large measure during digestion. It is the Ego that drives the blood into the brain. Whenever the body goes to sleep, the table will invariably tip towards the feet, raising the head. During coition the blood is centered in the sex organs, etc. All these examples tend to prove that during the waking hours, the Ego works in and controls the dense body by means of the blood. The larger portion of the total amount goes to that part of the body where at any given time, the Ego unfolds any particular activity. The reconstruction of the dense body in the Saturn Revolution of the Earth Period was for the purpose of rendering it capable of inter-penetration by the mind. It gave the first impulse to the building of the frontal part of the brain; also the incipient division in the nervous system which has since become apparent in its subdivisions--the voluntary and the sympathetic. The latter was the only one provided for in the Moon Period. The voluntary nervous system (which has transformed the dense body from a mere automaton acting under stimuli from without, to an extraordinary adaptable instrument capable of being guided and controlled by an Ego from within) was not added until the present Earth Period. The principal art of the reconstructive work was done by the Lords of Form. They are the Creative Hierarchy which is most active in the Earth Period, as were the Lords of Flame in the Saturn Period, the Lords of Wisdom in the Sun Period, and the Lords of Individuality in the Moon Period. The Earth Period is pre-eminently the Period of Form, for there the form or matter side of evolution reaches its greatest and most pronounced state. Here spirit is more helpless and suppressed and Form is the most dominant factor--hence the prominence of the Lords of Form. The Sun Revolution of the Earth Period During this Revolution the vital body was reconstructed to accommodate the germinal mind. The vital body was fashioned more in the likeness of the dense body, so that it could become fitted for use as the densest vehicle during the Jupiter Period, when the dense body will have become spiritualized. The Angels, the humanity of the Moon Period, were aided by the Lords of Form in reconstruction. The organization of the vital body is now next in efficiency to the dense body. Some writers on this subject call the former a link, and contend that it is simply a mold of the dense body, and not a separate vehicle. While not desiring to criticize, and admitting that this contention is justified by the fact that man, at his present stage of evolution, cannot ordinarily use the vital body as a separate vehicle--because it always remains with the dense body and to extract it in toto would cause death of the dense body--yet there was a time when it was not so firmly incorporated with the latter, as we shall presently see. During those epochs of our Earth's history which have already been mentioned as the Lemurian and the Atlantean, man was involuntarily clairvoyant, and it was precisely this looseness of connection between the dense and the vital bodies that made him so. (The Initiators of that time helped the candidate to loosen the connection still further, as in the voluntary clairvoyant.) Since then the vital body has become much more firmly interwoven with the dense body in the majority of people, but in all sensitives it is loose. It is that looseness which constitutes the difference between the psychic and the ordinary person who is unconscious of all but the vibrations contacted by means of the five senses. All human beings have to pass through this period of close connection of the vehicles and experience the consequent limitation of consciousness. There are, therefore, two classes of sensitives, those who have not become firmly enmeshed in matter, such as the majority of the Hindus, the Indians, etc., who possess a certain low grade of clairvoyance, or are sensitive to the sounds of nature, and those who are in the vanguard of evolution. The latter are emerging from the acme of materiality, and are again divisible into two kinds, one of which develops in a passive, weak-willed manner. By the help of others they re-awaken the solar plexus or other organs in connection with the involuntary nervous system. These are therefore involuntary clairvoyants, mediums who have no control of their faculty. They have retrograded. The other kind is made up of those who by their own wills unfold the vibratory powers of the organs now connected with the voluntary nervous system and thus become trained occultists, controlling their own bodies and exercising the clairvoyant faculty as they will to do. They are called voluntary or trained clairvoyants. In the Jupiter Period man will function in his vital body as he now does in his dense body; and as no development in nature is sudden, the process of separating the two bodies has already commenced. The vital body will then attain a much higher degree of efficiency than the dense body of today. As it is a much more pliable vehicle, the spirit will then be able to use it in a manner impossible of realization in the case of the present dense vehicle. The Moon Revolution of the Earth Period Here the Moon Period was recapitulated, and much the same conditions prevailed (on an advanced scale) as obtained on Globe D of that Period. There was the same kind of fire-fog atmosphere; the same fiery core; the same division of the Globe into two parts, in order to allow the more highly evolved beings a chance to progress at the proper rate and pace, which it would be impossible for beings such as our humanity to equal. In that Revolution the Archangels (humanity of the Sun Period) and the Lords of Form took charge of the reconstruction of the desire body, but they were not alone in that work. When the separation of the Globe into two parts occurred, there was a similar division in the desire bodies of some of the evolving beings. We have already noted that where this division took place, the form was ready to become the vehicle of an indwelling spirit, and in order to further this purpose the Lords of Mind (humanity of the Saturn Period) took possession of the higher part of the desire body and implanted in it the separate selfhood, without which the present man with all his glorious possibilities, could never have existed. Thus in the latter part of the Moon Revolution the first germ of separate personality was implanted in the higher part of the desire body by the Lords of Mind. The Archangels were active in the lower part of the desire body, giving it the purely animal desires. They also worked in the desire bodies where there was no division. Some of these were to become the vehicles of the animal group-spirits, which work on them from without, but do not enter wholly into the animal forms, as the individual spirit does into the human body. The desire body was reconstructed to render it capable of being interpenetrated by the germinal mind which, during the Earth Period, will be implanted in all those desire bodies in which it was possible to make the before-mentioned division. As has been previously explained, the desire body is an unorganized ovoid, holding the dense body as a dark spot within its center, as the white of an egg surrounds the yolk. There are a number of sense centers in the ovoid, which have appeared since the beginning of the Earth Period. In the average human being these centers appear merely as eddies in a current and are not now awake, hence his desire body is of no use to him as a separate vehicle of consciousness; but when the sense centers are awakened they look like whirling vortices. Rest Periods Between Revolutions Hitherto we have noted only the Cosmic Nights between Periods. We saw that there was an interval of rest and assimilation between the Saturn and the Sun Periods; another Cosmic Night between the Sun and the Moon Periods, etc. But in addition to these, there are also rests between the Revolutions. We might liken the Periods to the different incarnations of man; the Cosmic Nights between them to the intervals between deaths and new births; and the rest between Revolutions would then be analogous to the rest of sleep between two days. When a Cosmic Night sets in, all manifested things are resolved into a homogenous mass--the Cosmos again becomes Chaos. This periodical return of matter to primordial substance is what makes it possible for the spirit to evolve. Were the crystallizing process of active manifestation to continue indefinitely it would offer an insurmountable barrier to the progress of Spirit. Every time matter has crystallized to such a degree that it becomes too hard for the spirit to work in, the latter withdraws to recuperate its exhausted energy, on the same principle that a power-drill which has stopped when boring in hard metals, is withdrawn to regain its momentum. It is then able to bore its way further into the metal. Freed from the crystallizing energy of the evolving spirits, the chemical forces in matter turn Cosmos to Chaos by restoring matter to its primordial state, that a new start may be made by the regenerated virgin spirits at the dawn of a new Day of Manifestation. The experience gained in former Periods and Revolutions enables the Spirit to build up to the point last reached, with comparative celerity, also to facilitate further progress by making such alterations as its cumulative experience dictates. Thus at the end of the Moon Revolution of the Earth Period, all the Globes and all life returned to Chaos, re-emerging therefrom at the beginning of the fourth Revolution. The Fourth Revolution of the Earth Period In the exceeding complexity of the scheme of evolution, there are always spirals within spirals, ad infinitum. So it will not be surprising to learn that in every Revolution the work of recapitulation and rest is applied to the different Globes. When the life wave reappeared on Globe A in this Revolution, it went though the development of the Saturn Period; then after a rest which, however did not involve the complete destruction of the Globe; but only an alteration, it appeared on Globe B, where the work of the Sun Period was recapitulated. Then after a rest, the life wave passed on to Globe C, and the work of the Moon Period was repeated. Finally, the life wave arrived on Globe D, which is our Earth, and not until then did the proper work of the Earth Period begin. Even then, the spiral within the spiral precluded its beginning immediately on the arrival of the life wave from Globe C, for the bestowal of the germ of mind did not actually take place until the fourth Epoch, the first three Epochs being still further recapitulations of the Saturn, Sun and Moon Periods, but always on a higher scale. Notice how the fourth period is different from the previous three in this Rosicrucian model.

According to Rosicrucianism Etheric Region: related to the etheric body; home of the Angels (seen as being one step beyond the human stage, as humans are a degree in advance of the animal evolution), astrologically associated to Aquarius; Akashic records in the reflecting ether (pictures at least several hundred years back or much more in some cases, almost as the pictures on a screen, scene shifts backward). The etheric region is subdivided in four regions according to the grades of density of the aether permeating our physical planet Earth; Reflecting Ether, Light Ether, Life Ether and Chemical Ether. Chemical Region; the physical Earth as perceived through the five senses enhanced by the current technological equipment, subdivided in three regions according to the four main states of matter: solid, liquid, gaseous, and plasma. It is the current home of the self-conscious humanity, astrologically associated to Pisces. The Chemical region of the physical world is home to four life waves, or kingdoms, at a different stage in the evolutionary path: mineral life is the first and lowest level of spiritual evolution on Earth; then comes plants, with actual life, then animals (cold-blooded animals, then warm-blooded), and finally the human being. The beings belonging to each life wave either evolve through the work of the individual Spirit (human being) or are yet evolving under a group spirit and have acquired more or less subtle bodies according to the development stage of each life wave.

The Rosy Cross (also called Rose Cross and Rose Croix) is a symbol largely associated with the semi-mythical Christian Rosenkreuz, Qabbalist and alchemist and founder of the Rosicrucian Order.[1][2] The Rose Cross is said to be a cross with a white rose at its centre[3] and symbolizes the teachings of a tradition formed within the Christian tenets

Els Quarte Gats is the name of a café in Barcelona, Spain that famously became a popular meeting place for famous artists throughout the modernist period in Cataluña. The café opened on June 12, 1897 in the famous Casa Martí, and served as a hostel, bar and cabaret until it eventually became a central meeting point for Barcelona’s most prominent modernist figures, such as Pablo Picasso and Ramon Casas I Carbó. The bar closed due to financial difficulties in June 1903, but was reopened and eventually restored to its original condition in 1989. List of Famous Patrons[edit] Ramon Casas (Artist) Santiago Rusiñol (Artist) Rubén Dario (Poet) Pablo Picasso (Artist) Isaac Albéniz (Pianist and composer) Enric Granados (Pianist and composer) Lluís Millet (Musician and composer) Antoní Gaudí (Architect) Ricard Opisso (Cartoonist, illustrator and painter) Miquel Utrillo (Artist) Julio González (Sculptor)[8]

The Four Little Girls (Les Quatre Petites Filles) is a play written in French by the painter Pablo Picasso. It is the second of two full-length plays written by Picasso, the first being Desire Caught by the Tail. Written between November 24, 1947, and August 13, 1948,[1] it was published in 1949. In 1952 Picasso wrote a second version of the play using the same title.[2] Both versions use a stream of consciousness narrative style, and many critics believe that Picasso never meant for the play to be staged, only read.

Blues lyrics of early traditional blues verses consisted of a single line repeated four times. It was only in the first decades of the 20th century that the most common current structure became standard: the so-called AAB pattern, consisting of a line sung over the four first bars, its repetition over the next four, and then a longer concluding line over the last bars. Early blues frequently took the form of a loose narrative, often relating troubles experienced within African American society. There are theories that the four-beats-per-measure structure of the blues might have its origins in the Native American tradition of pow wow drumming.

The I–V–vi–IV progression is a common chord progression popular across several genres of music. It involves the I, V, vi, and IV chords; for example, in the key of C major, this would be: C–G–Am–F.[1] The V is often replaced by iii ("Price Tag"), III ("If We Ever Meet Again" chorus), ii ("Halo"), I ("Doesn't Mean Anything"), II ("Try Too Hard" by P!nk), or IV ("I Gotta Feeling"). A 2009 song by the comedy group The Axis of Awesome, called "Four Chords", parodied the ubiquity of the progression in popular music. It was written in E major (thus using the chords E major, B major, C# minor, and A major) and was subsequently published on YouTube.[2] As of August 2015, the most popular version has been viewed over 35 million times.

Three-chord tunes are more common than simple progressions, since a melody may then dwell on any note of the scale. They are often presented as successions of four chords, in order to produce a binary harmonic rhythm, but two of the four chords are then the same. Often the chords may be selected to fit a pre-conceived melody, but just as often it is the progression itself that gives rise to the melody. I - IV - V - V. I - I - IV - V. I - IV - I - V. (Common in Elizabethan music (Scholes 1977), this also underpins the American college song "Goodnight Ladies",[citation needed] is the exclusive progression used in Kwela.[10] I - IV - V - IV.

The twelve bar blues and its many variants use an elongated, three-line form of the I - IV - V progression that has also generated countless hit records, including the most significant output of rock and rollers such as Chuck Berry and Little Richard. In its most elementary form (there are many variants) the chords progress as follows: I - I - I - I IV - IV - I - I V - IV - I - I

Another common way of extending the I - IV - V sequence is by adding the chord of the sixth scale degree, giving the sequence I - vi - IV - V or I - vi - ii - V, sometimes called the 50s progression. 50s progression in C, ending with C About this sound Play (help·info) In fact this sequence had been in use from the earliest days of classical music (used often by Wolfgang Amadeus Mozart[citation needed]), but after generating popular hits such as Rodgers and Hart's "Blue Moon" (1934),[citation needed] Jerome Kern and Dorothy Fields' 1936 "The Way You Look Tonight",[citation needed] and Hoagy Carmichael's "Heart and Soul" (1938),[12] it became associated with the black American vocal groups of the 1940s, The Ink Spots and The Mills Brothers ("Till Then"),[citation needed] and thus later became the entire basis of the 1950s doo-wop genre, a typical example being The Monotones' "The Book of Love".[citation needed] Taken up into the pop mainstream, for example with Felice and Boudleaux Bryant's "All I Have to Do Is Dream",[citation needed] a hit for The Everly Brothers, in the 1960s it continued to generate records as otherwise disparate as The Paris Sisters' "I Love How You Love Me" (written by Mann and Kolber) and Boris Pickett's "Monster Mash".[citation needed] It continued to be used sectionally, as in the last part of The Beatles' "Happiness Is a Warm Gun",[13] and also to form the harmonic basis of further new songs for decades ("Every Breath You Take" by The Police).[citation needed]

Similar strategies to all the above work equally well in minor modes: there have been one-, two- and three-minor-chord songs, minor blues. A notable example of a descending minor chord progression is the four-chord Andalusian cadence, i - VII - VI - V.

Picassos cubist painting makes me think about quadrants with the lines intersecting. Picasso was known for his cubist paintings with. I discussed how science and art are connected in that science is the first square form of inquiry and art is the third square. Picasso paintings coincided with Einsteins theory of relativity. The question is which came first? It was Picasso's cubism, which distorted time and space. You would think it would be the other way around.

Robert Delaunay, Simultaneous Windows on the City, 1912, 46 x 40 cm, Hamburger Kunsthalle, an example of Abstract Cubism. It is a cubist work that kind of seems to have quadrants within it due to the lines of cubism.

The Three Dancers (French: Les Trois Danseuses[1]) is a painting by Spanish artist Pablo Picasso, painted in June 1925. It is an oil on canvas and measures 84.8 in x 56 in (215.3 cm x 142.2 cm).

It was suggested that one of the dancers was being crucified, and that was what Picasso was portraying

Picasso's Guernica has a mother carrying her dead trial. It is argued that this is a continuation of the Madonna motif in which Mary holds Jesus after his crucifixion. The cross never leaves art even in so called secular art. For instance, in Van Gohs secular paintings he purposefully places crosses, like in window, where in previous eras there would have been actual crosses. This was intentional according to art historians. Andy Warhol is a modern artist who would make pictures of celebrities as if they were iconic religious images. It is not a coincidence growing up he was a devout catholic, and his modern images the feel as though the celebrities were religious icons, while also their cartoonish character undermined it.

Salvador Dali's paintings reflected the nature of dreams with distorted time and space and surrealist content. Art is the third quadrant form of inquiry, the third quadrant itself in the quadrant model is the dreaming quadrant. And recall that Art, the third square field of inquiry, is connected to religion, the second. Dali did not stray completely from the religious motifs that dominated art history. The Temptation of St. Anthony shows St. Anthony naked holding up a cross, and he also has a picture of the Madonna with the baby Jesus, but unlike earlier periods of religious paintings, his were absurd. He even has a painting of Christ on a hovering cross from the perspective looking down from a birds eye view on the cross and a landscape in the background

Crucifixion (Corpus Hypercubus) is a 1954 oil-on-canvas painting by Salvador Dalí which depicts the Crucifixion of Jesus, though it deviates from traditional portrayals of the Crucifixion by depicting Christ on the polyhedron net of a hypercube and adding elements of Surrealism. It is one of his most well known paintings from the later period of his career. Dalí’s inspiration for Corpus Hypercubus came from his change in artistic style during the 1940s and 1950s. Around that time, his interest in surrealism diminished and he became fascinated with nuclear science, feeling that “thenceforth, the atom was [his] favorite food for thought.” His interest grew from the bombing of Hiroshima at the end of World War II which left a lasting impression on him. In his 1951 essay “Mystical Manifesto”, he introduced an art theory he called “nuclear mysticism” that combined Dalí’s interests in Catholicism, mathematics, science, and Catalan culture in an effort to reestablish Classical values and techniques, which he extensively utilizes in Corpus Hypercubus.[1] That same year, to promote nuclear mysticism and explain the “return to spiritual classicism movement” in modern art,[2] he traveled throughout the United States giving lectures. Before painting Corpus Hypercubus, Dalí announced his intention to portray an exploding Christ using both classical painting techniques along with the motif of the cube and he declared that “this painting will be the great metaphysical work of [his] summer.” Juan de Herrera’s Treatise on Cubic Forms was particularly influential to Dalí.[3] Dali was inspired by nuclear physics, and once again, art was influenced by science, and Dali began making paintings that he felt reflected the quantum world. In his painting of Christ on the hypercube, he sought to portray the fourth dimension time in the painting. Remember that time is an illusion. The fourth square is always different. The first three spatial dimensions are similar, but the fourth, time, is different.

In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubical cells. The tesseract is one of the six convex regular 4-polytopes, along with the 16 cell. A net of a tesseract is what Dali portrayed as Jesus's cross.

The tetradic (double complementary) colors scheme is the richest of all the schemes because it uses four colors arranged into two complementary color pairs. This scheme is hard to harmonize and requires a color to be dominant or subdue the colors.; if all four colors are used in equal amounts, the scheme may look unbalanced. Rectangle The rectangle color scheme uses four colors arranged intotwo complementary pairs and offers plenty of possibilities for variation. Rectangle color schemes work best when one color is dominant. Square The square color scheme The fourth is always different.

Recent developments in primary colors[edit] Some recent TV and computer displays are starting to add a fourth "primary" of yellow, often in a four-point square pixel area, to get brighter pure yellows and larger color gamut.[14] Even the four-primary technology does not yet reach the range of colors the human eye is theoretically capable of perceiving (as defined by the sample-based estimate called the Pointer Gamut[15]), with 4-primary LED prototypes providing typically about 87% and 5-primary prototypes about 95%. Several firms, including Samsung and Mitsubishi, have demonstrated LED displays with five or six "primaries", or color LED point light sources per pixel.[16] A recent academic literature review claims a gamut of 99% can be achieved with 5-primary LED technology.[17] While technology for achieving a wider gamut appears to be within reach, other issues remain; for example, affordability, dynamic range, and brilliance. In addition, there exists hardly any source material recorded in this wider gamut, nor is it currently possible to recover this information from existing visual media. Regardless, industry is still exploring a wide variety of "primary" active light sources (per pixel) with the goal of matching the capability of human color perception within a broadly affordable price. One example of a potentially affordable but yet unproven active light hybrid places a LED screen over a plasma light screen, each with different "primaries". Because both LED and plasma technologies are many decades old (plasma pixels going back to the 1960s), both have become so affordable that they could be combined.

In the printing industry, to produce the varying colors the subtractive primaries cyan, magenta, and yellow are applied together in varying amounts. Before the color names cyan and magenta were in common use, these primaries were often known as blue-green and purple, or in some circles as blue and red, respectively, and their exact color has changed over time with access to new pigments and technologies.[18] Subtractive color mixing – the magenta and cyan primaries are sometimes called purple and blue-green, or red and blue. Mixing yellow and cyan produces green colors; mixing yellow with magenta produces reds, and mixing magenta with cyan produces blues. In theory, mixing equal amounts of all three pigments should produce grey, resulting in black when all three are applied in sufficient density, but in practice they tend to produce muddy brown colors. For this reason, and to save ink and decrease drying times, a fourth pigment, black, is often used in addition to cyan, magenta, and yellow. The resulting model is the so-called CMYK color model. The abbreviation stands for cyan, magenta, yellow, and key—black is referred to as the key color, a shorthand for the key printing plate that impressed the artistic detail of an image, usually in black ink.[19] In practice, colorant mixtures in actual materials such as paint tend to be more complex. Brighter or more saturated colors can be created using natural pigments instead of mixing, and natural properties of pigments can interfere with the mixing. For example, mixing magenta and green in acrylic creates a dark cyan—something which would not happen if the mixing process were perfectly subtractive. In the subtractive model, adding white to a color, whether by using less colorant or by mixing in a reflective white pigment such as zinc oxide, does not change the color's hue but does reduce its saturation. Subtractive color printing works best when the surface or paper is white, or close to it. A system of subtractive color does not have a simple chromaticity gamut analogous to the RGB color triangle, but a gamut that must be described in three dimensions. There are many ways to visualize such models, using various 2D chromaticity spaces or in 3D color spaces.

Painters have long used more than three "primary" colors in their palettes—and at one point considered red, yellow, blue, and green to be the four primaries.[23] Red, yellow, blue, and green are still widely considered the four psychological primary colors,[24] though red, yellow, and blue are sometimes listed as the three psychological primaries,[25] with black and white occasionally added as a fourth and fifth.

Ewald Hering proposed opponent color theory in 1892.[6] He thought that the colors red, yellow, green, and blue are special in that any other color can be described as a mix of them, and that they exist in opposite pairs. That is, either red or green is perceived and never greenish-red; although yellow is a mixture of red and green in the RGB color theory, the eye does not perceive it as such. Most artists used yellow and blue highlights. Matisse changed painting by using green and red.

Matisse was known in his later life for spearheading art that was extremely simple. One of his later paintings was simply squares, like the squares of the quadrant model of reality. The quadrant model itself is extremely simple. Just think, 16 squares, four quadrants, but it can explain all of reality, and it is not just that it can explain all of reality, it does. The last olympic logo had four parts to it, inspired by Matisse's later cut out works. In some of Matisses' later works he merely cut out squares of different colors. At the end of Matisse's life he became religious. His whole life he was an atheist but he had surgery and at the end of his life he made a sort of cathedral where he tried to represent color in its purest form just through light, and he has an image of Jesus in the Cathedral and a cross.

Psychiatry is bull shit. I want to show you all an example of how bull shit psychiatry is. These are the symptoms for PTSD. Let's analyze them. One is reexperienceing symptoms where you relive trauma. I don't relive any trauma never have. The only trauma I might relive is the trauma of being injected in the psych ward with a drug that almost killed me. There is no such thing as an illness called PTSD. Reexperiencing a traumatic experience is called a symptom of PTSD. That is not an illness. That is just a regular consequence of experiencing something very traumatic. Just imagine if you were in Vietnam and you saw your friend explode next to you. If you did not relive that experience then you have something wrong with you. It's not an illness it's just what happens. People relive all types of experiences all the time, just the dude in Vietnam had experiences that were extraordinary. He doesn't have a disease called PTSD he just experienced something incredible and a normal consequence of that is remembering it like people remember things all the time. If it is an incredible memory it might make you sweat ok big deal. Bad dreams and frightening thoughts. Everybody has bad dreams and frightening thoughts. Staying away from places, events or objects that are reminders of the experience. That is just probably a normal consequence of a very tragic thing. Imagine you had your friend killed in Vietnam and he carried a teddy bear. You might not want to carry a teddy bear. That's normal. All of these symptoms are very vague. Psychiatrists stop trying to symptomize and encapsulate normal things into "diseases" it's really annoying and you have a whole populace of people who love it and are into it because they are dumb sheep who can't think deeply about things so they like to categorize people and create self fulfilling prophecies with their categorizations. "feeling emotionally numb". What does that even mean? Maybe if somebody saw his friend blow up he might sometimes feel emotionally sad when he thinks about it. That's a normal reaction to experiencing something sad its not a disorder. You act like this is a real thing like it is a real disorder. No PTSD is just fancy language for normal reactions to experiencing something very sad. PTSD can cause many symptoms. These symptoms can be grouped into three categories: 1. Re-experiencing symptoms Flashbacks—reliving the trauma over and over, including physical symptoms like a racing heart or sweating Bad dreams Frightening thoughts. Re-experiencing symptoms may cause problems in a person’s everyday routine. They can start from the person’s own thoughts and feelings. Words, objects, or situations that are reminders of the event can also trigger re-experiencing. 2. Avoidance symptoms Staying away from places, events, or objects that are reminders of the experience Feeling emotionally numb Feeling strong guilt, depression, or worry Losing interest in activities that were enjoyable in the past Having trouble remembering the dangerous event. Things that remind a person of the traumatic event can trigger avoidance symptoms. These symptoms may cause a person to change his or her personal routine. For example, after a bad car accident, a person who usually drives may avoid driving or riding in a car. 3. Hyperarousal symptoms Being easily startled Feeling tense or “on edge” Having difficulty sleeping, and/or having angry outbursts. Hyperarousal symptoms are usually constant, instead of being triggered by things that remind one of the traumatic event. They can make the person feel stressed and angry. These symptoms may make it hard to do daily tasks, such as sleeping, eating, or concentrating. It’s natural to have some of these symptoms after a dangerous event. Sometimes people have very serious symptoms that go away after a few weeks. This is called acute stress disorder, or ASD. When the symptoms last more than a few weeks and become an ongoing problem, they might be PTSD. Some people with PTSD don’t show any symptoms for weeks or months. Do children react differently than adults? Children and teens can have extreme reactions to trauma, but their symptoms may not be the same as adults. In very young children, these symptoms can include: Bedwetting, when they’d learned how to use the toilet before Forgetting how or being unable to talk Acting out the scary event during playtime Being unusually clingy with a parent or other adult. Older children and teens usually show symptoms more like those seen in adults. They may also develop disruptive, disrespectful, or destructive behaviors. Older children and teens may feel guilty for not preventing injury or deaths. They may also have thoughts of revenge. For more information, see the NIMH booklets on helping children cope with violence and disasters. (from Post-Traumatic Stress Disorder (PTSD) )

Being is the 17th square. Being represents God. According to Leo Kass the patriarch Jacob lived 17 years after his son Joseph went missing and presumed dead, and lived 17 years after their reunion in Egypt, and the lifespans of Abraham aged 175, Isaac aged 180, and Jacob aged 147 are not a coincidence. "(The sum of the factors in all three cases is 17; of what possible significance this is, I have no idea.)" Leon Kass, The beginning of wisdom: reading Genesis,(Simon and Schuster, 2003)

sudoku is made up of quadrants and is a very popular mind game

In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. It is quadrants

In the design of experiments, Latin squares are a special case of row-column designs for two blocking factors:[5] Many row-column designs are constructed by concatenating Latin squares.[6] In algebra, Latin squares are generalizations of groups; in fact, Latin squares are characterized as being the multiplication tables (Cayley tables) of quasigroups. A binary operation whose table of values forms a Latin square is said to obey the Latin square property.

The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square.

Latin squares have been used as the basis for several board games, notably the popular abstract strategy game Kamisado.

Sudoku imposes the additional restriction that nine particular 3×3 adjacent subsquares must also contain the digits 1–9 (in the standard version). The more recent KenKen puzzles are also examples of Latin squares

Euler diagram for P, NP, NP-complete, and NP-hard set of problems. The left side is valid under the assumption that P≠NP, while the right side is valid under the assumption that P=NP (except that the empty language and its complement are never NP-complete This is one of the biggest problems in math and is divided into p no np complete and np hard, those four choices. In computational complexity theory, a decision problem is NP-complete when it is both in NP and NP-hard. The set of NP-complete problems is often denoted by NP-C or NPC. The abbreviation NP refers to "nondeterministic polynomial time". Although any given solution to an NP-complete problem can be verified quickly (in polynomial time), there is no known efficient way to locate a solution in the first place; indeed, the most notable characteristic of NP-complete problems is that no fast solution to them is known. That is, the time required to solve the problem using any currently known algorithm increases very quickly as the size of the problem grows. As a consequence, determining whether or not it is possible to solve these problems quickly, called the P versus NP problem, is one of the principal unsolved problems in computer science today.

A lot of people do crossword puzzles. I know my Dad did every morning A crossword is a word puzzle that normally takes the form of a square or a rectangular grid of white and black shaded squares. The goal is to fill the white squares with letters, forming words or phrases, by solving clues which lead to the answers. In languages that are written left-to-right, the answer words and phrases are placed in the grid from left to right and from top to bottom. The shaded squares are used to separate the words or phrases. Crossword puzzles are made of quadrants

One of the smallest crosswords in general distribution is a 4×4 crossword compiled daily by John Wilmes, distributed online by USA Today as "QuickCross" and by Universal Uclick as "PlayFour." A four by four grid is the quadrant model

A crossnumber (also known as a cross-figure) is the numerical analogy of a crossword, in which the solutions to the clues are numbers instead of words. Clues are usually arithmetical expressions, but can also be general knowledge clues to which the answer is a number or year. There are also numerical fill-in crosswords. The Daily Mail Weekend magazine used to feature crossnumbers under the misnomer Number Word. This kind of puzzle should not be confused with a different puzzle that the Daily Mail refers to as Cross Number.

Kakuro or Kakkuro (Japanese: カックロ) is a kind of logic puzzle that is often referred to as a mathematical transliteration of the crossword The canonical Kakuro puzzle is played in a grid of filled and barred cells, "black" and "white" respectively. Puzzles are usually 16×16 in size, although these dimensions can vary widely.

Killer sudoku (also killer su doku, sumdoku, sum doku, sumoku, addoku, or samunamupure) is a puzzle that combines elements of sudoku and kakuro. Despite the name, the simpler killer sudokus can be easier to solve than regular sudokus, depending on the solver's skill at mental arithmetic; the hardest ones, however, can take hours to crack.

In sudoku although the 9×9 grid with 3×3 regions is by far the most common, many other variations exist. Sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with pentomino regions have been published under the name Logi-5; the World Puzzle Championship has featured a 6×6 grid with 2×3 regions and a 7×7 grid with six heptomino regions and a disjoint region. Larger grids are also possible. The Times offers a 12×12-grid "Dodeka Sudoku" with 12 regions of 4×3 squares. Dell Magazines regularly publishes 16×16 "Number Place Challenger" puzzles (using the numbers 1-16 or the letters A-P). Nikoli offers 25×25 Sudoku the Giant behemoths. A 100×100-grid puzzle dubbed Sudoku-zilla was published in 2010.[12]

Alphabetical variations have emerged, sometimes called Wordoku; there is no functional difference in the puzzle unless the letters spell something. Some variants, such as in the TV Guide, include a word reading along a main diagonal, row, or column once solved; determining the word in advance can be viewed as a solving aid. A Wordoku might contain words other than the main word.

"Quadratum latinum" is a Sudoku variation with Latin numbers (I, II, III, IV, ..., IX) proposed by Hebdomada aenigmatum, a monthly magazine of Latin puzzles and crosswords. Like the "Wordoku", the "Quadratum latinum" presents no functional difference with a normal Sudoku but adds the visual difficulty of using Latin numbers.

Hypersudoku is one of the most popular variants. It is published by newspapers and magazines around the world and is also known as "NRC Sudoku", "Windoku", "Hyper-Sudoku", and "4 Square Sudoku". The layout is identical to a normal Sudoku, but with additional interior squares defined in which the numbers 1 to 9 must appear. The solving algorithm is slightly different from the normal Sudoku puzzles because of the emphasis on the overlapping squares. This overlap gives the player more information to logically reduce the possibilities in the remaining squares. The approach to playing is similar to Sudoku but with possibly more emphasis on scanning the squares and overlap rather than columns and rows. Hypersudoku is so popular because there are four squares in the game and a quadrant is explicitly images in the game

If you unfold a cube you get the image of a cross

V-Cube also produces a 2×2×2, 3×3×3 and a 4x4x4 rubiks cubes.

Rubiks cube is another enormously popular game that reflects the image of quadrants

The unit circle for trigonometry is fundamental to trigonometry. It is a circle with a quadrant inside of it. The four angle of the unit circle are 0, pi over 2, pi, and 3pi over 2 in radian angles.

In quadrant 1 cosine and sin are positive. In quadrant 2 cosine is negative and sign is positive. In quadrant 3 of the cartesian coordinate system cosine is negative and sin is negative. In quadrant 4 of the unit circle cosine is positive and sin is negative.

Cosine is the x coordinate and sine is the y coordinate.

The unit circle is the basis for trigonometry and it reflects the quadrant model image.

graphs of sine, cosine and tangent are made from information in the unit circle.

Although the lattice method for multiplication is no longer being used right now in school, it is easy understand I will illustrate with two good examples. Study them carefully and follow the steps exactly as shown Example #1: Multiply 42 and 35 Arrange 42 and 35 around a 2 × 2 grid as shown below: Draw the diagonals of the small squares as shown below: Multiply 3 by 4 to get 12 and put 12 in intersection of the first row and the first column as show below. Notice that 3 is located in the first row and 4 in the first column. That is why the answer goes in the intersection. By the same token, multiply 5 and 2 and put the answer in the intersection of second row and the second column And so forth... Then, going from right to left, add the numbers down the diagonals as indicated with the arrows. The first diagonal has only 0. Bring zero down. The second diagonal has 6, 1, 0. Add these numbers to get 7 and bring it down. And so forth... After the grid is completed, what you see in red is the answer that is 1470 Example #2: Multiply 658 and 47 Arrange 657 and 47 around a 3 × 2 grid as shown below: Draw the diagonals of the small squares, find products, and put the answers in intersecting rows and columns as already demonstrated: Then, going from right to left, add the numbers down the diagonals as shown before. The first diagonal has only 6. Bring 6 down. The second diagonal has 2, 5, and 5. Add these numbers to get 12. Bring 2 down and carry the 1 over to the next diagonal. The third diagonal has 3, 0, 3, and 2. Add these numbers to get 8 and add 1 (your carry) to 8 to get 9. and so forth... After the grid is completed, what you see in red is the answer to the multiplication that is 30926 I understand that this may be your first encounter with the lattice method for multiplication. It may seem that it is tough. Just practice with other examples and you will be fine. Any questions about the lattice method for multiplication? Just contact me The lattice method employs quadrants and many math teachers think it is the ideal way to solve mathematical problems

In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials—hence the method may be referred to as the FOIL method. The word FOIL is an acronym for the four terms of the product: First ("first" terms of each binomial are multiplied together) Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second) Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second) Last ("last" terms of each binomial are multiplied) The general form is: (a+b)(c+d)=\underbrace {ac} _{\mathrm {first} }+\underbrace {ad} _{\mathrm {outside} }+\underbrace {bc} _{\mathrm {inside} }+\underbrace {bd} _{\mathrm {last} } Note that a is both a "first" term and an "outer" term; b is both a "last" and "inner" term, and so forth. The order of the four terms in the sum is not important, and need not match the order of the letters in the word FOIL. The Foil method is the basis for algebra and multiplying binomials. It has four components fitting the quadrant model image. Even if you represent the foil model pictorially, you get four squares/ the quadrant model image.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side a + b can be cut into a cube of side a, a cube of side b, three a×a×b rectangular boxes, and three a×b×b rectangular boxes. In calculus, this picture also gives a geometric proof of the derivative (x^{n})'=nx^{n-1}:[9] if one sets a=x and b=\Delta x, interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, (x+\Delta x)^{n}, where the coefficient of the linear term (in \Delta x) is nx^{n-1}, the area of the n faces, each of dimension (n-1): (x+\Delta x)^{n}=x^{n}+nx^{n-1}\Delta x+{\tbinom {n}{2}}x^{n-2}(\Delta x)^{2}+\cdots . Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms – (\Delta x)^{2} and higher – become negligible, and yields the formula (x^{n})'=nx^{n-1}, interpreted as "the infinitesimal change in volume of an n-cube as side length varies is the area of n of its (n-1)-dimensional faces". If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral \textstyle {\int x^{n-1}\,dx={\tfrac {1}{n}}x^{n}} – see proof of Cavalieri's quadrature formula for details.[9] Notice how this geometric proof, fundamental to algebra and geometry, reflects the four squares of the quadrant model image.

Look at the geometric proof. That proof is the foundation of Calculus it is what Newton used to prove Calculus. The geometric proof involves four sections of a square. It is the quadrant model image.

In the 11th century, the Islamic mathematician Ibn al-Haytham (known as Alhazen in Europe) computed the integrals of cubics and quartics (degree three and four) via mathematical induction, in his Book of Optics

In algebra, a quartic function, is a function of the form f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e, where a is nonzero, which is defined by a polynomial of degree four, called quartic polynomial. Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form f(x)=ax^{4}+cx^{2}+e. A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form ax^{4}+bx^{3}+cx^{2}+dx+e=0, where a ≠ 0. The derivative of a quartic function is a cubic function. Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum. The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals. Notice how four degrees is the highest degree. And it took a ton of effort to discover the quartic equation. It has been proven that five degrees is impossible. Any higher than five it is assume it has been impossible but it has not been proven. Four is always different. Five is ultra transcendent.

Lagrange's four-square theorem is a special case of the Fermat polygonal number theorem and Waring's problem. Another possible generalisation is the following problem: Given natural numbers a,b,c,d, can we solve n=ax_{1}^{2}+bx_{2}^{2}+cx_{3}^{2}+dx_{4}^{2} for all positive integers n in integers x_{1},x_{2},x_{3},x_{4}? The case a=b=c=d=1 is answered in the positive by Lagrange's four-square theorem. The general solution was given by Ramanujan. He proved that if we assume, without loss of generality, that a\leq b\leq c\leq d then there are exactly 54 possible choices for a,b,c,d such that the problem is solvable in integers x_{1},x_{2},x_{3},x_{4} for all n. (Ramanujan listed a 55th possibility a=1,b=2,c=5,d=5, but in this case the problem is not solvable if n=15.[8])

In additive number theory, Pierre de Fermat's theorem on sums of two squares states that an odd prime p is expressible as p=x^{2}+y^{2},\, with x and y integers, if and only if p\equiv 1{\pmod {4}}. For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways: 5=1^{2}+2^{2},\quad 13=2^{2}+3^{2},\quad 17=1^{2}+4^{2},\quad 29=2^{2}+5^{2},\quad 37=1^{2}+6^{2},\quad 41=4^{2}+5^{2}. On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Albert Girard was the first to make the observation, describing all positive integral numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published posthumously in 1634.[1] Fermat was the first to claim a proof of it; he announced this theorem in a letter to Marin Mersenne dated December 25, 1640: for this reason this theorem is sometimes called Fermat's Christmas Theorem. Since the Brahmagupta–Fibonacci identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer n, we see that if all the prime factors of n congruent to 3 modulo 4 occur to an even exponent, then n is expressible as a sum of two squares. The converse also holds.[2] This equivalence provides the characterization Girard guessed.

The 2-torus double-covers the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points. In fact, the conformal type of the torus is determined by the cross-ratio of the four points.

The Procuress is a 1656 oil-on-canvas painting by the 24-year-old Jan Vermeer. It can be seen in the Gemäldegalerie Alte Meister in Dresden. It is his first genre painting and shows a scene of contemporary life, an image of mercenary love[1] perhaps in a brothel. It differs from his earlier biblical and mythological scenes. It is one of only three paintings Vermeer signed and dated (the other two are The Astronomer and The Geographer). It seems Vermeer was influenced by earlier works on the same subject by Gerard ter Borch, and The Procuress (c. 1622) by Dirck van Baburen, which was owned by Vermeer's mother-in-law Maria Thins and hung in her home.[2] The procuress is a painting by vermeer that reflects the quadrant model pattern. There are three figures very close. Two are interlocked, a man and a woman. That is the duality. There is a third the triad, very close to them. The fourth figure is close but a little bit farther.

Another painting by Vermeer in which one can perceive the quadrant model pattern is Diana and Her Companions. Again there are three girls very close to each other. There is a fourth a little bit farther away with her back turned. Finally in the back there is a fifth who is in the dark. The fifth is ultra transcendent (and sometimes leads to a new quadrant).

Of the many types of perspective drawings, the most common categorizations of artificial perspective are one-, two- and three-point. The names of these categories refer to the number of vanishing points in the perspective drawing. There is also four point perspective though, although this is a lot different from the previous three. The fourth is always different/ transcendent. The fifth is always ultra transcendent/questionable.

The way that one point perspective is pictorially represented is usually by drawing an X on the paper, with the midpoint of the X representing the horizon/ vanishing point. The X is the quadrant.

Perspective is very important in drawing and painting and revolutionized painting, especially during the Renaissance.

The Rosicrucian quadrant

The primary principle of the Prenatal Epoch has been stated by Max Heindel in The Message of the Stars, where he says that the body is the product of lunar forces and that the position of the Ascendant, or its opposite, at birth, is the Moon's position at conception. The keyword of the Moon is fecundation or fertility, and it is Jehovah and the lunar Angels that preside at the birth of a child. This is stated in the Cosmo-Conception and other works of the Rosicrucian Philosophy. We thus see that the Moon has primary influence over the formation of the physical body, and that the Ascendant represents merely the transference of the Moon's position from conception to birth.

This law was known to the ancients as the "Truitine of Hermes," from Hermes Trismegistus, who first correctly formulated and stated the law as follows: "The place of the Moon at conception becomes the birth ascendant or its opposite point."

"But this proved to be but one-half of a very important law, for while the Ascendant at birth was the place of the Moon at a certain Epoch, the Ascendant or its opposite point at this Epoch was the place of the Moon at birth -- a very remarkable interchange of factors." --E.H. Bailey.

According to the Ancient Wisdom, "The World-Breath has a definite and periodic pulsation, a systole and diastole action, whereby birth and death are controlled." This idea of periodicity, well established by modern science, furthers the idea that birth can take place only in respect to any single locality at intervals, that these intervals are in accord with lunar motion, and that only every seventh impulse of the World-Breath permits of human births.

The modern version of the Prenatal Epoch we first established by the English astrologer known to the astrological world as Sepharial, in the year 1886. It was published by him in 1890. In this he had the collaboration of a trained and veteran scientist, a doctor, who helped him to establish the primary laws of the Prenatal Epoch by years of painstaking research and actual experiments. This doctor was an expert obstetrician and proved the laws of the Prenatal Epoch by actual firsthand data.

These laws have been further verified, extended, and complemented by the painstaking researches of E.H. Bailey, to whom great credit is due for his many and exact proofs of the Prenatal Epoch. His book upon this subject is considered standard authority, and we are in the main following his very worthy contribution to the subject and are extending him full credit.

One of the primary uses of the Prenatal Epoch is the correction or rectification of the birth time when only the approximate time is given. Another is its utility in determining correctly the sex of the native. Finally, it gives sidelights on the character and inner nature of the individual as fundamental as those of the birth chart.

"As births are brought about in exact harmony with lunar laws, it is shown that intrauterine life is in direct relation with the sidereal world without, that the great fact of maternity is capable of purely astronomical measurement and rule .... The law is nothing less than a mathematical measurement of human life, a stupendous natural fact; nothing more exactly mathematical and matter of fact is to be found in the records of scientists than this record of intra- uterine life, for only through its study will the laws of generation be fully understood." -- Sepharial.

"In the measurement of the intra-uterine period we actually measure the whole future of the individual; alter this one fact -- the moment of conception (or its spiritual counterpart, the Epoch) -- and you change the whole course of the progeny's destiny. If we accept the occult theory that the Prenatal Epoch is the descent of the Ego to the Desire World, then it must show the inherent character of the Ego about to incarnate. It may be stated that the Epoch has a more intimate relationship with the individual than the horoscope at birth, the latter appearing to reject the personality and its heredity and environment. In other words, the Epoch represents the man about to manifest in the flesh, the horoscope denotes actual personal conditions and environments into which he is born. Every birth is directly connected with the Epoch, and every authentic natural birth will, within the limits of an error of observation, yield an Epoch in accordance with the rules to be given."-- Bailey.

For summary, let us restate the fundamental principle of the Prenatal Epoch known as the Truitine of Hermes: "The Ascendant at birth is the place of the Moon at a certain Epoch, ant the Ascendant or its opposite point at Epoch was the place of the Moon at birth."

This yields the:

Four Laws of the Epoch

1. When the Moon at birth increases in light, it will be on the ascending degree of Epoch, and the Moon at Epoch will be on the ascending degree at birth.

2. When the Moon at birth decrease in light, it will be on the decreasing degree at Epoch, and the Moon at Epoch will be on the descending degree at birth.

3. When the Moon at birth is (a) increasing in light and below the horizon, or (b) decreasing in light and above the horizon, the period of gestation is longer than the norm.

4. When the Moon at birth is (a) increasing in light and above the horizon or (b) decreasing in light and below the horizon, the period of gestation is shorter than the norm.

From these four laws we deduce the following:  

Four Orders of Epoch

1. Moon above horizon and increasing in light.......... 273 days minus x.

2. Moon above horizon and decreasing in light.......... 273 days plus x.

3. Moon below horizon and increasing in light.......... 273 days plus x.

4. Moon below horizon and decreasing in light.......... 273 days minus x.

It is to be understood that the 273 days referred to in the above table is the normal period of gestation, or nine solar or ten lunar months. This normal period is increased or decreased in accordance with the distance of the Moon from either the Ascendant or Descendant, and "x" is a certain number of days corresponding to this distance obtained by dividing the distance in degrees by thirteen degrees, the latter being the average daily motion of the Moon.

When making the count, count to the Ascendant (AC) when the Moon is increasing in light, and to the Descendant (DC) when the Moon is decreasing in light. Another more definite way of stating this would be: In orders Nos. 1 and 4 the distance in degrees of the Moon from the horizon last crossed (AC or DC), divided by thirteen, gives "x", or the number of days by which this period is decreased; and in orders Nos. 2 and 3 the distance of the Moon in degrees from the horizon which it is approaching, divided by thirteen, gives the number of days by which this period is increased. These rules are illustrated by the following examples