Quadrant Model of Reality Book 7 Art and Philosophy More

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.