Quadrant Model of Reality: Quadrant Model of Reality Book 4 Philosophy

Philosophy Chapter

QMRIn mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of an odd integer; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is

H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.

H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by Hurwitz (1919).

A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers. The set of all Lipschitz quaternions

L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}

forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that you can perform Euclidean division on them, obtaining a small remainder.

Examples of real quadratic integer rings[edit]

Powers of the golden ratio

For D > 0, ω is a positive irrational real number, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the Pell's equation X2 − D Y2 = 1, a Diophantine equation that has been widely studied, are the units of these rings, for D ≡ 2, 3 (mod 4).

For D = 5, ω =

1+√5

is the golden ratio. This ring was studied by Peter Gustav Lejeune Dirichlet. Its invertible elements have the form ±ωn, where n is an arbitrary integer. This ring also arises from studying 5-fold rotational symmetry on Euclidean plane, for example, Penrose tilings.[4]

Indian mathematician Brahmagupta treated the Pell's equation X2 − 61 Y2 = 1, corresponding to the ring is Z[√61]. Some results were presented to European community by Pierre Fermat in 1657.[which?]

Principal rings of quadratic integers[edit]

Unique factorization property is not always verified for rings of quadratic integers, as seen above for the case of Z[√−5]. However, as for every Dedekind domain, a ring of quadratic integers is a unique factorization domain if and only if it is a principal ideal domain. This occurs if and only if the class number of the corresponding quadratic field is one.

The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are \mathcal{O}_{\mathbf{Q}(\sqrt{D})} for

D = −1, −2, −3, −7, −11, −19, −43, −67, −163.

This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967. (See Stark–Heegner theorem.) This is a special case of the famous class number problem.

There are many known positive integers D > 0, for which the ring of quadratic integers is a principal ideal ring. However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.

Euclidean rings of quadratic integers[edit]

See also: Euclidean domain § Norm-Euclidean fields

When a ring of quadratic integers is a principal ideal domain, it is interesting to know if it is a Euclidean domain. This problem has been completely solved as follows.

Equipped with the norm N(a + b\sqrt{D}) = a^2 - Db^2, as an Euclidean function, \mathcal{O}_{\mathbf{Q}(\sqrt{D})} is an Euclidean domain for negative D when

D = −1, −2, −3, −7, −11, [5]

and, for positive D, when

D = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 (sequence A048981 in OEIS).

There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function.[6]

For negative D, a ring of quadratic integers is Euclidean if and only if the norm is a Euclidean function for it. It follows that, for

D = −19, −43, −67, −163,

the four corresponding rings of quadratic integers are among the rare known examples of principal ideal domains that are not Euclidean domains.

On the other hand, the generalized Riemann hypothesis implies that a ring of real quadratic integers that is a principal ideal domain is also a Euclidean domain for some Euclidean function.[citation needed

Quadratic integer rings[edit]

Every square-free integer (different of 0 and 1) D defines a quadratic integer ring, which is the integral domain of the algebraic integers contained in \mathbf{Q}(\sqrt{D}). It is the set Z[ω] =a + ωb : a, b ∈ Z, where ω is defined as above. It is called the ring of integers of Q(√D) and often denoted \mathcal{O}_{\mathbf{Q}(\sqrt{D})}. By definition, it is the integral closure of Z in \mathbf{Q}(\sqrt{D}).

The properties of the quadratic integers (and more generally of algebraic integers) has been a long standing problem, which has motivated the elaboration of the notions of ring and ideal. In particular the fundamental theorem of arithmetic is not true in many rings of quadratic integers. However there is a unique factorization for ideals, which is expressed by the fact that every ring of algebraic integers is a Dedekind domain.

Quadratic integer rings and their associated quadratic fields are thus commonly the starting examples of most studies of algebraic number fields.

The quadratic integer rings divide in two classes depending on the sign of D. If D > 0, all elements of \mathcal{O}_{\mathbf{Q}(\sqrt{D})} are real, and the ring is a real quadratic integer ring. If D < 0, the only real elements of \mathcal{O}_{\mathbf{Q}(\sqrt{D})} are the ordinary integers, and the ring is a complex quadratic integer ring.

Examples of complex quadratic integer rings[edit]

Gaussian integers

Eisenstein primes

For D < 0, ω is a complex (imaginary or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic complex numbers.

A classic example is \mathbf{Z}[\sqrt{-1}], the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.[3]

The elements in \mathcal{O}_{\mathbf{Q}(\sqrt{-3})} = \mathbf{Z}\left[{{1 + \sqrt{-3}} \over 2}\right] are called Eisenstein integers.

Both rings mentioned above are rings of integers of cyclotomic fields Q(ζ4) and Q(ζ3) correspondingly. In contrast, Z[√−3] is not even a Dedekind domain.

Both above examples are principal ideal rings and also Euclidean domains for the norm. This is not the case for

\mathcal{O}_{\mathbf{Q}(\sqrt{-5})} = \mathbf{Z}\left[\sqrt{-5}\right],

which is not even a unique factorization domain. This can be shown as follows.

In \mathcal{O}_{\mathbf{Q}(\sqrt{-5})}, we have

9 = 3\cdot3 = (2+\sqrt{-5})(2-\sqrt{-5}).

The factors 3, 2+\sqrt{-5} and 2-\sqrt{-5} are irreducible, as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of ±1 being at least 4. Thus the factorization of 9 into irreducible factors is not unique.

The ideals \langle 3, 1+\sqrt{-5}\rangle and \langle 3, 1-\sqrt{-5}\rangle are not principal, as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible.

Norm and conjugation[edit]

A quadratic integer in \mathbb{Q}(\sqrt{D}) may be written

a + b√D,

where either a and b are either integers, or, only if D ≡ 1 (mod 4), halves of odd integers. The norm of such a quadratic integer is

N(a + b√D) = a2 – b2D.

The norm of a quadratic integer is always an integer. If D < 0, the norm of a quadratic integer is the square of its absolute value as a complex number (this is false if D > 0). The norm is a completely multiplicative function, which means that the norm of a product of quadratic integers is always the product of their norms.

Every quadratic integer a + b√D has a conjugate

\overline{a+b\sqrt{D}} = a-b\sqrt{D}.

An algebraic integer has the same norm as its conjugate, and this norm is the product of the algebraic integer and its conjugate. The conjugate of a sum or a product of algebraic integers it the sum or the product (respectively) of the conjugates. This means that the conjugation is an automorphism of the ring of the integers of \mathbb{Q}(\sqrt{D}).

Units[edit]

A quadratic integer is a unit in the ring of the integers of \mathbb{Q}(\sqrt{D}) if and only if its norm is 1 or –1. In the first case its multiplicative inverse is its conjugate. It is the opposite of its conjugate in the second case.

If D < 0, the ring of the integers of \mathbb{Q}(\sqrt{D}) has at most six units. In the case of the Gaussian integers (D = –1), the four units are 1, –1, √–1, –√–1. In the case of the Eisenstein integers (D = –3), the six units are ±1,

±1 ± √–3

. For all other negative D, there are only two units that are 1 and –1.

If D > 0, the ring of the integers of \mathbb{Q}(\sqrt{D}) has infinitely many units that are equal to ±ui, where i is an arbitrary integer, and u is a particular unit called a fundamental unit. Given a fundamental unit u, there are three other fundamental units, its conjugate \overline{u}, and also -u and -\overline{u}. Commonly, one calls the fundamental unit, the unique one which has an absolute value greater than 1 (as a real number). It is the unique fundamental unit that may be written a + b√D, with a and b positive (integers or halves of integers).

The fundamental units for the 10 smallest positive square-free D are 1 + √2, 2 + √3,

1 + √5

(the golden ratio), 5 + 2√6, 8 + 3√7, 3 + √10, 10 + 3√11,

3 + √13

, 15 + 4√14, 4 + √15. For larger D, the coefficients of the fundamental unit may be very large. For example, for D = 19, 31, 43, the fundamental units are respectively 170 + 39 √19, 1520 + 273 √31 and 3482 + 531 √43.

Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same D, which allowed them to solve some cases of Pell's equation.[citation needed]

The characterization[clarification needed] of the quadratic integers was first given by Richard Dedekind in 1871.[1][2]

Definition[edit]

A quadratic integer is a complex number which is a solution of an equation of the form

x2 + Bx + C = 0

with B and C integers. In other words, a quadratic integer is an algebraic integer in a quadratic field. Each quadratic integer that is not an integer lies in a uniquely determined quadratic field, namely, the extension of \mathbb{Q} generated by the square-root of B2-4C, which can always be written in the form \mathbb{Q}(\sqrt{D}), where D is the unique square-free integer for which B2 – 4C = DE2 for some integer E.

The quadratic integers (including the ordinary integers), which belong to a quadratic fieds \mathbb{Q}(\sqrt{D}), form a integral domain called ring of integers of \mathbb{Q}(\sqrt{D}).

Here and in the following, D is supposed to be a square-free integer. This does not restricts the generality, as the equality √a2D = a√D (for any positive integer a) implies \mathbb{Q}(\sqrt{D})=\mathbb{Q}(\sqrt{a^2D}).

Every quadratic integer may be written a + ωb , where a and b are integers, and where ω is defined by:

\omega =

\begin{cases}

\sqrt{D} & \mbox{if }D \equiv 2, 3 \pmod{4} \\

{{1 + \sqrt{D}} \over 2} & \mbox{if }D \equiv 1 \pmod{4}

\end{cases}

(as D has been supposed square-free the case D \equiv 0\pmod{4} is impossible, since it would imply that D would be divisible by the square 4).

Although the quadratic integers belonging to a given quadratic field form a ring, the set of all quadratic integers is not a ring, because it is not closed under addition, as \sqrt{2}+\sqrt{3} is an algebraic integer, which has a minimal polynomial of degree four.

QMRIn number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are the solutions of equations of the form

x2 + Bx + C = 0

with B and C integers. They are thus algebraic integers of the degree two. When algebraic integers are considered, usual integers are often called rational integers.

Common examples of quadratic integers are the square roots of integers, such as √2, and the complex number i = √–1, which generates the Gaussian integers. Another common example is the non-real cubic root of unity

-1 + √–3

, which generates the Eisenstein integers.

Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations. The study of rings of quadratic integers is basic for many questions of algebraic number theory.

QMRAdditive number theory[edit]

Main article: Additive number theory

One of the most important problems in additive number theory is Waring's problem, which asks whether it is possible, for any k ≥ 2, to write any positive integer as the sum of a bounded number of kth powers,

n=x_1^k+\cdots+x_\ell^k. \,

The case for squares, k = 2, was answered by Lagrange in 1770, who proved that every positive integer is the sum of at most four squares. The general case was proved by Hilbert in 1909, using algebraic techniques which gave no explicit bounds. An important breakthrough was the application of analytic tools to the problem by Hardy and Littlewood. These techniques are known as the circle method, and give explicit upper bounds for the function G(k), the smallest number of kth powers needed, such as Vinogradov's bound

G(k)\leq k(3\log k+11). \,

QMRPell's equation, first misnamed by Euler.[53] He wrote on the link between continued fractions and Pell's equation.[54]

First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.[55]

QMRPell's equation (also called the Pell–Fermat equation) is any Diophantine equation of the form

x^2-ny^2=1\,

where n is a given positive nonsquare integer and integer solutions are sought for x and y. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.

Recall that to square something is to make it a quadrant

QMRClassical Chinese is distinguished from written vernacular Chinese in its style, which appears extremely concise and compact to modern Chinese speakers, and to some extent in the use of different lexical items (vocabulary). An essay in Classical Chinese, for example, might use half as many Chinese characters as in vernacular Chinese to relate the same content.

In terms of conciseness and compactness, Classical Chinese rarely uses words composed of two Chinese characters; nearly all words are of one syllable only. This stands directly in contrast with modern Northern Chinese varieties including Mandarin, in which two-syllable, three-syllable, and four-syllable words are extremely common, whilst although two-syllable words are also quite common within modern Southern Chinese varieties, they are still more archaic in that they use more one-syllable words than Northern Chinese varieties. This phenomenon exists, in part, because polysyllabic words evolved in Chinese to disambiguate homophones that result from sound changes. This is similar to such phenomena in English as the pen–pin merger of many dialects in the American south: because the words "pin" and "pen" sound alike in such dialects of English, a certain degree of confusion can occur unless one adds qualifiers like "ink pen" and "stick pin." Similarly, Chinese has acquired many polysyllabic words in order to disambiguate monosyllabic words that sounded different in earlier forms of Chinese but identical in one region or another during later periods. Because Classical Chinese is based on the literary examples of ancient Chinese literature, it has almost none of the two-syllable words present in modern Chinese varieties.

QMRThe Han dynasty formally recognized four sources of law: lü (律: "codified laws"), ling (令: "the emperor's order"), ke (科: "statutes inherited from previous dynasties") and bi (比: "precedents"), among which ling has the highest binding power over the other three. Most legal professionals were not lawyers but generalists trained in philosophy and literature. The local, classically trained, Confucian gentry played a crucial role as arbiters and handled all but the most serious local disputes.

QMRDeng Xiaoping Theory (simplified Chinese: 邓小平理论; traditional Chinese: 鄧小平理論; pinyin: Dèng Xiǎopíng Lǐlùn), also known as Dengism, is the series of political and economic ideologies first developed by Chinese leader Deng Xiaoping. The theory does not reject Marxism–Leninism or Mao Zedong Thought but instead seeks to adapt them to the existing socio-economic conditions of China.

China's economic growth largely owes its success to this pragmatism of Deng Xiaoping's theory. The task faced by Deng was twofold: to promote modernization while preserving the ideological unity of the CPC and its control of the difficult process of reforms.

It was generalized by the concept of the Four Modernizations.

This became the main motivation for ideological conservatism of Deng Xiaoping Theory: "Four Cardinal Principles" which the Communist Party must uphold, namely,

Upholding the basic spirit of Communism

Upholding the People's democratic dictatorship political system

Upholding the leadership of the Communist Party

Upholding Marxism–Leninism and Mao Zedong Thought

They were introduced as early as January 1963: at the Conference on Scientific and Technological Work held in Shanghai that month, Zhou Enlai called for professionals in the sciences to realize "the Four Modernizations."[2] In February 1963, at the National Conference on Agricultural Science and Technology Work, Nie Rongzhen specifically referred to the Four Modernizations as comprising agriculture, industry, national defense, and science and technology.[3] In 1975, in one of his last public acts, Zhou Enlai made another pitch for the Four Modernizations at the 4th National People's Congress. After Zhou's death and Mao’s soon thereafter, Deng Xiaoping assumed control of the party in late 1978. In December 1978 at the Third Plenum of the 11th Central Committee, Deng Xiaoping announced the official launch of the Four Modernizations, formally marking the beginning of the reform era.

The science and technology modernization although understood by Chinese leaders as being key to the transformation of industry and the economy, proved to be more of a theoretical goal versus an achievable objective. This was primarily due to decades-long isolation of Chinese scientists from the western international community, outmoded universities, and an overall lack of access to advanced scientific equipment, information technology, and management knowhow. Recognizing the need for technical assistance to spur this most important modernization, the Chinese Government elicited the support of the United Nations Development Programme (UNDP) in the fall of 1978 to scope out and provide financial resources for the implementation of an initial complement of targeted projects. The initial projects from 1979–1984 included the establishment of overseas on-the-job training and academic programs, set-up of information processing centers at key government units, and the development of methods to make informed decisions within the Chinese context based on market principles. The key advisor to the Chinese Government on behalf of the UNDP was Jack Fensterstock of the United States. This first technical assistance effort (CPR/79-001) by the UNDP led to the entry of large-scale multilateral funding agencies including the World Bank and the Asian Development Bank.

The Four Modernizations were designed to make China a great economic power by the early 21st century. These reforms essentially stressed economic self-reliance. The People's Republic of China decided to accelerate the modernization process by stepping up the volume of foreign trade by opening up its markets, especially the purchase of machinery from Japan and the West. By participating in such export-led growth, China was able to speed up its economic development through foreign investment, a more open market, access to advanced technologies, and management experience.

Controversy[edit]

On December 5, 1978 in Beijing, former red guard Wei Jingsheng posted on the Democracy Wall the Fifth Modernization as being "democracy". He was arrested a few months later and jailed for 15 years.[4]

QMRThe Four Modernizations were goals first set forth by Zhou Enlai in 1963, and enacted by Deng Xiaoping, starting in 1978, to strengthen the fields of agriculture, industry, national defense, and science and technology in China.[1] The Four Modernizations were adopted as a means of rejuvenating China's economy in 1978 following the death of Mao Zedong, and were among the defining features of Deng Xiaoping's tenure as head of the party

Shanghai has four major railway stations: Shanghai Railway Station, Shanghai South Railway Station, Shanghai West Railway Station, and Shanghai Hongqiao Railway Station. Three are connected to the metro network and serve as hubs in the railway network of China. Two main railways terminate in Shanghai: Jinghu Railway from Beijing, and Huhang Railway from Hangzhou. Hongqiao Station also serves as the main Shanghai terminus of three high-speed rail lines: the Shanghai–Hangzhou High-Speed Railway, the Shanghai–Nanjing High-Speed Railway, and the Beijing–Shanghai High-Speed Railway.

QMRThe 2010 census put Shanghai's total population at 23,019,148, a growth of 37.53% from 16,737,734 in 2000.[82][83] 20.6 million of the total population, or 89.3%, are urban, and 2.5 million (10.7％) are rural.[84] Based on the population of its total administrative area, Shanghai is the second largest of the four direct-controlled municipalities of China, behind Chongqing, but is generally considered the largest Chinese city because Chongqing's urban population is much smaller

QMRThe Four Lords of the Warring States were four powerful aristocrats of the late Warring States period of Chinese history who exerted a strong influence on the politics of their respective states in the third century BCE.[1]

During this time, the Zhou king was a mere figurehead, and seven states led by aristocratic families competed for real power. Although they were not themselves monarchs, four aristocrats stood out because of their tremendous military power and wealth: Lord Mengchang (d. 279 bce) of Qi, Lord Xinling (d. 242 bce) of Wei, Lord Pingyuan (d. 251 bce) of Zhao and Lord Chunshen (d. 237 bce) of Chu.[2]

All four were renowned for their activity in the politics of their era as well as being the persona of their state respectively at the time; they also wielded influence via the cultivation and housing of many talented house-guests, who often included learned men and tacticians. As such, they came to be the most prominent patrons of the shi (士) or scholar-knights, stimulating the intellectual life of the time. Their prestige became the inspiration for Lü Buwei when he created his academic analogue in Qin. Lord Mengchang[edit]

Main article: Lord Mengchang

Lord Mengchang was an aristocrat of the State of Qi. He was born Tian Wen, son of Tian Ying and the grandson of King Wei of Qi. He succeeded his father's fief in Xue.

Lord Xinling[edit]

Main article: Lord Xinling

Born as Wei Wuji, he was the son of King Zhao of the State of Wei and younger half-brother to King Anxi of Wei. In 277 bce, King Anxi assigned Wei Wuji the fief of Xinling.

At the height of his career, he was the supreme commander of the armed forces of the Kingdom of Wei. After stepping down, Lord Xinling became dispirited and died in 243 bce.

Lord Pingyuan[edit]

Main article: Lord Pingyuan

Born Zhao Sheng, he was a son of King Wuling of Zhao, brother of King Huiwen and uncle to King Xiaocheng. During his life, he was thrice appointed the Prime Minister of the State of Zhao.

Zhao Sheng's fief was the City of Dongwu. Lord Pingyuan was his title, and some of his famous retainers included the philosophers Xun Kuang and Gongsun Long, the Yin and Yang master Zou Yan, and the diplomat Mao Sui.

Lord Chunshen[edit]

Main article: Lord Chunshen

Born Huang Xie, he was originally a government official working for King Qingxiang of Chu, and later followed Crown Prince Wan when he spent ten years as a hostage in the Kingdom of Qin.

After the death of King Qingxiang, Prince Wan and Huang Xie returned to the Kingdom of Chu. Prince Wan was enthroned as King Kaolie of Chu, while Huang Xie was appointed Prime Minister and received the title of Lord Chunshen. For the next 25 years, Lord Chunshen remained Prime Minister of Chu, until his assassination by Li Yuan in 238 bce.

These four lords are paralleled in some books of the Records of the Grand Historian, the first of the Twenty-Four Histories of China.

In the Biographies of Lord Pingyuan and Yu Qing,[3]

At this time, [in addition to Lord Pingyuan in Zhao,] in Qi lived Mengchang, in Wei Xinling, and in Chu Chunshen. They competed to invite shi (talents).

In the Biography of Lord Chunshen,[4]

Lord Chunshen now stood as the prime minister of the Kingdom of Chu. At this time, in Qi lived Lord Mengchang, in Zhao Lord Pingyuan, and in Wei Lord Xinling. They competed to humble themselves before shi (talents) [to hire them], invited brilliant guests, and tried to defeat each other. They sustained their states and held the real power.

QMRMaths24 is a competitive, arithmetical card game aimed predominantly at primary and high school pupils. Although it can be played informally, the game was organised and operated within Southern Africa in a series of interschool, geographically increasing tournaments. The game experienced its peak during the 1990s, and is now no longer produced or played in any official manner.

History[edit]

The Maths24 game was devised, sponsored and officiated by the Old Mutual insurance company. It first began appearing in schools in 1989, and was even sold commercially to the public for a short period in the early 1990s.

Old Mutual devised the game and competitions chiefly as a promotional activity to publicise the company. The project aimed to introduce pupils and encourage their participation in mathematics via entertaining activities.

Interest in Maths24 began to taper off towards the end of the 1990s. Although official sponsorship and promotion of the game stopped, many schools and individuals continue to play the game informally.

Due to its unconventional release and propagation, the game went through a variety of rules, playing-styles and even names, being known for some time as The I-Got-It Game.

Card Description[edit]

The cards are double-sided, thin cardboard squares with sides measuring approximately 10 cm. The conventional cards bare the Old Mutual logo in the centres of each side, however with the green and white inverted on one side to differentiate. Later variations of the cards bore red, blue and black logos.

The conventional card displays four numbers, each a single digit from 1 to 9. Numbers may repeat. The cards are designed to be viewed from any angle.

The card difficulty is ranked by displaying one, two or three dots in each corner of the card in white, red and yellow respectively, as the difficulty increases.

Rules[edit]

Although official rules were later published, the game evolved with common basic rules, and many smaller variations.

Any number of competitors (usually four at most) sit around a table. The cards are placed, one at a time, in the centre of the table. The first person to cover the card with their hand and claim to have the solution would then be given the first opportunity to give their answer.

Cards are solved by using the numbers, applying only the addition, subtraction, multiplication and division operations to achieve a final mathematical solution of 24.

All four numbers must be included. The numbers can only be used once. No other mathematical operations are allowed. There may be more than one way to solve each card.

Tournament Play[edit]

The Maths24 game was intended and mostly played in tournament scenarios, ranging from school to international levels.

Competitors are initially distributed into tables of four, each with their own adjudicator. The game is played in rounds, with participants competing for points. Points are earned by solving cards, with one, two or three points assigned to cards of increasing difficulty. After claiming a solution to a card, if a participant failed to give a correct or legal solution their points would be deducted according to the difficulty. Rounds continue for a predefined length of time, at the end of which the points are tallied and the winners proceed.

Participants claim a card by covering it with their hand after it has been placed onto the table, and audibly declaring "I've got it". Later, due to minor injuries, the rules enforced covering only the centre of the card with only the index and middle fingers.

At the top competition levels, there would typically be almost no delay between a card's placement and a participant's claiming it. This is because there are only a finite number of cards that can be made, and participant's ability to give a solution from memory. Also, participants could exploit the small delay, instead, between claiming a card and giving its solution, to work out the solution then.

Variations[edit]

Because of the finite extent of the basic cards, many variations and adaptations of the game were introduced to add complexity to tournaments.

Combinations[edit]

One[1] of the officially recognized, international top-twenty Maths24 participants has shown that although the total possible number of four repeatable single-digit combinations, where order does not matter is 495, there are only 403 legally solvable possible combinations.

All 403 possible cards, in ascending order, with one possible solution are given below:

The game has been played in Shanghai since the 1960s,[citation needed] using playing cards. It is similar to the card game Maths24.

Mental arithmetic and fast thinking are necessary skills for competitive play. Pencil and paper will slow down a player, and are generally not allowed during play anyway.

In the original version of the game played with a standard 52-card deck, there are \tbinom {4+13-1}4=1820 four-card combinations.[citation needed]

Additional operations, such as square root and factorial, allow more possible solutions to the game. For instance, a set of 1,1,1,1 would be impossible to solve with only the five basic operations. However, with the use of factorials, it is possible to get 24 as (1+1+1+1)!=24.

QMRThe 24 Game is an arithmetical card game in which the objective is to find a way to manipulate four integers so that the end result is 24. For example, for the card with the numbers 4, 7, 8, 8, a possible solution is (7-(8\div8))\times4=24.

The game has been played in Shanghai since the 1960s,[citation needed] using playing cards. It is similar to the card game Maths24.

Buffet way four principals

Business management financial market tenets these are buffets four groups of tenets

Saw both of these on my timeline one after the other thought it was a cool coincidence something like that happened a lot lately

QMRThe Rieger–Nishimura lattice. Its nodes are the propositional formulas in one variable up to intuitionistic logical equivalence, ordered by intuitionistic logical implication.

QMRIn logic, a four-valued logic is any logic with four truth values. Multiple such logics were invented to deal with various practical problems.

Applications[edit]

Four-valued logic taught on technical schools is used to model signal values in digital circuits: the four values are 1, 0, Z and X. 1 and 0 stand for boolean true and false, Z stands for high impedance or open circuit and X stands for unknown value (e. g. still undefined part of the circuit design). This logic is itself a subset of 9-valued logic standard by the IEEE called IEEE 1164 and implemented e. g. in in VHDL's std_logic.

Another four-valued logic is Belnap's relevance logic. Its possible values are true, false, both (true and false), and neither (true nor false). Belnap's logic is designed to cope with multiple information sources such that if only true is found then true is assigned, if only false is found then false is assigned, if some sources say true and others say false then both is assigned, and if no information is given by any information source then neither is assigned.

There is also a SAE J1939 standard, used for CAN data transmission in heavy road vehicles, which has four logical (boolean) values: False, True, Error Condition, and Not installed (represented by values 0-3). Error Condition means there is a technical problem obstacling data acquisition. The logics for that is for example True and Error Condition=Error Condition. Not installed is used for a feature which does not exist in this vehicle, and should be disregarded for logical calculation. On CAN, usually fixed data messages are sent containing many signal values each, so a signal representing a not-installed feature will be sent anyway.

Driving needs[edit]

Each of the four corner temperaments has a driving need that energizes its behavior.

For the Melancholic, the motivation is fear of rejection and/or the unknown. They have a low self-esteem and, figuring that others do not like them, they reject others first.[12]

The Supine also has low self-esteem, but is driven to try to gain acceptance by liking and serving others.[13]

The Sanguine is driven by the need for attention, and tries to sell themselves through their charm, and accepts others before those others can reject them. Their self-esteem crashes if they are nevertheless rejected. Yet, they will regain the confidence to keep trying to impress others.

The Choleric is motivated by their goals, in which other people are tools to be used.[14]

The Phlegmatic's lack of a motivation becomes their driving need: to protect their low energy reserve.[15]

QMRThe Christian cross symbol is often shown in different shapes and sizes, in many different styles. It may be used in personal jewelry, or used on top of church buildings. It is shown both empty and in crucifix form, that is, with a figure of Christ, often referred to as the corpus (Latin for "body"), affixed to it.

The Greek cross is the most common Christian forms, the cross with arms of equal length, in common use by the 4th century. The standard Latin cross (with an elongated descending arm) is encoded at U+271D ✝ latin cross.

Roman Catholic, Anglican and Lutheran depictions of the cross are often crucifixes, in order to emphasize that it is Jesus that is important, rather than the cross in isolation. Large crucifixes are a prominent feature of some Lutheran churches, as illustrated in the article Rood. However, some other Protestant traditions depict the cross without the corpus, interpreting this form as an indication of belief in the resurrection rather than as representing the interval between the death and the resurrection of Jesus.

Crosses are a prominent feature of Christian cemeteries, either carved on gravestones or as sculpted stelas. Because of this, planting small crosses is sometimes used in countries of Christian culture to mark the site of fatal accidents, or to protest alleged deaths.

In Catholic countries, crosses are often erected on the peaks of prominent mountains, such as the Zugspitze or Mount Royal, so as to be visible over the entire surrounding area.

Contents [hide]

1 List of variants

1.1 Basic forms

1.2 Association with saints

1.3 Confessional or regional variants

1.4 Modern innovations

2 Types of artifacts

3 See also

4 References

List of variants[edit]

Basic forms[edit]

Basic variants, or early variants widespread since antiquity.

Image Name Description

Greek cross.svg Greek cross With arms of equal length. One of the most common Christian forms, in common use by the 4th century.

Cross with a longer descending arm. Along with the Greek cross, it is the most common form. It represents the cross of Jesus' crucifixion.

Patriarchal or Archbishop Cross.svg Patriarchal cross (three-bar cross) Also called an archiepiscopal cross or a crux gemina. A double cross, with the two crossbars near the top. The upper one is shorter, representing the plaque nailed to Jesus' cross. Similar to the Cross of Lorraine, though in the original version of the latter, the bottom arm is lower. The Eastern Orthodox cross adds a slanted bar near the foot.

Heraldic Globus cruciger.svg Globus cruciger Globe cross. An orb surmounted by a cross; used in royal regalia.

Cross of the pope.svg Papal cross A cross with three bars near the top. The bar are of unequal length, each one shorter than the one below.

Coa Illustration Cross Staurogram.svg Monogrammatic Cross, or Staurogram or Tau-Rho Cross The earlier visual image of the cross, already present in New Testament manuscripts as P66, P45 and P75.[1]

Calvary cross.svg Stepped cross A cross resting on a base with three steps, also called a graded or a Calvary cross.

Jerusalem Cross Also known as the Crusader's Cross. A large cross with a smaller cross in each of its angles. It was used as a symbol of the kingdom of Jerusalem

Association with saints[edit]

Image Name Description

Peter's Cross.svg Cross of St. Peter A cross with the crossbeam placed near the foot, that is associated with Saint Peter because of the tradition that he was crucified head down.

Te cross.svg Tau cross (Anthony's cross) A T-shaped cross. Also called the Saint Anthony's cross and crux commissa.

Saint Andrew's cross.svg Saltire or crux decussata An X-shaped cross associated with St. Andrew, patron of Scotland, and so a national symbol of that country. The shape is that of the cross on which Saint Andrew is said to have been martyred. Also known as St. Andrew's Cross or Andrew Cross.

Armenian cross Symbol of the Armenian Apostolic Church, and a typical feature of khachkars. Also known as the "Blooming Cross" owing to the trefoil emblems at the ends of each branch.

Ascension of the Cross bas-relief, Jvari Monastery.jpg Bolnisi cross Ancient Georgian cross and national symbol from the 5th century AD.

Caucasian Albanian Cross.svg Caucasian Albanian cross Ancient Caucasian Albanian cross and national symbol from the 4th century AD.

Croix copte égyptienne.svg Coptic ankh Shaped like the letter T surmounted by an oval or circle. Originally the Egyptian symbol for "life", it was adopted by the Copts (Egyptian Christians). Also called a crux ansata, meaning "cross with a handle".

Armenian Catholicossate of Cilicia - khatchkar.jpg Armenian cross-stone (Khachkar) A khachkar (cross-stone) is a popular symbol of Armenians.

Cantercross.svg Canterbury cross A cross with four arms of equal length which widen to a hammer shape at the outside ends. Each arm has a triangular panel inscribed in a triquetra (three-cornered knot) pattern. There is a small square panel in the center of the cross. A symbol of the Anglican and Episcopal Churches.

Monasterboice 12.jpg Celtic Cross Essentially a Latin cross, with a circle enclosing the intersection of the upright and crossbar, as in the standing High crosses.

Coptic cross The original Coptic cross has its origin in the Coptic ankh.

Coptic cross.svg New Coptic Cross This new Coptic Cross is the cross currently used by the Coptic Catholic Church and the Coptic Orthodox Church of Alexandria. It evolved from the older Coptic Crosses depicted above. A gallery of Coptic Crosses can be found here.

SaintNinoCross.jpg Grapevine cross Also known as the cross of Saint Nino of Cappadocia, who Christianised Georgia.

Original Coptic cross.svg Gnostic cross Cross used by the early Gnostic sects.

Cathar cross.svg Occitan cross Based on the counts of Toulouse's traditional coat of arms, it soon became the symbol of Occitania as a whole.

RosecrossPlexi.jpg Rose Cross A cross with a rose blooming at the center. The central symbol to all groups embracing the Esoteric Christian philosophy of the Rosicrucians.

Cross of the Russian Orthodox Church 01.svg Russian Orthodox cross (See Suppedaneum cross, below).

Serbian cross A Greek cross with 4 Cyrillic S's (C) in each of its angles, which represent the imperial motto of the Palaiologos dynasty when he resurrected the Byzantine Empire: King of Kings, Ruling Over Kings (βασιλεὺς βασιλέων, βασιλεύων βασιλευόντων - Basileus Basileōn, Basileuōn Basileuontōn). A national symbol of Serbia and symbol of the Serbian Orthodox Church.

Cross of the Russian Orthodox Church 01.svg Suppedaneum cross Also known as Russian cross, Slavic or Slavonic cross. A three-barred cross in which the short top bar represents the inscription over Jesus' head, and the lowest (usually slanting) short bar, placed near the foot, represents his footrest (in Latin, suppedaneum). This cross existed in a slightly different form (with the bottom crossbeam pointing upwards) in Byzantium, and it was changed and adopted by the Russian Orthodox Church and especially popularized in the East Slavic countries.

Mar Thoma Sliva.jpg Saint Thomas Cross The ancient cross used by Saint Thomas Christians (also known as Syrian Christians or Nasrani) in Kerala, India.[2]

Macedonian cross.svg Macedonian cross, also known as Veljusa Cross (Вељушки крст). Macedonian Christian symbol, symbol of the Macedonian Orthodox Church.

Anuradhapura Cross-Vector.svg Anuradhapura cross A symbol of Christianity in Sri Lanka.

Modern innovations[edit]

Image Name Description

Marian Cross.svg Marian Cross A term invented to refer to Pope John Paul II's combination of a Latin cross and the letter M, representing Mary being present on Calvary.

Christian Universalist symbol.svg Off Center Cross of Christian Universalism. The off-center cross was invented in late April, 1946, in a hotel room in Akron, Ohio, during the Universalist General Assembly, where a number of Universalist ministers pooled their ideas.[3]

Crucifix A cross with a representation of Jesus' body hanging from it. It is primarily used in Catholic, Anglican, Lutheran, and Eastern Orthodox churches (where the figure is painted), and it emphasizes Christ's sacrifice— his death by crucifixion.

Echmiatsin altair.jpg Altar cross A cross on a flat base to rest upon the altar of a church. The earliest known representation of an altar cross appears in a miniature in a 9th-century manuscript. By the 10th century such crosses were in common use, but the earliest extant altar cross is a 12th-century one in the Great Lavra on Mt. Athos. Mass in the Roman Rite requires the presence of a cross (more exactly, a crucifix) "on or close to" the altar.[4] Accordingly, the required cross may rest on the reredos rather than on the altar, or it may be on the wall behind the altar or be suspended above the altar.

AbunaPaulos.jpg Blessing cross Used by priests of the Eastern Catholic, Eastern Orthodox and Oriental Orthodox Churches to bestow blessings upon the faithful.

Kirkkoliput.jpg Processional cross Used to lead religious processions; sometimes, after the procession it is placed behind the altar to serve as an altar cross.

Lotharkreuz, Kaiserseite, Aachener Dom, Juni 2008.jpg Crux gemmata A cross inlaid with gems. Denotes a glorification of the cross, this form was inspired by the cult of the cross that arose after Saint Helena's discovery of the True Cross in Jerusalem in 327.

Thure Annerstedt.JPG Pectoral cross A large cross worn in front of the chest (in Latin, pectus) by some clergy.

Gotland-Stenkumla-Kirche 09.jpg Rood Large crucifix high in a church; most medieval Western churches had one, often with figures of the Virgin Mary and John the Evangelist alongside, and often mounted on a rood screen

Nasrani Syrian Christian Minnu.jpg Nasrani Minnu Wedding pendant with cross made of 21 beads used by Saint Thomas Christians of India

QMRThe Patriarchal cross is a variant of the Christian cross, the religious symbol of Christianity. Similar to the familiar Latin cross, the Patriarchal cross possesses a smaller crossbar placed above the main one, so that both crossbars are near the top. Sometimes the patriarchal cross has a short, slanted crosspiece near its foot (Orthodox cross). This slanted, lower crosspiece often appears in Byzantine Greek and Eastern European iconography, as well as Eastern Orthodox churches.

The Byzant Christianization came to the Morava empire in the year 863, provided at the request of Prince Rastic sent Byzantine Emperor Michael III.[1] The symbol, often referred to as the patriarchal cross, appeared in the Byzantine Empire in large numbers in the 10th century. For a long time, it was thought to have been given to Saint Stephen by the pope as the symbol of the apostolic Kingdom of Hungary. The two-barred cross is one of the main elements in the coats of arms of Hungary since 1190. It appeared during the reign of King Béla III, who was raised in the Byzantine court. Béla was the son of Russian princess Eufrosina Mstislavovna. The cross appears floating in the coat of arms and on the coins from this era. In medieval Kingdom of Hungary was extended Byzantine Cyril-Methodian and western Latin church was expanded later.[2]

The two-barred cross in the Hungarian coat of arms comes from the same source of Byzantine (Eastern Roman) Empire in the 12th century. Unlike the ordinary Christian cross, the symbolism and meaning of the double cross is not well understood.

The Patriarchal cross is a variant of the Christian cross, the religious symbol of Christianity. Similar to the familiar Latin cross, the Patriarchal cross possesses a smaller crossbar placed above the main one, so that both crossbars are near the top. Sometimes the patriarchal cross has a short, slanted crosspiece near its foot (Orthodox cross). This slanted, lower crosspiece often appears in Byzantine Greek and Eastern European iconography, as well as Eastern Orthodox churches.

Imagery[edit]

The top beam represents the plaque bearing the inscription "Jesus of Nazareth, King of the Jews" (often abbreviated in the Latinate "INRI", and in the Greek as "INBI"). A popular view is that the slanted bottom beam is a foot rest, however there is no evidence of foot rests ever being used during crucifixion, and it has a deeper meaning. The bottom beam may represent a balance of justice. Some sources suggest that, as one of the thieves being crucified with Jesus repented of his sin and believed in Jesus as the Messiah and was thus with Christ in Paradise, the other thief rejected and mocked Jesus and therefore descended into Hades.

Many symbolic interpretations of the double cross have been put forth. One of them says that the first horizontal line symbolized the secular power and the other horizontal line the ecclesiastic power of Byzantine emperors.[citation needed] Also, that the first cross bar represents the death and the second cross the resurrection of Jesus Christ.

Other variations[edit]

The Russian cross can be considered a modified version of the Patriarchal cross, having two smaller crossbeams, one at the top and one near the bottom, in addition to the longer crossbeam. One suggestion is the lower crossbeam represents the footrest (suppendaneum) to which the feet of Jesus were nailed. In some earlier representations (and still currently in the Greek Church) the crossbar near the bottom is straight, or slanted upwards. In later Slavic and other traditions, it came to be depicted as slanted, with the side to the viewer's left usually being higher. During 1577–1625 the Russian use of the cross was between the heads of the double-headed eagle in the coat of arms of Russia.

One tradition says that this comes from the idea that as Jesus Christ took his last breath, the bar to which his feet were nailed broke, thus slanting to the side. Another tradition holds that the slanted bar represents the repentant thief and the unrepentant thief that were crucified with Christ, the one to Jesus' right hand repenting and rising to be with God in Paradise, and one on his left falling to Hades and separation from God. In this manner it also reminds the viewer of the Last Judgment.

Still another explanation of the slanted crossbar would suggest the Cross Saltire, as tradition holds that the Apostle St. Andrew introduced Christianity to lands north and west of the Black Sea: today's Ukraine, Russia, Belarus, Moldova, and Romania.

Another form of the cross was used by the Jagiellonian dynasty in Poland. This cross now features on the coat of arms of Lithuania, where it appears on the shield of the knight. It is also the badge of the Lithuanian Air Force and forms the country's highest award for bravery, the Order of the Cross of Vytis.

The Patriarchal Cross appears on the Pahonia, used at various times as the coat of arms of Belarus.

QMROrder of the Cross of Vytis

From Wikipedia, the free encyclopedia

Order of the Cross of Vytis

Cross of Vytis.jpg

Sash, badge, and star of the Order of the Cross of Vytis

Award of Lithuania

Type State Decoration

Awarded for Heroic defense of Lithuania's freedom and independence.

Established 15 January 1991

Precedence

Next (higher) Order of Vytautas the Great

Next (lower) Order of the Lithuanian Grand Duke Gediminas

Related Medal of the Order of the Cross of Vytis

Antanas Smetona decorated with the interwar Cross of Vytis, alongside other decorations

The Order of the Cross of Vytis (Lithuanian: Vyčio Kryžiaus ordinas) is a Lithuanian Presidential Award conferred on people who heroically defended Lithuania’s freedom and independence.

History[edit]

The Order of the Cross of Vytis was the first state decoration of the pre-war Lithuania reinstated on 15 January 1991. The first to receive the First Class Order of the Cross of Vytis in the re-established Independent State of Lithuania were the victims of the 1991 January Events in Vilnius and Medininkai.

The Order of the Cross of Vytis is conferred on persons for acts of bravery performed in defending the freedom and independence of the Republic of Lithuania. According to the Law of 3 July 1997 on the Legal Status of Persons Who Participated in the Resistance to Occupations of 1940-1990 and on the Recognition of the Ranks of Volunteer Militaries Equivalent to the Ranks of the Land and Air Force Servicemen, the renewed Order "is conferred on volunteer soldiers of the armed resistance and participants of non-violent resistance to occupations."

After the restoration of independence, the leaders and members of the Senior Committee for a Free Lithuania and other outstanding representatives of the world Lithuanian community were honoured with the Order of the Cross of Vytis for services to the cause of restoration of the Independent State of Lithuania. The Order of the Cross of Vytis was also awarded to the officers of the Lithuanian Armed Forces for contributing to the withdrawal of the foreign army from Lithuania, and on the Lithuanian policemen and prosecutors for distinction in the fight against organised crime.

November 23 is a holiday in honour of the Order of the Cross of Vytis. The cross of Vytis is basically of the shape of one of the varieties of the Cross of Lorraine, namely the Cross of Jagiellons.

The Order of the Cross of Vytis is awarded in five classes:[1]

LTU Order of the Cross of Vytis - Grand Cross BAR.svg Grand Cross

LTU Order of the Cross of Vytis - Commander's Grand Cross BAR.svg Grand Commander

LTU Order of the Cross of Vytis - Commander's Cross BAR.svg Commander

LTU Order of the Cross of Vytis - Officer's Cross BAR.svg Officer

LTU Order of the Cross of Vytis - Knight's Cross BAR.svg Knight

Order statutes[edit]

The statutes of the order are as follows:[1]

Article 27. Bestowal of the Order of the Cross of Vytis

1. The Order of the Cross of Vytis shall be:conferred on the persons who heroically defended Lithuania’s freedom and independence

1) for extraordinary fortitude, clever conspiracy and command, endurance and self-sacrifice during the fights of armed and unarmed resistance to the 1940–1990 occupation, for Lithuania’s freedom and resistance or during the imprisonment and other repressions;

2) for exceptional determination, self-sacrifice and loyalty to the duty during the restoration of the Statehood of Lithuania in 1988–1990;

3) removing the threat to the Statehood of Lithuania or its integrity after 11 March 1990.

2. This Order may also be conferred on the persons who suffered the death of a hero, as well as posthumously.

3. The Order of the Cross of Vytis shall also be conferred upon for:

1) extraordinary acts of bravery performed during the fights;

2) especially clever command of fighting operations carried out by national defence units;

3) exceptional courage and fortitude when defending the country’s territory, protecting the State border, important national, economic and other civil objects;

4) exceptional bravery during the prevention of ecological disasters, natural disasters, accidents and distinction during other extraordinary circumstances when human life is in danger;

5) extraordinary bravery and fortitude when preserving public order, peace and civil rights and freedoms;

6) valour and self-sacrifice when carrying out State tasks in difficult circumstances and in the event of a great danger;

7) scientific work especially significant for the national defence;

8) especially significant projects related to the organisation of the national defence.

Article 28. Structure of the Order of the Cross of Vytis

1. The Cross of Vytis shall be of five classes.

2. Badges of the Order of the Cross of Vytis shall be as follows: the Grand Cross, the Grand Cross of Commander, the Cross of Commander, the Cross of Officer and the Cross of the Knight.

3. The Grand Cross of the Order of the Cross of Vytis shall consist of:

1) the Cross – silver, double, 50 mm high and 30 mm wide, covered with black enamel, its edges – with white enamel. In the middle of the obverse – white Vytis on the red shield. The crossed golden swords shall be behind the shield. On the reverse – inscription “For bravery“ and the year when the Order of the Cross of Vytis was instituted - 1919;

2) the Star – silver, nine rays, 85 mm in diameter. In its centre – the reduced Order of the Cross of Vytis;

3) the sash – red moiré, 100 mm wide (for women – 65 mm wide), with four black stripes on the edges.

4. The Grand Cross of Commander of the Order of the Cross of Vytis shall consist of:

1) the Cross – same as that of the Grand Cross;

2) the Star – same as that of the Grand Cross;

3) the ribbon – same as the sash of the Grand Cross, but 40 mm wide.

5. The Cross of Commander of the Order of the Cross of Vytis shall consist of:

1) the Cross – same as that of the Grand Cross;

2) the ribbon – same as the sash of the Grand Cross, but 40 mm wide.

6. the Cross of Officer of the Order of the Cross of Vytis shall consist of:

1) the Cross – same as that of the Grand Cross, but 42 mm high and 25 mm wide;

2) the breast ribbon – same as the sash of the Grand Cross, but 32 mm wide, with two silver oak twigs.

7. the Cross of the Knight of the Order of the Cross of Vytis shall consist of:

1) the Cross – same as that of the Grand Cross, but 42 mm high and 25 mm wide;

2) the breast ribbon – same as the sash of the Grand Cross, but 32 mm wide, with one silver oak twig.

QMRThe Orthodox, Byzantine[1][2][3][need quotation to verify] or Russian (Orthodox) Cross,[4][5] also known as the Suppedaneum cross,[6] is a variation of the Christian cross, commonly[quantify] found in Eastern Orthodox Churches, as well as the Eastern Catholic Churches of Byzantine rite and the Society for Eastern Rite Anglicanism. The cross has three horizontal crossbeams—the top one represents the plate inscribed with INRI, and the bottom one, a footrest. In the Russian Orthodox tradition, the lower beam is slanted: the side to Christ's right is usually higher. This is because the footrest slants upward toward the penitent thief St. Dismas, who was (according to tradition[citation needed]) crucified on Jesus' right, and downward toward impenitent thief Gestas. It is also a common perception that the foot-rest points up, toward Heaven, on Christ’s right hand-side, and downward, to Hades, on Christ’s left. One of the Orthodox Church’s Friday prayers clearly explains the meaning: "In the midst, between two thieves, was Your Cross found as the balance-beam of righteousness; For while one was led down to hell by the burden of his blaspheming, the other was lightened of his sins unto the knowledge of things divine, O Christ God glory to You."[citation needed] The earliest version of a slanted footstool can be found in Jerusalem, but throughout the Eastern Christian world until the 17th century, the footstool is slanted the other way, pointing upwards rather than downwards, making the downward footstool a Russian innovation. In the Greek and most other Orthodox Churches, the footrest remains straight, as in earlier representations. Common variations include the "Cross over Crescent" and the "Calvary cross".

One variation of the Orthodox Cross is the 'Cross over Crescent', which is sometimes accompanied by "Gabriel perched on the top of the Cross blowing his trumpet."[7] Didier Chaudet, in the academic journal China and Eurasia Forum Quarterly, writes that an "emblem of the Orthodox Church is a cross on top on a crescent. It is said that this symbol was devised by Ivan the Terrible, after the conquest of the city of Kazan, as a symbol of the victory of Christianity over Islam through his soldiers"; the Orthodox World Encyclopaedia concurs with this view.[8][9][10] However, B.A. Uspensky offers another view, stating that in pre-Christian times, the 'Cross over Crescent' symbolized the sun and the moon, and that in the Christian Era, the cross is a symbol of Christ and the moon is a symbol of the Virgin Mary.[11]

In Russia, the top crossbeam can be absent; however, in the Russian North it can be attached on top of the vertical beam.[12]

A variation is a monastic "Calvary Cross", in which the cross is situated atop the hill of Calvary, its slopes symbolized by steps. To the viewer's left is the Holy Lance, with which Jesus was wounded in his side, and to the right, a pole topped by a vinegared hyssop sponge. Under Calvary are Adam's skull and bones;[2] the right-arm bone is usually above the left one, and believers fold their arms across their chests in this way during Orthodox communion. Around the cross are abbreviations in Church Slavonic. This type of cross is usually embroidered on a schema-monk's robe.

Between 1577–1625, the Russian Orthodox Cross was depicted between the heads of a double-headed eagle in the coat of arms of Russia. It was drawn on military banners until the end of the 17th century.[13]

QMRThe Holy Lance (German: Heilige Lanze), also known as the Holy Spear, the Spear of Destiny, or the Lance of Longinus, is the name given to the lance that pierced the side of Jesus as he hung on the cross, according to the Gospel of John. Several churches across the world claim to possess this lance.

This was during Jesus crucifixion

QMRCross over Crescent variation of the Orthodox Cross at the Ss. Peter and Paul Cathedral

The Descent of Christ into hell, a mosaic from Hosios Loukas in Greece, the 11th century

Constantine and Helena in the Exaltation of the Cross, a mosaic from Hosios Loukas in Greece, the 11th century

The Crucifixion, a 13th-century mosaic from Constantinople

15th-century Russian depiction in which the traditional INRI plank is instead marked with "ЦР҃Ь СЛ҃ВЫ", standing for "King of Glory"

Coat of arms of Russia from the seal of Ivan IV (the Terrible), 1577

Coat of arms of Russia from the seal of Fyodor I, 1589

A rider with the banner from an icon Blessed Be the Host of the King of Heaven (Church Militant), 1550s

A 17th-century miniature of the Battle of Kulikovo (1380). A warrior bears a red banner with a cross

A copper cross typical for Old believers

A cross of a Russian Orthodox priest

A modern memorial to Ss. Cyril and Methodius in Khanty-Mansiysk, Russia

Coat of arms of Kherson Governorate, Russian Empire, 1878

Sainte-Geneviève-des-Bois Russian Cemetery, the resting place of many eminent Russian émigrés.

QMRTwo-barred cross

From Wikipedia, the free encyclopedia

A two-barred cross is like a Latin cross with an extra bar added. The lengths and placement of the bars (or "arms") vary, and most of the variations are interchangeably called either of cross of Lorraine, the patriarchal cross or the archiepiscopal cross.[1]

Contents [hide]

1 The two bars

2 Decorations

3 Heraldic use

4 In print

5 In medicine and botany

6 In chess

7 References

The two bars[edit]

The two bars can be placed tight together (condensed) or far apart. They can be symmetrically spaced either around the middle, or above or below the middle. One asymmetrical variation has one bar near the top and the other just below the middle. Finally the bars can be of equal length, or with one shorter than the other.

Decorations[edit]

The ends of the arms can be decorated according to different styles. A style with round or rounded ends is called treflée or botonée (from French bouton) in heraldic use. The same style is called budded, apostles' or cathedral cross in religious use.[2] A straight and pointy style called pattée also includes maltese cross variations,[3] and finally a pointed style called aiguisé.[4]

Heraldic use[edit]

The crosses appear in heraldic use in the second century A.D.[5] A balanced cross is used in the Coat of arms of Hungary as well as in several small shields within shields of Vytis. An outlined balanced cross (equal length outlined bars on equal distances) is used on coat of arms shields and order medals [6]

In Slovakia both the flag, their coat or arms and several municipal symbols include a double cross, where graded bars are more common than equally long bars, and balanced distances along the vertical line are more common.[7]

In print[edit]

In typography the double cross is called double dagger, double obelisk and diesis.[8]

In medicine and botany[edit]

The International Union Against TB and Lung Diseases is since 1902[9] using a red cross variation in its logotype. The two equally long bars are on the upper half of the cross and all six ends are aiguisé.[10]

In botany a balanced cross (equal length bars on equal distances) is used to mark very poisonous plants [11]

In chess[edit]

Used to symbolize checkmate [12]

QMRThe Cross of Lorraine (French: Croix de Lorraine) was originally a heraldic cross. The two-barred cross consists of a vertical line crossed by two shorter horizontal bars. In most renditions, the horizontal bars are "graded" with the upper bar being the shorter, though variations with the bars of equal length are also seen. The Lorraine name has come to signify several cross variations, including the patriarchal cross with its bars near the top.

Design[edit]

The Cross of Lorraine consists of one vertical and two horizontal bars. It is a heraldic cross, used by the Dukes of Lorraine. Duke René, who reigned between 1431 and 1453, "was a major transmitter of the Hermetic tradition in Italy and had the cross of Lorraine as his personal sigil".[1] This cross is related to the Crusader's cross, and the six globes of the Medici family.

History[edit]

The Lorraine cross was carried to the Crusades by the original Knights Templar, granted to them for their use by the Patriarch of Jerusalem.[2][3]

In the Catholic Church, an equal-armed Lorraine Cross denotes the office of archbishop.

Symbol in France[edit]

The flag of Free France is a regular flag of France that has been defaced with a Lorraine cross.

In France, the Cross of Lorraine was the symbol of Free France during World War II, the liberation of France from Nazi Germany, and Gaullism and includes several variations of a two barred cross.

The Cross of Lorraine is an emblem of Lorraine in eastern France. Between 1871 and 1918 (and again between 1940 and 1944), the northern third of Lorraine was annexed to Germany, along with Alsace. During that period the Cross served as a rallying point for French ambitions to recover its lost provinces. This historical significance lent it considerable weight as a symbol of French patriotism. During the War, Capitaine de corvette Thierry d'Argenlieu suggested the Cross of Lorraine as the symbol of the Free French Forces led by Charles de Gaulle as an answer to the Nazi swastika.

The Cross was displayed on the flags of Free French warships, and the fuselages of Free French aircraft. The medal of the Order of Liberation bears the Cross of Lorraine.

De Gaulle himself is memorialised by a 43-metre (141 ft) high Cross of Lorraine in his home village of Colombey-les-Deux-Églises. The Cross of Lorraine was later adopted by Gaullist political groups such as the Rally for the Republic.

The French frigate Aconit, named after the corvette Aconit of the Free French Navy, flies the Cross of Lorraine on her foremast.

The tampion of the Rubis features the Cross of Lorraine in honour of the Free French submarine Rubis.

The Free French naval jack and French naval honour jack.

New World[edit]

French Jesuit missionaries and settlers to the New World carried the Cross of Lorraine c. 1750–1810. The symbol was said to have helped the missionaries to convert the native peoples they encountered, because the two-armed cross resembled existing local imagery.[4]

European heraldry[edit]

The flag and the coat of arms of Slovakia both include the double cross. It was introduced to the territory of today's Slovakia by Constantine (Cyril) and Methodius, who brought Christianity to Slavic empire of Great Moravia in the 9th century. In Slovakia, the double cross as a symbol of Lorraine is considered to have arisen when the Great Moravian king Svätopluk I "passed" it to Zwentibold of Lorraine, the godchild of Svätopluk and son of the emperor Arnulf of Carinthia.[citation needed]

The coat of arms of Hungary depicts a double cross, which is often attributed to Byzantine influence as King Béla III of Hungary was raised in the Byzantine Empire in the 12th century, and it was during his rule when the double cross became a symbol of Hungary. Also the 'dual cross' is the consonant 'gy' in ancient Hungarian runic writing which reads "egy" (one) when it stands alone mostly, if not always, with "God" meaning.

A golden double cross with equal bars, known as the Cross of Jagiellons, was used by Grand Duke of Lithuania and King of Poland Jogaila since his conversion to Christianity in 1386, as a personal insignia and was introduced in the Coat of Arms of Lithuania. Initially, the lower bar of the cross was longer than the upper, since it originates from the Hungarian type of the double cross. It later became the symbol of Jagiellon dynasty and is one of the national symbols of Lithuania, featured in the Order of the Cross of Vytis and the badge of the Lithuanian Air Force.

The double-barred cross is one of the national symbols in Belarus, both as the Jagiellon Cross and as the Cross of St. Euphrosyne of Polatsk, an important religious artifact. The symbol is supposed to have Byzantine roots and is used by the Belarusian Greek Catholic Church as a symbol uniting Eastern-Byzantine and Western-Latin church traditions. The Belarusian Cross can be found on the traditional coat of arms of Belarus, the Pahonia.

Hungarian arms, depicting the cross on the sinister side.

Order of the Cross of Vytis, a Lithuanian presidential award

The Cross of Saint Euphrosyne

Miscellaneous uses[edit]

The cross is used as an emblem by the American Lung Association and related organizations through the world, and as such is familiar from their Christmas Seals program. Its use was suggested in 1902 by Paris physician Gilbert Sersiron as a symbol for the "crusade" against tuberculosis.[5][6]

For its defense of France in World War I, the American 79th Infantry Division was nicknamed the "Cross of Lorraine" Division; its insignia is the cross. The German 79th Infantry Division of World War II used the cross of Lorraine as its insignia because its first attack was in the Lorraine region. The insignia was redesignated effective December 1, 2009, for the 79th US Army Reserve Sustainment Support Command in Los Alamitos, California.[7]

In the television series Magnum, P.I., Thomas Magnum and his Vietnam War comrades were all shown to wear rings that bore the cross of Lorraine.

Ironically, the cross is also used as the symbol of the fascist Norsefire party in the film version of the graphic novel V for Vendetta.

The cross of Lorraine was previously used in the SABRE, Apollo, and Worldspan global distribution systems (GDS) as a delimiter in various input formats, however, the latest version of the Graphical User Interface for each system uses a different symbol: Apollo displays it as a plus sign, Worldspan as a number sign, and Sabre as a yen symbol.

The "Cross of Lorraine" symbol appears in Unicode as U+2628 ☨ cross of lorraine (HTML ☨). It is not to be confused with U+2021 ‡ double dagger (HTML ‡ · &Dagger;).

The Cross of Lorraine was noted as a symbol of the Free French in the film Casablanca. A ring bearing the Cross was worn by Norwegian underground agent Berger and shown to one of the movies heroes (Victor Laszlo) as proof of loyalty.

It has also been used as a symbol for the city Roeselare (black cross) and Ypres (red cross) in Belgium.

282 (East Ham) Squadron, Air Training Corps have the "Cross of Lorraine" on their unit crest in honour of their previous Squadron President, Odette Hallowes who worked for an independent French section of the Special Operations Executive during World War II.

The Cross is also used as the symbol for the Celebritarian Corporation/art movement, created and led by musician/painter Marilyn Manson.

The Anglo-Canadian hardcore punk band Gallows use the cross as a symbol, as well as having named a song after it.

The Lorraine Cross is referenced in Sigma Phi Lambda Fraternity's brothers' song. The local fraternity was established in 1935 on the campus of LaSalle University, a Roman-Catholic university run by the De LaSalle Christian Brothers located in Philadelphia, PA, USA. Sigma Phi Lambda recently celebrated its 80th Anniversary.

QMRWriting in the Roman era, Clement of Alexandria gives some idea of the importance of astronomical observations to the sacred rites:

And after the Singer advances the Astrologer (ὡροσκόπος), with a horologium (ὡρολόγιον) in his hand, and a palm (φοίνιξ), the symbols of astrology. He must know by heart the Hermetic astrological books, which are four in number. Of these, one is about the arrangement of the fixed stars that are visible; one on the positions of the sun and moon and five planets; one on the conjunctions and phases of the sun and moon; and one concerns their risings.[31]

The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:[1]:p. 78

Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY.

Contents [hide]

1 Proof

2 History

3 References

4 Bibliography

5 External links

Proof[edit]

A formal proof of the theorem is as follows: Let the perpendiculars XX′ and XX″ be dropped from the point X on the straight lines AM and DM respectively. Similarly, let YY′ and YY″ be dropped from the point Y perpendicular to the straight lines BM and CM respectively.

Proof of Butterfly theorem

Now, since

\triangle MXX' \sim \triangle MYY',\,

{MX \over MY} = {XX' \over YY'},

\triangle MXX'' \sim \triangle MYY'',\,

{MX \over MY} = {XX'' \over YY''},

\triangle AXX' \sim \triangle CYY'',\,

{XX' \over YY''} = {AX \over CY},

\triangle DXX'' \sim \triangle BYY',\,

{XX'' \over YY'} = {DX \over BY},

From the preceding equations, it can be easily seen that

\left({MX \over MY}\right)^2 = {XX' \over YY' } {XX'' \over YY''},

{} = {AX.DX \over CY.BY},

{} = {PX.QX \over PY.QY},

{} = {(PM-XM).(MQ+XM) \over (PM+MY).(QM-MY)},

{} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2},

since PM = MQ.

Now,

{ (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}.

So, it can be concluded that MX = MY, or M is the midpoint of XY.

An alternate proof can be found using projective geometry.[2]

History[edit]

Proving the butterfly theorem was posed as a problem by William Wallace in The Gentlemen's Mathematical Companion (1803). Three solutions were published in 1804, and in 1805 Sir William Herschel posed the question again in a letter to Wallace. Rev. Thomas Scurr asked the same question again in 1814 in the Gentlemen's Diary or Mathematical Repository.[3]

The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:[1]:p. 78

Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY.

Contents [hide]

1 Proof

2 History

3 References

4 Bibliography

5 External links

Proof[edit]

Proof of Butterfly theorem

Now, since

\triangle MXX' \sim \triangle MYY',\,

{MX \over MY} = {XX' \over YY'},

\triangle MXX'' \sim \triangle MYY'',\,

{MX \over MY} = {XX'' \over YY''},

\triangle AXX' \sim \triangle CYY'',\,

{XX' \over YY''} = {AX \over CY},

\triangle DXX'' \sim \triangle BYY',\,

{XX'' \over YY'} = {DX \over BY},

From the preceding equations, it can be easily seen that

\left({MX \over MY}\right)^2 = {XX' \over YY' } {XX'' \over YY''},

{} = {AX.DX \over CY.BY},

{} = {PX.QX \over PY.QY},

{} = {(PM-XM).(MQ+XM) \over (PM+MY).(QM-MY)},

{} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2},

since PM = MQ.

Now,

{ (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}.

So, it can be concluded that MX = MY, or M is the midpoint of XY.

An alternate proof can be found using projective geometry.[2]

History[edit]

Other properties[edit]

In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect.[19]

Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side.[19]

If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side.[19]

In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect.[19]

Brahmagupta quadrilaterals[edit]

A Brahmagupta quadrilateral[24] is a cyclic quadrilateral with integer sides, integer diagonals, and integer area. All Brahmagupta quadrilaterals with sides a, b, c, d, diagonals e, f, area K, and circumradius R can be obtained by clearing denominators from the following expressions involving rational parameters t, u, and v:

a=[t(u+v)+(1-uv)][u+v-t(1-uv)]

b=(1+u^2)(v-t)(1+tv)

c=t(1+u^2)(1+v^2)

d=(1+v^2)(u-t)(1+tu)

e=u(1+t^2)(1+v^2)

f=v(1+t^2)(1+u^2)

K=uv[2t(1-uv)-(u+v)(1-t^2)][2(u+v)t+(1-uv)(1-t^2)]

4R=(1+u^2)(1+v^2)(1+t^2).

Properties of cyclic quadrilaterals that are also orthodiagonal[edit]

Circumradius and area[edit]

For a cyclic quadrilateral that is also orthodiagonal (has perpendicular diagonals), suppose the intersection of the diagonals divides one diagonal into segments of lengths p1 and p2 and divides the other diagonal into segments of lengths q1 and q2. Then[25] (the first equality is Proposition 11 in Archimedes' Book of Lemmas)

D^2=p_1^2+p_2^2+q_1^2+q_2^2=a^2+c^2=b^2+d^2

where D is the diameter of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. These equations imply that the circumradius R can be expressed as

R=\tfrac{1}{2}\sqrt{p_1^2+p_2^2+q_1^2+q_2^2}

or, in terms of the sides of the quadrilateral, as

R=\tfrac{1}{2}\sqrt{a^2+c^2}=\tfrac{1}{2}\sqrt{b^2+d^2}.

It also follows that

a^2+b^2+c^2+d^2=8R^2.

Thus, according to Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals p and q, and the distance x between the midpoints of the diagonals as

R=\sqrt{\frac{p^2+q^2+4x^2}{8}}.

A formula for the area K of a cyclic orthodiagonal quadrilateral in terms of the four sides is obtained directly when combining Ptolemy's theorem and the formula for the area of an orthodiagonal quadrilateral. The result is

K=\tfrac{1}{2}(ac+bd).

Anticenter and collinearities[edit]

Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent.[19]:p.131;[20] These line segments are called the maltitudes,[21] which is an abbreviation for midpoint altitude. Their common point is called the anticenter. It has the property of being the reflection of the circumcenter in the "vertex centroid". Thus in a cyclic quadrilateral, the circumcenter, the "vertex centroid", and the anticenter are collinear.[20]

If the diagonals of a cyclic quadrilateral intersect at P, and the midpoints of the diagonals are M and N, then the anticenter of the quadrilateral is the orthocenter of triangle MNP. The vertex centroid is the midpoint of the line segment joining the midpoints of the diagonals.[20]

In a cyclic quadrilateral, the "area centroid" Ga, the "vertex centroid" Gv, and the intersection P of the diagonals are collinear. The distances between these points satisfy[22]

PG_a = \tfrac{4}{3}PG_v.

Other properties[edit]

Japanese theorem

In a cyclic quadrilateral ABCD, the incenters in triangles ABC, BCD, CDA, and DAB are the vertices of a rectangle. This is one of the theorems known as the Japanese theorem. The orthocenters of the same four triangles are the vertices of a quadrilateral congruent to ABCD, and the centroids in those four triangles are vertices of another cyclic quadrilateral.[3]

In a cyclic quadrilateral ABCD with circumcenter O, let P be the point where the diagonals AC and BD intersect. Then angle APB is the arithmetic mean of the angles AOB and COD. This is a direct consequence of the inscribed angle theorem and the exterior angle theorem.

There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression.[23]

If a cyclic quadrilateral has side lengths that form an arithmetic progression the quadrilateral is also ex-bicentric.

If the opposite sides of a cyclic quadrilateral are extended to meet at E and F, then the internal angle bisectors of the angles at E and F are perpendicular.[8]

Parameshvara's formula[edit]

A cyclic quadrilateral with successive sides a, b, c, d and semiperimeter s has the circumradius (the radius of the circumcircle) given by[11][18]

R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}.

This was derived by the Indian mathematician Vatasseri Parameshvara in the 15th century.

Using Brahmagupta's formula, Parameshvara's formula can be restated as

4KR=\sqrt{(ab+cd)(ac+bd)(ad+bc)}

where K is the area of the cyclic quadrilateral

Angle formulas[edit]

For a cyclic quadrilateral with successive sides a, b, c, d, semiperimeter s, and angle A between sides a and d, the trigonometric functions of A are given by[17]

\cos A = \frac{a^2 + d^2 - b^2 - c^2}{2(ad + bc)},

\sin A = \frac{2\sqrt{(s-a)(s-b)(s-c)(s-d)}}{(ad+bc)},

\tan \frac{A}{2} = \sqrt{\frac{(s-a)(s-d)}{(s-b)(s-c)}}.

The angle θ between the diagonals satisfies[4]:p.26

\tan \frac{\theta}{2} = \sqrt{\frac{(s-b)(s-d)}{(s-a)(s-c)}}.

If the extensions of opposite sides a and c intersect at an angle φ, then

\cos{\frac{\varphi}{2}}=\sqrt{\frac{(s-b)(s-d)(b+d)^2}{(ab+cd)(ad+bc)}}

where s is the semiperimeter.[4]:p.31

Diagonals[edit]

In a cyclic quadrilateral with successive vertices A, B, C, D and sides a = AB, b = BC, c = CD, and d = DA, the lengths of the diagonals p = AC and q = BD can be expressed in terms of the sides as[4]:p.25,[11][12]:p. 84

p = \sqrt{\frac{(ac+bd)(ad+bc)}{ab+cd}} and q = \sqrt{\frac{(ac+bd)(ab+cd)}{ad+bc}}

so showing Ptolemy's theorem

pq = ac+bd.

According to Ptolemy's second theorem,[4]:p.25,[11]

\frac {p}{q}= \frac{ad+bc}{ab+cd}

using the same notations as above.

For the sum of the diagonals we have the inequality[13]

p+q\ge 2\sqrt{ac+bd}.

Equality holds if and only if the diagonals have equal length, which can be proved using the AM-GM inequality.

Moreover,[14]:p.64,#1639

p+q)^2 \leq (a+c)^2+(b+d)^2.

In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.

If M and N are the midpoints of the diagonals AC and BD, then[15]

\frac{MN}{EF}=\frac{1}{2}\left |\frac{AC}{BD}-\frac{BD}{AC}\right|

where E and F are the intersection points of the extensions of opposite sides.

If ABCD is a cyclic quadrilateral where AC meets BD at E, then[16]

\frac{AE}{CE}=\frac{AB}{CB}\cdot\frac{AD}{CD}.

A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common.[12]:p. 84

Area[edit]

The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula[4]:p.24

K=\sqrt{(s-a)(s-b)(s-c)(s-d)} \,

where s, the semiperimeter, is s =

(a + b + c + d). It is a corollary to Bretschneider's formula since opposite angles are supplementary. If also d = 0, the cyclic quadrilateral becomes a triangle and the formula is reduced to Heron's formula.

The cyclic quadrilateral has maximal area among all quadrilaterals having the same sequence of side lengths. This is another corollary to Bretschneider's formula. It can also be proved using calculus.[7]

Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals,[8] which by Brahmagupta's formula all have the same area. Specifically, for sides a, b, c, and d, side a could be opposite any of side b, side c, or side d.

The area of a cyclic quadrilateral with successive sides a, b, c, d and angle B between sides a and b can be expressed as[4]:p.25

K = \tfrac{1}{2}(ab+cd)\sin{B}

or[4]:p.26

K = \tfrac{1}{2}(ac+bd)\sin{\theta}

where θ is either angle between the diagonals. Provided A is not a right angle, the area can also be expressed as[4]:p.26

K = \tfrac{1}{4}(a^2-b^2-c^2+d^2)\tan{A}.

Another formula is[9]:p.83

\displaystyle K=2R^2\sin{A}\sin{B}\sin{\theta}

where R is the radius of the circumcircle. As a direct consequence,[10]

K\le 2R^2

where there is equality if and only if the quadrilateral is a square.

Special cases[edit]

Any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles. A bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential.

Characterizations[edit]

A cyclic quadrilateral ABCD

A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent. This common point is the circumcenter.[1]

A convex quadrilateral ABCD is cyclic if and only if its opposite angles are supplementary, that is[1]

\alpha + \gamma = \beta + \delta = \pi = 180^{\circ}.

The direct theorem was Proposition 22 in Book 3 of Euclid's Elements.[2] Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle.Another necessary and sufficient condition for a convex quadrilateral ABCD to be cyclic is that an angle between a side and a diagonal is equal to the angle between the opposite side and the other diagonal.[3] That is, for example,

\angle ACB = \angle ADB.

Ptolemy's theorem expresses the product of the lengths of the two diagonals e and f of a cyclic quadrilateral as equal to the sum of the products of opposite sides:[4]:p.25

\displaystyle ef = ac + bd.

The converse is also true. That is, if this equation is satisfied in a convex quadrilateral, then it is a cyclic quadrilateral.

If two lines, one containing segment AC and the other containing segment BD, intersect at X, then the four points A, B, C, D are concyclic if and only if[5]

\displaystyle AX\cdot XC = BX\cdot XD.

The intersection X may be internal or external to the circle. In the former case, the cyclic quadrilateral is ABCD, and in the latter case, the cyclic quadrilateral is ABDC. When the intersection is internal, the equality states that the product of the segment lengths into which X divides one diagonal equals that of the other diagonal. This is known as the intersecting chords theorem since the diagonals of the cyclic quadrilateral are chords of the circumcircle.

Yet another characterization is that a convex quadrilateral ABCD is cyclic if and only if[6]

\tan{\frac{\alpha}{2}}\tan{\frac{\gamma}{2}}=\tan{\frac{\beta}{2}}\tan{\frac{\delta}{2}}=1.

QMR In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

The word cyclic is from the Ancient Greek κύκλος (kuklos) which means "circle" or "wheel".

All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle.

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

The word cyclic is from the Ancient Greek κύκλος (kuklos) which means "circle" or "wheel".

Related theorems[edit]

Heron's formula for the area of a triangle is the special case obtained by taking d = 0.

The relationship between the general and extended form of Brahmagupta's formula is similar to how the law of cosines extends the Pythagorean theorem.

Non-trigonometric proof[edit]

An alternative, non-trigonometric proof utilizes two applications of Heron's triangle area formula on similar triangles.[1]

Extension to non-cyclic quadrilaterals[edit]

In the case of non-cyclic quadrilaterals, Brahmagupta's formula can be extended by considering the measures of two opposite angles of the quadrilateral:

K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\theta}

where θ is half the sum of any two opposite angles. (The choice of which pair of opposite angles is irrelevant: if the other two angles are taken, half their sum is 180° − θ. Since cos(180° − θ) = −cos θ, we have cos2(180° − θ) = cos2 θ.) This more general formula is known as Bretschneider's formula.

It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 180°. Consequently, in the case of an inscribed quadrilateral, θ is 90°, whence the term

abcd\cos^2\theta=abcd\cos^2 \left(90^\circ\right)=abcd\cdot0=0, \,

giving the basic form of Brahmagupta's formula. It follows from the latter equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.

A related formula, which was proved by Coolidge, also gives the area of a general convex quadrilateral. It is[2]

K=\sqrt{(s-a)(s-b)(s-c)(s-d)-\textstyle{1\over4}(ac+bd+pq)(ac+bd-pq)}\,

where p and q are the lengths of the diagonals of the quadrilateral. In a cyclic quadrilateral, pq = ac + bd according to Ptolemy's theorem, and the formula of Coolidge reduces to Brahmagupta's formula.

Proof[edit]

Diagram for reference

Trigonometric proof[edit]

Here the notations in the figure to the right are used. The area K of the cyclic quadrilateral equals the sum of the areas of △ADB and △BDC:

= \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin C.

But since ABCD is a cyclic quadrilateral, ∠DAB = 180° − ∠DCB. Hence sin A = sin C. Therefore,

K = \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin A

K^2 = \frac{1}{4} (pq + rs)^2 \sin^2 A

4K^2 = (pq + rs)^2 (1 - \cos^2 A) = (pq + rs)^2 - (pq + rs)^2 \cos^2 A.\,

Solving for common side DB, in △ADB and △BDC, the law of cosines gives

p^2 + q^2 - 2pq\cos A = r^2 + s^2 - 2rs\cos C. \,

Substituting cos C = −cos A (since angles A and C are supplementary) and rearranging, we have

2 (pq + rs) \cos A = p^2 + q^2 - r^2 - s^2. \,

Substituting this in the equation for the area,

4K^2 = (pq + rs)^2 - \frac{1}{4}(p^2 + q^2 - r^2 - s^2)^2

16K^2 = 4(pq + rs)^2 - (p^2 + q^2 - r^2 - s^2)^2.

The right-hand side is of the form a2 − b2 = (a − b)(a + b) and hence can be written as

[2(pq + rs) - p^2 - q^2 + r^2 +s^2][2(pq + rs) + p^2 + q^2 -r^2 - s^2] \,

which, upon rearranging the terms in the square brackets, yields

= [ (r+s)^2 - (p-q)^2 ][ (p+q)^2 - (r-s)^2 ] \,

= (q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s). \,

Introducing the semiperimeter S =

p + q + r + s

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Proof[edit]

Diagram for reference

Trigonometric proof[edit]

Here the notations in the figure to the right are used. The area K of the cyclic quadrilateral equals the sum of the areas of △ADB and △BDC:

= \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin C.

But since ABCD is a cyclic quadrilateral, ∠DAB = 180° − ∠DCB. Hence sin A = sin C. Therefore,

K = \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin A

K^2 = \frac{1}{4} (pq + rs)^2 \sin^2 A

4K^2 = (pq + rs)^2 (1 - \cos^2 A) = (pq + rs)^2 - (pq + rs)^2 \cos^2 A.\,

Solving for common side DB, in △ADB and △BDC, the law of cosines gives

p^2 + q^2 - 2pq\cos A = r^2 + s^2 - 2rs\cos C. \,

Substituting cos C = −cos A (since angles A and C are supplementary) and rearranging, we have

2 (pq + rs) \cos A = p^2 + q^2 - r^2 - s^2. \,

Substituting this in the equation for the area,

4K^2 = (pq + rs)^2 - \frac{1}{4}(p^2 + q^2 - r^2 - s^2)^2

16K^2 = 4(pq + rs)^2 - (p^2 + q^2 - r^2 - s^2)^2.

The right-hand side is of the form a2 − b2 = (a − b)(a + b) and hence can be written as

[2(pq + rs) - p^2 - q^2 + r^2 +s^2][2(pq + rs) + p^2 + q^2 -r^2 - s^2] \,

which, upon rearranging the terms in the square brackets, yields

= [ (r+s)^2 - (p-q)^2 ][ (p+q)^2 - (r-s)^2 ] \,

= (q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s). \,

Introducing the semiperimeter S =

p + q + r + s

16K^2 = 16(S-p)(S-q)(S-r)(S-s). \,

Taking the square root, we get

K = \sqrt{(S-p)(S-q)(S-r)(S-s)}.

Formula[edit]

Brahmagupta's formula gives the area K of a cyclic quadrilateral whose sides have lengths a, b, c, d as

K=\sqrt{(s-a)(s-b)(s-c)(s-d)}

where s, the semiperimeter, is defined to be

s=\frac{a+b+c+d}{2}.

This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.

If the semiperimeter is not used, Brahmagupta's formula is

K=\frac{1}{4}\sqrt{(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)}.

Another equivalent version is

K=\frac{\sqrt{(a^2+b^2+c^2+d^2)^2+8abcd-2(a^4+b^4+c^4+d^4)}}{4}\cdot

QMRIn Euclidean geometry, Brahmagupta's formula finds the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides.

Proof[edit]

Proof of the theorem.

We need to prove that AF = FD. We will prove that both AF and FD are in fact equal to FM.

To prove that AF = FM, first note that the angles FAM and CBM are equal, because they are inscribed angles that intercept the same arc of the circle. Furthermore, the angles CBM and CME are both complementary to angle BCM (i.e., they add up to 90°), and are therefore equal.Finally, the angles CME and FMA are the same. Hence, AFM is an isosceles triangle, and thus the sides AF and FM are equal.

The proof that FD = FM goes similarly: the angles FDM, BCM, BME and DMF are all equal, so DFM is an isosceles triangle, so FD = FM. It follows that AF = FD, as the theorem claims.

Quadrant Model of Reality

Tuesday, May 3, 2016

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