Quadrant Model of Reality Book 17 Philosophy and Math

Philosophy Chapter

QMROctane is a hydrocarbon and an alkane with the chemical formula C8H18, and the condensed structural formula CH3(CH2)6CH3. Octane has many structural isomers that differ by the amount and location of branching in the carbon chain. One of these isomers, 2,2,4-trimethylpentane (isooctane) is used as one of the standard values in the octane rating scale. Octane is a component of gasoline (petrol).... See More

Quattro Arcade[edit] Quattro Arcade Quattro Arcade Cover.jpg Developer(s) Codemasters Publisher(s) Camerica Platform(s) Nintendo Entertainment System Release date(s) 1992 Genre(s) Platform, compilation Mode(s) Multi-player Quattro Arcade is a collection of 4 platform action video games released for the NES in 1992. It is made up of CJ's Elephant Antics, Stunt Buggies, F16 Renegade, and Go! Dizzy Go! (part of the Dizzy series).

Quattro Sports[edit] Quattro Sports Quattro Sports Cover.jpg NES cover Developer(s) Codemasters Publisher(s) Camerica Series Quattro Platform(s) Amiga, Nintendo Entertainment System, ZX Spectrum Release date(s) 1990 Genre(s) Sports, compilation Quattro Sports is a video game made by Codemasters and released for the Nintendo Entertainment System (NES) and Amiga. The NES version was not licensed by Nintendo. It features four sports games, Tennis Simulator, BMX Simulator, Soccer Simulator, and Pro Baseball.

QMRQuattro ("four" in Italian) is a series of video game compilations (each with four games) released in the 1990s. They consisted of games developed by Codemasters and were published by Camerica except for Power Machines. The NES versions of the games were released as multicarts.

QMRQuattro (meaning four in Italian) is the sub-brand used by the car brand Audi AG to indicate that all-wheel drive (AWD) technologies or systems are used on specific models of its Audi automobiles.[1] It is also Audi's actual performance division alongside S and RS The word "quattro" is a registered trademark of Audi AG, a subsidiary of the German automotive concern, Volkswagen Group.[1] Quattro was first introduced in 1980 on the permanent four wheel drive Audi Quattro model, often referred to as the Ur-Quattro ("Ur-" is a German prefix that means "original" or "first"). The term quattro has since been applied to all subsequent Audi AWD models. Due to the nomenclature rights derived from the trademark, the word quattro is now always spelled with a lower case "q", in honour of its former namesake. Other companies in the Volkswagen Group have used different trademarks for their 4WD vehicles. While Audi has always used the term "quattro", Volkswagen-branded cars initially used "syncro", but more recently, VW uses "4motion". Škoda simply uses the nomenclature "4x4" after the model name, whereas SEAT uses merely "4". None of the above trademarks or nomenclatures defines the operation or type of 4WD system, as detailed below.

quattro GmbH is a wholly owned private subsidiary of the German automobile manufacturer AUDI AG,[1][3][4][5] part of the Volkswagen Group. Founded in October 1983,[1] it primarily specialises in producing high performance Audi cars[2] and components,[1] along with purchaser specified customisations.[1] The company takes its name as an homage to Audi's original four-wheel drive rally-inspired road car – the Audi Quattro, and in bowing to this, always spells its 'quattro GmbH' name with a leading lower-case 'q'.

quattro GmbH specialises in four 'key' areas,[1] including the design, testing and production of specialist and high performance Audi automobiles, such as the Audi RS3, Audi RS4, Audi RS5, Audi RS6, Audi RS7, Audi RS Q3, Audi TTRS and the Audi R8. They also design and specify roadwheels, and design and produce sports suspension,[1] and the specialist car body parts (such as front bumpers and splitters, side skirts, rear bumpers and diffusers, and rear spoilers) which are mainly used on the Audi "S line" trim specification available on most of the model range.

Characteristics[edit] Historically, Quattro Pro used keyboard commands close to Lotus 1-2-3. While it is commonly said to have been the first program to use tabbed sheets, Boeing Calc actually utilized tabbed sheets earlier.[1][2] It currently runs under the Windows operating system. For years Quattro Pro had a comparative advantage, in regard to maximum row and column limits, (allowing a maximum worksheet size of one million rows by 18,276 columns). This avoided the 65,536 row by 256 column spreadsheet limitations inherent to Microsoft Excel, (prior to Excel 2007). Even with the maximum row advantage, Quattro Pro has been a distant second to Excel, in regard to the number of sales, since approximately 1996 to the present. When version 1.0 was in development, it was codenamed "Buddha" since it was meant to "assume the Lotus position", #1 in the market. When the product was launched in 1988, its original name, suggested to Mr. Kahn by Senior VP, Spencer Leyton at a Vietnamese restaurant in Santa Cruz, was Quattro (the Italian word for "four", a play on being one step ahead of "1-2-3"). Borland changed the name to Quattro Pro for its 1990 release. The common file extension of Quattro Pro spreadsheet file is .qpw, which it has used since version 9. Quattro Pro versions 7 and 8 used .wb3, version 6 used .wb2, version 5 used .wb1, and DOS versions used .wq2 and .wq1.[3] Origins

QMRQuattro Pro is a spreadsheet program developed by Borland and now sold by Corel, most often as part of Corel's WordPerfect Office suite.

QMRThe Four big families of Hong Kong (Chinese: 香港四大家族)[1] are an initial group of Chinese families. Today, the Big Four families and their descendants are Fortune Global 500 companies with multinational corporations and listings from the London Stock Exchange to the New York Stock Exchange. Along with their European mostly British rivals in colonial Hong Kong, they were responsible for much of the foundations of the territory including its economy, finance, and trade, whilst the British government provided the first codified legal system which until then and in most of China today, was lacking. The four big families begin with the Chinese surname Li, Ho, Hui and Lo. Respectively the head of four families are Li Sek-peng (李石朋), Robert Ho (何東), Hui Oi-chow (許愛周) and Lo Man Kam (羅文錦).[1] Today the two families most recognizable by regular HK citizens are the Li and Ho group descendants.[1]

Families[edit] The following families and their descendants are listed below. Each indentation is at least one generation down, but not necessarily the next generation. Not all the descendants are shown. Most members of these families have reached tycoon status. Li family notables[edit] Li Shek-pang (李石朋, 1863–1916) also known as Li Pui-choi (李佩材) - Originally from Guangdong Li Koon-chun (李冠春, 1887–1966) - Founder of Bank of East Asia, Director of Tung Wah Group of Hospitals and Po Leung Kuk[2] Li Fook-shu (李福樹, 1912–95) - Council member of Chinese University of Hong Kong[3] David Li Kwok-po (李國寶, 1939–) - Chairman, Bank of East Asia, "[1]." Member of LegCo, ExCo, and HK Basic Law Drafting Committee Adrian Li (李民橋, 1973–) Co-CEO, Bank of East Asia, CPPC Delegate Brian Li (李民斌, 1974–) Co-CEO, Bank of East Asia, CPPC Delegate Arthur Li Kwok-cheung (李國章, 1945–) - Hong Kong Secretary for Education & Manpower (2002-2007), ExCo Member, CPPC Delegate, Council member of the University of Hong Kong Simon Li (李福善, 1922–2013) - CPPC Delegate, HK Basic Law Drafting Committee, Candidate for HK Chief Executive (1996) Gladys Li (李志喜, 1948–) - Senior Counsel, former Chairwoman, HK Bar Association Ronald Li (李福兆, 1929–2014) - Founder & Chairman of Hong Kong Stock Exchange[4] Alfred Ronald Li (李國麟, 1952–) Li Tse-fong (李子方, 1891–1976) Li Fook-wo (李福和, 1916–2014) Li Fook-kow (李福逑, 1922–2011) Andrew Li (李國能, 1948–) - Chief Justice of the Court of Final Appeal of Hong Kong (1997–2010) Li Lan-sang (李蘭生, 1900–69) Alan Li (李福深)

Ho family notables[edit] Robert Hotung (何東, 1862–1956) - Businessman, philanthropist Victoria Hotung (何錦姿, 1897–?) - married to Man-kam Lo Robert Ho (何世禮, 1906–98) George Ho (何佐芝, 1918–2014) - Founder of the Commercial Radio Hong Kong[5] George Joseph Ho (何驥 1950–) - Chairman of the Commercial Radio Hong Kong Ho Fook (何福, 1863–1926) - Businessman, philanthropist Ho Kwong (何世光) Stanley Ho (何鴻燊, 1921–) - Head of Macau casinos Pansy Ho (何超瓊, 1962–) - Actress and managing director of various casinos Josie Ho (何超儀, 1974–) - Singer, actress Lawrence Ho (何猷龍, 1976–) - Chief exec of Melco International development[6] Ho Sai-chuen (何世全, 1891–1938) - Doctor and member of the Sanitary Board Ho Kom-tong (何甘棠, 1866–1950) - Businessman, philanthropist Grace Ho (何愛瑜) - married to Lee Hoi-chuen Bruce Lee (李小龍, 1940–73) - Movie star and martial artist Brandon Lee (李國豪, 1965–93) - Actor and martial artist Hui family notables[edit] Hui Oi-chow (許愛周) - Business man Victor Hui (許晉奎) - Chairman of Hong Kong Football Association, vice-president of Sports Federation and Olympic Committee of Hong Kong, China[7][8] Julian Hui (許晉亨) - Business man

Lo family notables[edit] Lo Cheung-shiu (羅長肇, 1867–1934) - Compradore of the Jardine, Matheson & Co. Lo Man-kam (羅文錦, 1893–1959) - Solicitor, founder of Lo and Lo law firm,[9] member of the Executive and Legislative Councils of Hong Kong, married to Victoria Hotung, daughter of Robert Hotung Lo Tak-shing (羅德丞, 1935–2006) - Solicitor, member of the Hong Kong Basic Law Drafting Committee, CPPCC[10] Executive and Legislative Councils of Hong Kong Lo Man-wai (羅文惠, 1895–1985) - Solicitor and member of the Executive and Legislative Councils. Enid Lo (羅德貞), woman tennis player, married to John L. Litton Henry Litton (列顯倫, 1934–) - Non-Permanent Judge of the Court of Final Appeal John Litton (烈宗仁) - Barrister, Hong Kong Bar (1990- )

QMRThe Four big families of the Republic of China (Chinese: 蔣宋孔陳四大家族; pinyin: Jiǎng-Sòng-Kǒng-Chén sì dà jiāzú)[1] are an initial group of families in the Republic of China era. They were responsible for much of China's management of finance, politics, economy, and law. The four big families begin with the Chinese surnames Chiang, Soong, Kung, and Chen.

Families[edit] The following families and their descendants are listed below. Each indentation is at least one generation down, but not necessarily the next generation. Not all the descendants are shown. Chiang family notables[edit] Chiang Kai-shek (蔣介石, 1887–1975), first President of the Republic of China[2] Chiang Ching-kuo (蔣經國, 1910–1988), President of the Republic of China Eddie Chiang (蔣孝勇, 1948–1996), politician Winston Chang (章孝慈, 1941–1996), Soochow University president John Chiang (蔣孝嚴, 1941–), politician Soong family notables[edit] Charlie Soong (宋嘉樹, 1863–1918), businessman, friend of Sun Yat-Sen T. V. Soong (宋子文, 1891–1971), businessman[1] Soong sisters Soong Ai-ling (宋藹齡, 1890–1973), married to H. H. Kung Soong Ching-ling (宋慶齡, 1893–1981), married to Sun Yat-sen, Honorary President of the People's Republic of China (1981), President of the People's Republic of China (1968–1972), Chairperson of the Standing Committee of the National People's Congress (1976–1978), Vice President of the People's Republic of China Soong Mei-ling (宋美齡, 1898–2003), married to Chiang Kai-shek, First Lady of the Republic of China Soong Zi-on (宋子安, 1906–1969), Guangzhou bank chairman

Kung family notables[edit] H. H. Kung (孔祥熙, 1881–1967), businessman, politician[1] Chen family notables[edit] Chen Qi-mei (陳其美, 1878–1916), politician[1] Chen Guo-fu (陳果夫, 1892–1951), politician Chen Li-fu (陳立夫, 1900–2001), politician

QMRAlthough there are thousands of Chinese family names, the 100 most common, which together make up less than 5% of those in existence, are shared by 85% of the population. The three most common surnames in Mainland China are Li, Wang and Zhang, which make up 7.9%, 7.4% and 7.1% respectively. Together they number close to 300 million and are easily the most common surnames in the world. In Chinese, the phrase "three Zhang, four Li" (Chinese: 张三李四; pinyin: zhāng sān lǐ sì) is used to say "just anybody". In a 1990 study, the top 200 family names accounted for over 96% of a random sample of 174,900 persons, with over 500 other names accounting for the remaining 4%. In a different study (1987), which combined data from Taiwan and China (sample size of 570,000 persons), the top 19 names covered 55.6%,[10] and the top 100 names covered 87% of the sample. Other data suggest that the top 50 names comprise 70% of the population.[11] Most commonly occurring Chinese family names have only one character; however, about twenty double-character family names have survived into modern times. These include Sima (司馬, simp. 司马), Zhuge (諸葛, simp. 诸葛), Ouyang (歐陽, simp. 欧阳), occasionally romanized as O'Young, suggesting an Irish origin to English-speakers, and Situ (or Sito 司徒). Sima, Zhuge, and Ouyong also happen to be the surnames of four extremely famous premodern Chinese historical figures. There are family names with three or more characters, but those are not ethnically Han Chinese. For example, Aixinjueluo (愛新覺羅, also romanized from the Manchu language as Aisin Gioro), was the family name of the Manchu royal family of the Qing dynasty.

QMRPorter five forces analysis is a framework that attempts to analyze the level of competition within an industry and business strategy development. It draws upon industrial organization (IO) economics to derive five forces that determine the competitive intensity and therefore attractiveness of an Industry. Attractiveness in this context refers to the overall industry profitability. An "unattractive" industry is one in which the combination of these five forces acts to drive down overall profitability. A very unattractive industry would be one approaching "pure competition", in which available profits for all firms are driven to normal profit. This analysis is associated with its principal innovator Michael E. Porter of Harvard University. Porter referred to these forces as the micro environment, to contrast it with the more general term macro environment. They consist of those forces close to a company that affect its ability to serve its customers and make a profit. A change in any of the forces normally requires a business unit to re-assess the marketplace given the overall change in industry information. The overall industry attractiveness does not imply that every firm in the industry will return the same profitability. Firms are able to apply their core competencies, business model or network to achieve a profit above the industry average. A clear example of this is the airline industry. As an industry, profitability is low and yet individual companies, by applying unique business models, have been able to make a return in excess of the industry average. Porter's five forces include - three forces from 'horizontal' competition: the threat of substitute products or services, the threat of established rivals, and the threat of new entrants; and two forces from 'vertical' competition: the bargaining power of suppliers and the bargaining power of customers. Porter developed his Five Forces analysis in reaction to the then-popular SWOT analysis, which he found unrigorous and ad hoc.[1] Porter's five forces is based on the Structure-Conduct-Performance paradigm in industrial organizational economics. It has been applied to a diverse range of problems, from helping businesses become more profitable to helping governments stabilize industries.[2] Other Porter strategic frameworks include the value chain and the generic strategies.

QMRStrategic analysis typically focuses on two views of organization: the industry-view and the Resource-Based View (RBV). These views analyse the organisation without taking into consideration relationship between the organizations strategic choice (i.e. Porter generic strategies) and institutional frameworks. The National Diamond' is a tool for analyzing the organization's task environment. The National Diamond highlights that strategic choices should not only be a function of industry structure and a firm's resources, it should also be a function of the constraints of the institutional framework. Institutional analysis (such as the National Diamond) becomes increasingly important as firms enter new operating environments and operate within new institutional frameworks. Michael Porter's National Diamond framework resulted from a study of patterns of comparative advantage among industrialized nations. It works to integrate much of Porter's previous work in his competitive five forces theory, his value chain framework as well as his theory of competitive advantage into a consolidated framework that looks at the sources of competitive advantage sourcable from the national context. It can be used both to analyze a firm's ability to function in a national market, as well as analyse a national markets ability to compete in an international market. It recognizes four pillars of research (factor conditions, demand conditions, related and supporting industries, firm structure, strategy and rivalry) that one must undertake in analysing the viability of a nation competing in a particular international market, but it also can be used as a comparative analysis tool in recognising which country a particular firm is suited to expanding into. Two of the aforementioned pillars focus on the (national) macroeconomics environment to determine if the demand is present along with the factors needed for production (i.e. both extreme ends of the value chain). Another pillar focuses on the specific relationships supporting industries have with the particular firm/nation/industry being studied. The last pillar it looks at the firm's strategic response (microeconomics) i.e. its strategy, taking into account the industry structure and rivalry (see five forces). In this way it tries to highlight areas of competitive advantage as well as competitive weakness, by looking at a companies/nations suitability to the particular conditions of a particular market.

Principles[edit] For analyzing national competitiveness, we need to focus upon firm performance. The role of the national environment is providing a context within which firms develop their identity, resources, capabilities, and managerial styles. For a country to sustain a competitive advantage in a particular industry sector requires dynamic advantage: firms must broaden and extend the basis of their competitive advantage by innovation and upgrading. The dynamic conditions that influence innovation and the upgrading are far more important than initial resource endowments in determining national patterns of competitiveness. Components[edit] The four different components of the framework are: Factor Endowment Related And Supporting Industries Demand Conditions Strategy, Structure, And Rivalry

QMRPorter five forces analysis is a framework that attempts to analyze the level of competition within an industry and business strategy development. It draws upon industrial organization (IO) economics to derive five forces that determine the competitive intensity and therefore attractiveness of an Industry. Attractiveness in this context refers to the overall industry profitability. An "unattractive" industry is one in which the combination of these five forces acts to drive down overall profitability. A very unattractive industry would be one approaching "pure competition", in which available profits for all firms are driven to normal profit. This analysis is associated with its principal innovator Michael E. Porter of Harvard University. Porter referred to these forces as the micro environment, to contrast it with the more general term macro environment. They consist of those forces close to a company that affect its ability to serve its customers and make a profit. A change in any of the forces normally requires a business unit to re-assess the marketplace given the overall change in industry information. The overall industry attractiveness does not imply that every firm in the industry will return the same profitability. Firms are able to apply their core competencies, business model or network to achieve a profit above the industry average. A clear example of this is the airline industry. As an industry, profitability is low and yet individual companies, by applying unique business models, have been able to make a return in excess of the industry average. Porter's five forces include - three forces from 'horizontal' competition: the threat of substitute products or services, the threat of established rivals, and the threat of new entrants; and two forces from 'vertical' competition: the bargaining power of suppliers and the bargaining power of customers. Porter developed his Five Forces analysis in reaction to the then-popular SWOT analysis, which he found unrigorous and ad hoc.[1] Porter's five forces is based on the Structure-Conduct-Performance paradigm in industrial organizational economics. It has been applied to a diverse range of problems, from helping businesses become more profitable to helping governments stabilize industries.[2] Other Porter strategic frameworks include the value chain and the generic strategies.

QMRThe Chinese Telegraph Code, Chinese Telegraphic Code, or Chinese Commercial Code (simplified Chinese: 中文电码; traditional Chinese: 中文電碼; pinyin: Zhōngwén diànmǎ or simplified Chinese: 中文电报码; traditional Chinese: 中文電報碼; pinyin: Zhōngwén diànbàomǎ)[1] is a four-digit decimal code (character encoding) for electrically telegraphing messages written with Chinese characters.

QMRThe Four-Corner Method (simplified Chinese: 四角号码检字法; traditional Chinese: 四角號碼檢字法; pinyin: sì jiǎo hàomǎ jiǎnzì fǎ; literally: "four corner code lookup-character method") is a character-input method used for encoding Chinese characters into either a computer or a manual typewriter, using four or five numerical digits per character. The Four-Corner Method is also known as the Four-Corner System. The four digits encode the shapes found in the four corners of the symbol, top-left to bottom-right. Although this does not uniquely identify a Chinese character, it leaves only a very short list of possibilities. A fifth digit can be added to describe an extra part above the bottom-right if necessary.

1,5,10,20 dollar bills are used the most. 50 and 100 dollar bills are transcendent

QMRFour Sages and Twelve Philosophers[edit] Other than Confucius himself, the most venerated Confucians are the "Four Sages" or "Correlates" and the "Twelve Philosophers". All together there are 16, which is the squares of the quadrant model

QMRThe Four Sages, Assessors,[1] or Correlates (Chinese: 四配; pinyin: Sì Pèi) are four eminent Chinese philosophers in the Confucian tradition. They are traditionally accounted a kind of sainthood and their spirit tablets are prominently placed in Confucian temples, two upon the east and two upon the west side of the Hall of the Great Completion (Dacheng Dian). The Four Sages are: Yan Hui, Confucius's favourite disciple Zengzi or Zeng Sheng, another disciple of Confucius and author of the Great Learning Zisi or Kong Ji, Confucius's grandson, student of Zengzi, and author of the Doctrine of the Mean Mencius or Master Meng, student of Zisi and author of the Mencius. Within a traditional Confucian temple, Yan Hui's tablet is placed first to the east of Confucius.[1]

QMRThe four coin types in common circulation today have not had their sizes or denominations changed in well over a century, although their weights have been reduced due to the substitution of cheaper metals in their manufacture. Businesses usually have to keep adequate amounts in coin on hand, so as to be able to make change in fractional dollar amounts. Since they do not receive the coins they need through regular trade, there is often a one-way flow of coins from the banks to the retailers, who often have to pay fees for it.[citation needed] Furthermore, apart from some dollar coins, U.S. coins do not indicate their value in numerals, but in English words, and the value descriptions do not follow a consistent pattern, referring to three different units, and expressions in fractions: "One Cent"; "Five Cents"; "One Dime"; "Quarter Dollar"; the values of the coins must therefore be learned. Due to the penny's low value, some efforts have been made to eliminate the penny as circulating coinage.

A lot of smaller pizzas are divided into four pieces/quadrants

QMRA quad-core processor is a chip with four independent units called cores that read and execute central processing unit (CPU) instructions such as add, move data, and branch. Within the chip, each core operates in conjunction with other circuits such as cache, memory management, and input/output (I/O) ports. The individual cores in a quad-core processor can run multiple instructions at the same time, increasing the overall speed for programs compatible with parallel processing. Manufacturers typically integrate the cores onto a single semiconductor wafer, or onto multiple semiconductor wafers within a single IC (integrated circuit) package. Although it's tempting to suppose that a quad-core processor would operate twice as fast as a dual-core processor and four times as fast as a single-core processor, things don't work out that simply. Results vary depending on the habits of the computer user, the nature of the programs being run, and the compatibility of the processor with other hardware in the system as a whole. Quad-core and higher multi-core processor configurations have become common for general-purpose computing, not only for PCs but for mobile devices such as smartphones and tablets. See a brief introduction to multi-core processor architecture:

QMrIn geometry and trigonometry, a right angle is an angle that bisects the angle formed by two adjacent parts of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.[1] As a rotation, a right angle corresponds to a quarter turn (that is, a quarter of a full circle).[2] Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles,[3] making the right angle basic to trigonometry. The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line. Quadrants are made up of right angles- four right angles

QMRIn Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the legs. The legs can be equal (see isosceles tangential trapezoid below), but they don't have to be.

QMRTangential quadrilateral

From Wikipedia, the free encyclopedia

An example of a tangential quadrilateral

In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides are all tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals.[1] Tangential quadrilaterals are a special case of tangential polygons.

Other, rarely used, names for this class of quadrilaterals are inscriptable quadrilateral, inscriptible quadrilateral, inscribable quadrilateral, circumcyclic quadrilateral, and co-cyclic quadrilateral.[1][2] Due to the risk of confusion with a quadrilateral that has a circumcircle, which is called a cyclic quadrilateral or inscribed quadrilateral, it is preferable not to use any of the last five names.[1]

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

Special cases[edit]

Examples of tangential quadrilaterals are the kites, which include the rhombi, which in turn include the squares. The kites are exactly the tangential quadrilaterals that are also orthodiagonal.[3] If a quadrilateral is both tangential and cyclic, it is called a bicentric quadrilateral.

Characterizations[edit] In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle. Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter.[4] According to the Pitot theorem, the two pairs of opposite sides in a tangential quadrilateral add up to the same total length, which equals the semiperimeter s of the quadrilateral: a + c = b + d = \frac{a + b + c + d}{2} = s. Conversely a convex quadrilateral in which a + c = b + d must be tangential.[1]:p.65[4] If opposite sides in a convex quadrilateral ABCD (that is not a trapezoid) intersect at E and F, then it is tangential if and only if either of[4] \displaystyle BE+BF=DE+DF or \displaystyle AE-EC=AF-FC. The second of these is almost the same as one of the equalities in Urquhart's theorem. The only differences are the signs on both sides; in Urquhart's theorem there are sums instead of differences. Another necessary and sufficient condition is that a convex quadrilateral ABCD is tangential if and only if the incircles in the two triangles ABC and ADC are tangent to each other.[1]:p.66 A characterization regarding the angles formed by diagonal BD and the four sides of a quadrilateral ABCD is due to Iosifescu. He proved in 1954 that a convex quadrilateral has an incircle if and only if[5] \tan{\frac{\angle ABD}{2}}\cdot\tan{\frac{\angle BDC}{2}}=\tan{\frac{\angle ADB}{2}}\cdot\tan{\frac{\angle DBC}{2}}. Further, a convex quadrilateral with successive sides a, b, c, d is tangential if and only if R_aR_c=R_bR_d where Ra, Rb, Rc, Rd are the radii in the circles externally tangent to the sides a, b, c, d respectively and the extensions of the adjacent two sides for each side.[6]:p.72 Several more characterizations are known in the four subtriangles formed by the diagonals. Special line segments[edit] The eight tangent lengths of a tangential quadrilateral are the line segments from a vertex to the points where the incircle is tangent to the sides. From each vertex there are two congruent tangent lengths. The two tangency chords of a tangential quadrilateral are the line segments that connect points on opposite sides where the incircle is tangent to these sides. These are also the diagonals of the contact quadrilateral. Diagonals[edit] If e, f, g and h are the tangent lengths from A, B, C and D respectively to the points where the incircle is tangent to the sides of a tangential quadrilateral ABCD, then the lengths of the diagonals p = AC and q = BD are[8]:Lemma3 \displaystyle p=\sqrt{\frac{e+g}{f+h}\Big((e+g)(f+h)+4fh\Big)}, \displaystyle q=\sqrt{\frac{f+h}{e+g}\Big((e+g)(f+h)+4eg\Big)}. Characterizations in the four subtriangles[edit] Chao and Simeonov's characterization in terms of the radii of circles within each of four triangles In the nonoverlapping triangles APB, BPC, CPD, DPA formed by the diagonals in a convex quadrilateral ABCD, where the diagonals intersect at P, there are the following characterizations of tangential quadrilaterals. Let r1, r2, r3, and r4 denote the radii of the incircles in the four triangles APB, BPC, CPD, and DPA respectively. Chao and Simeonov proved that the quadrilateral is tangential if and only if[25] \frac{1}{r_1}+\frac{1}{r_3}=\frac{1}{r_2}+\frac{1}{r_4}. This characterization had already been proved five years earlier by Vaynshtejn.[16]:p.169[26] In the solution to his problem, a similar characterization was given by Vasilyev and Senderov. If h1, h2, h3, and h4 denote the altitudes in the same four triangles (from the diagonal intersection to the sides of the quadrilateral), then the quadrilateral is tangential if and only if[5][26] \frac{1}{h_1}+\frac{1}{h_3}=\frac{1}{h_2}+\frac{1}{h_4}. Another similar characterization concerns the exradii ra, rb, rc, and rd in the same four triangles (the four excircles are each tangent to one side of the quadrilateral and the extensions of its diagonals). A quadrilateral is tangential if and only if[1]:p.70 \frac{1}{r_a}+\frac{1}{r_c}=\frac{1}{r_b}+\frac{1}{r_d}. If R1, R2, R3, and R4 denote the radii in the circumcircles of triangles APB, BPC, CPD, and DPA respectively, then the quadrilateral ABCD is tangential if and only if[27]:pp. 23–24 R_1+R_3=R_2+R_4. In 1996, Vaynshtejn was probably the first to prove another beautiful characterization of tangential quadrilaterals, that has later appeared in several magazines and websites.[1]:pp. 72–73 It states that when a convex quadrilateral is divided into four nonoverlapping triangles by its two diagonals, then the incenters of the four triangles are concyclic if and only if the quadrilateral is tangential. In fact, the incenters form an orthodiagonal cyclic quadrilateral.[1]:p.74 A related result is that the incircles can be exchanged for the excircles to the same triangles (tangent to the sides of the quadrilateral and the extensions of its diagonals). Thus a convex quadrilateral is tangential if and only if the excenters in these four excircles are the vertices of a cyclic quadrilateral.[1]:p. 73 A convex quadrilateral ABCD, with diagonals intersecting at P, is tangential if and only if the four excenters in triangles APB, BPC, CPD, and DPA opposite the vertices B and D are concyclic.[1]:p. 79 If Ra, Rb, Rc, and Rd are the exradii in the triangles APB, BPC, CPD, and DPA respectively opposite the vertices B and D, then another condition is that the quadrilateral is tangential if and only if[1]:p. 80 \frac{1}{R_a}+\frac{1}{R_c}=\frac{1}{R_b}+\frac{1}{R_d}. Further, a convex quadrilateral ABCD with diagonals intersecting at P is tangential if and only if[5] \frac{a}{\triangle(APB)}+\frac{c}{\triangle(CPD)}=\frac{b}{\triangle(BPC)}+\frac{d}{\triangle(DPA)} where ∆(APB) is the area of triangle APB. Denote the segments that the diagonal intersection P divides diagonal AC into as AP = p1 and PC = p2, and similarly P divides diagonal BD into segments BP = q1 and PD = q2. Then the quadrilateral is tangential if and only if any one of the following equalities are true:[28] ap_2q_2 + cp_1q_1 = bp_1q_2 + dp_2q_1 or[1]:p. 74 \frac{(p_1+q_1-a)(p_2+q_2-c)}{(p_1+q_1+a)(p_2+q_2+c)}=\frac{(p_2+q_1-b)(p_1+q_2-d)}{(p_2+q_1+b)(p_1+q_2+d)} or[1]:p. 77 \frac{(a+p_1-q_1)(c+p_2-q_2)}{(a-p_1+q_1)(c-p_2+q_2)}=\frac{(b+p_2-q_1)(d+p_1-q_2)}{(b-p_2+q_1)(d-p_1+q_2)}. Conditions for a tangential quadrilateral to be another type of quadrilateral[edit] A tangential quadrilateral is a rhombus if and only if its opposite angles are equal.[29] If the incircle is tangent to the sides AB, BC, CD, DA at W, X, Y, Z respectively, then a tangential quadrilateral ABCD is also cyclic (and hence bicentric) if and only if any of[2][3]:p.124[19] WY is perpendicular to XZ \frac{AW}{WB}=\frac{DY}{YC} \frac{AC}{BD}=\frac{AW+CY}{BX+DZ} The first of these three means that the contact quadrilateral WXYZ is an orthodiagonal quadrilateral. A tangential quadrilateral is bicentric if and only if its inradius is greater than that of any other tangential quadrilateral having the same sequence of side lengths.[30]:pp.392-393 A tangential quadrilateral is a kite if and only if any one of the following conditions is true:[16] The area is one half the product of the diagonals The diagonals are perpendicular The two line segments connecting opposite points of tangency have equal length One pair of opposite tangent lengths have equal length The bimedians have equal length The products of opposite sides are equal The center of the incircle lies on the diagonal that is the axis of symmetry.

QMRIn geometry, the Pitot theorem, named after the French engineer Henri Pitot, states that in a tangential quadrilateral (i.e. one in which a circle can be inscribed) the two sums of lengths of opposite sides are the same. The theorem is a consequence of the fact that two tangent line segments from a point outside the circle to the circle have equal lengths. There are four equal pairs of tangentsegments, and both sums of two sides can be decomposed into sums of these four tangent segment lengths. The converse is also true: a circle can be inscribed into every convex quadrilateral in which the lengths of opposite sides sum to the same value. Henri Pitot proved his theorem in 1725, whereas the converse was proved by the Swiss mathematician Jakob Steiner in 1846.

QMRRight tangential trapezoid[edit] A right tangential trapezoid. A right tangential trapezoid is a tangential trapezoid where two adjacent angles are right angles. If the bases have lengths a and b, then the inradius is[5] r=\frac{ab}{a+b}. Thus the diameter of the incircle is the harmonic mean of the bases. The right tangential trapezoid has the area[5] \displaystyle K=ab and its perimeter P is[5] \displaystyle P=2(a+b). Isosceles tangential trapezoid[edit] Every isosceles tangential trapezoid is bicentric. An isosceles tangential trapezoid is a tangential trapezoid where the legs are equal. Since an isosceles trapezoid is cyclic, an isosceles tangential trapezoid is a bicentric quadrilateral. That is, it has both an incircle and a circumcircle. If the bases are a and b, then the inradius is given by[6] r=\tfrac{1}{2}\sqrt{ab}. To derive this formula was a simple Sangaku problem from Japan. From Pitot's theorem it follows that the lengths of the legs are half the sum of the bases. Since the diameter of the incircle is the square root of the product of the bases, an isosceles tangential trapezoid gives a nice geometric interpretation of the arithmetic mean and geometric mean of the bases as the length of a leg and the diameter of the incircle respectively.

Characterization[edit] A convex quadrilateral is tangential if and only if opposite sides satisfy Pitot's theorem: AB+CD=BC+DA. In turn, a tangential quadrilateral is a trapezoid if and only if either of the following two properties hold (in which case they both do): It has two adjacent angles that are supplementary (then this is also true for the other two angles). Specifically, a tangential quadrilateral ABCD is a trapezoid with parallel bases AB and CD if and only if A+D=B+C=\pi. The product of two adjacent tangent lengths equals the product of the other two tangent lengths. Specifically, if e, f, g, h are the tangent lengths emanating from A, B, C, D respectively in a tangential quadrilateral ABCD, then AB and CD are the parallel bases of a trapezoid if and only if[1]:Thm. 2 eh=fg.

QMRIn Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the legs. The legs can be equal (see isosceles tangential trapezoid below), but they don't have to be.

Special cases[edit] Examples of tangential trapezoids are rhombi and squares.

QMRIn Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the legs. The legs can be equal (see isosceles tangential trapezoid below), but they don't have to be.

The diagonals of an isosceles trapezoid have the same length; that is, every isosceles trapezoid is an equidiagonal quadrilateral. Moreover, the diagonals divide each other in the same proportions. As pictured, the diagonals AC and BD have the same length (AC = BD) and divide each other into segments of the same length (AE = DE and BE = CE). The ratio in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, that is, \frac{AE}{EC} = \frac{DE}{EB} = \frac{AD}{BC}. The length of each diagonal is, according to Ptolemy's theorem, given by p=\sqrt{ab+c^2} where a and b are the lengths of the parallel sides AD and BC, and c is the length of each leg AB and CD. The height is, according to the Pythagorean theorem, given by h=\sqrt{p^2-\left(\frac{a+b}{2}\right)^2}=\tfrac{1}{2}\sqrt{4c^2-(a-b)^2}. The distance from point E to base AD is given by d=\frac{ah}{a+b} where a and b are the lengths of the parallel sides AD and BC, and h is the height of the trapezoid.

QMR Self-intersections[edit] Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite.[3] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms, crossed quadrilaterals in which opposite sides have equal length. Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid.[4]

Characterizations[edit] If a quadrilateral is known to be a trapezoid, it is not necessary to check that the legs have the same length in order to know that it is an isosceles trapezoid (nor, under the definitions given in Wikipedia, is it sufficient, since a rhombus is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry); any of the following properties also distinguishes an isosceles trapezoid from other trapezoids: The diagonals have the same length. The base angles have the same measure. The segment that joins the midpoints of the parallel sides is perpendicular to them. Opposite angles are supplementary, which in turn implies that isosceles trapezoids are cyclic quadrilaterals. The diagonals divide each other into segments with lengths that are pairwise equal; in terms of the picture below, AE = DE, BE = CE (and AE ≠ CE if one wishes to exclude rectangles). If rectangles are included in the class of trapezoids then one may concisely define an isosceles trapezoid as "a cyclic quadrilateral with equal diagonals"[5] or as "a cyclic quadrilateral with a pair of parallel sides" or as "a convex quadrilateral with a line of symmetry through the mid-points of opposite sides".

QMR Self-intersections[edit] Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite.[3] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms, crossed quadrilaterals in which opposite sides have equal length. Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid.[4]

QMRIn Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. In any isosceles trapezoid two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (properties shared with the parallelogram). The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base).

Special cases[edit] Special cases of isosceles trapezoids Rectangles and squares are usually considered to be special cases of isosceles trapezoids though some sources would exclude them. Another special case is a 3-equal side trapezoid, sometimes known as a trilateral trapezoid[1] or a trisosceles trapezoid.[2] They can also be seen dissected from regular polygons of 5 sides or more as a truncation of 4 sequential vertices. The isosceles trapezoid is also rarely known as a symtra because of its symmetry.[3]

QMRIn geometry, the perpendicular bisector construction of a quadrilateral is a construction which produces a new quadrilateral from a given quadrilateral using the perpendicular bisectors to the sides of the former quadrilateral. This construction arises naturally in an attempt to find a replacement for the circumcenter of a quadrilateral in the case that is non-cyclic.

Definition of the construction[edit] Suppose that the vertices of the quadrilateral Q are given by Q_1,Q_2,Q_3,Q_4 . Let b_1,b_2,b_3,b_4 be the perpendicular bisectors of sides Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 respectively. Then their intersections Q_i^{(2)}=b_{i+2}b_{i+3} , with subscripts considered modulo 4, form the consequent quadrilateral Q^{(2)} . The construction is then iterated on Q^{(2)} to produce Q^{(3)} and so on. First iteration of the perpendicular bisector construction An equivalent construction can be obtained by letting the vertices of Q^{(i+1)} be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of Q^{(i)} .

QMRVarignon's theorem is a statement in Euclidean geometry by Pierre Varignon that was first published in 1731. It deals with the construction of a particular parallelogram (Varignon parallelogram) from an arbitrary quadrilateral (quadrangle). The midpoints of the sides of an arbitrary quadrangle form a parallelogram. If the quadrangle is convex or reentrant, i.e. not a crossing quadrangle, then the area of the parallelogram is half the area of the quadrangle. If one introduces the concept of oriented areas for n-gons, then the area equality above holds for crossed quadrilaterals as well.[1] The Varignon parallelogram exists even for a skew quadrilateral, and is planar whether or not the quadrilateral is planar. It can be generalized to the midpoint polygon of an arbitrary polygon.

Special cases[edit] The Varignon parallelogram is a rhombus if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an equidiagonal quadrilateral.[2] The Varignon parallelogram is a rectangle if and only if the diagonals of the quadrilateral are perpendicular, that is, if the quadrilateral is an orthodiagonal quadrilateral.[3]:p.14 If a crossing quadrilateral is formed from either pair of opposite parallel sides and the diagonals of a parallelogram, the Varignon parallelogram has a side of length zero and is a line segment.

Proof[edit] Varignon's theorem is easily proved as a theorem of affine geometry organized as linear algebra with the linear combinations restricted to coefficients summing to 1, also called affine or barycentric coordinates. The proof applies even to skew quadrilaterals in spaces of any dimension. Any three points E, F, G are completed to a parallelogram (lying in the plane containing E, F, and G) by taking its fourth vertex to be E − F + G. In the construction of the Varignon parallelogram this is the point (A + B)/2 − (B + C)/2 + (C + D)/2 = (A + D)/2. But this is the point H in the figure, whence EFGH forms a parallelogram. In short, the centroid of the four points A, B, C, D is the midpoint of each of the two diagonals EG and FH of EFGH, showing that the midpoints coincide. A second proof requires less algebra. By drawing in the diagonals of the quadrilateral, we notice two triangles are created for each diagonal. And by the Midline Theorem, the segment containing two midpoints of adjacent sides is both parallel and half the respective diagonal. Therefore, the sum of the diagonals is equal to the perimeter of the quadrilateral formed. Secondly, we can use vectors 1/2 the length of each side to first determine the area of the quadrilateral, and then to find areas of the four triangles divided by each side of the inner parallelogram.

QMRIn Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.[1]

Examples of equidiagonal quadrilaterals include the isosceles trapezoids, rectangles and squares. An equidiagonal kite that maximizes the ratio of perimeter to diameter, inscribed in a Reuleaux triangle Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, and 5π/12.[2]

Relation to other types of quadrilaterals[edit] A parallelogram is equidiagonal if and only if it is a rectangle,[6] and a trapezoid is equidiagonal if and only if it is an isosceles trapezoid. The cyclic equidiagonal quadrilaterals are exactly the isosceles trapezoids. There is a duality between equidiagonal quadrilaterals and orthodiagonal quadrilaterals: a quadrilateral is equidiagonal if and only if its Varignon parallelogram is orthodiagonal (a rhombus), and the quadrilateral is orthodiagonal if and only if its Varignon parallelogram is equidiagonal (a rectangle).[3] Equivalently, a quadrilateral has equal diagonals if and only if it has perpendicular bimedians, and it has perpendicular diagonals if and only if it has equal bimedians.[7] Silvester (2006) gives further connections between equidiagonal and orthodiagonal quadrilaterals, via a generalization of van Aubel's theorem.[8] Quadrilaterals that are both orthodiagonal and equidiagonal, and in which the diagonals are at least as long as all of the quadrilateral's sides, have the maximum area for their diameter among all quadrilaterals, solving the n = 4 case of the biggest little polygon problem. The square is one such quadrilateral, but there are infinitely many others. Equidiagonal, orthodiagonal quadrilaterals have been referred to as midsquare quadrilaterals [4]:p. 137 because they are the only ones for which the Varignon parallelogram (with vertices at the midpoints of the quadrilateral's sides) is a square. Such a quadrilateral, with successive sides a, b, c, d, has area[4]:Thm. 16 K=\frac{a^2+c^2+\sqrt{4(a^2c^2+b^2d^2)-(a^2+c^2)^2}}{4}.

QMRProperties of the diagonals in some quadrilaterals[edit] In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are perpendicular, and if their diagonals have equal length.[19] The list applies to the most general cases, and excludes named subsets. Quadrilateral Bisecting diagonals Perpendicular diagonals Equal diagonals Trapezoid No See note 1 No Isosceles trapezoid No See note 1 Yes Parallelogram Yes No No Kite See note 2 Yes See note 2 Rectangle Yes No Yes Rhombus Yes Yes No Square Yes Yes Yes Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named quadrilateral. Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but there is an infinite number of (non-similar) kites in which the diagonals are equal in length (and the kites are not any other named quadrilateral).

Diagonals and bimedians[edit] A corollary to Euler's quadrilateral theorem is the inequality a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2 where equality holds if and only if the quadrilateral is a parallelogram. Euler also generalized Ptolemy's theorem, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that pq \le ac + bd where there is equality if and only if the quadrilateral is cyclic.[18]:p.128–129 This is often called Ptolemy's inequality. In any convex quadrilateral the bimedians m, n and the diagonals p, q are related by the inequality pq \leq m^2+n^2, with equality holding if and only if the diagonals are equal.[31]:Prop.1 This follows directly from the quadrilateral identity m^2+n^2=\frac{1}{2}(p^2+q^2).

QMRIn Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

Special cases[edit] A kite is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals.[1] A rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral that is also a parallelogram). A square is a limiting case of both a kite and a rhombus. Orthodiagonal equidiagonal quadrilaterals in which the diagonals are at least as long as all of the quadrilateral's sides have the maximum area for their diameter among all quadrilaterals, solving the n = 4 case of the biggest little polygon problem. The square is one such quadrilateral, but there are infinitely many others.

QMRPolyhedra appeared in early architectural forms such as cubes and cuboids, with the earliest four-sided pyramids of ancient Egypt also dating from the Stone Age. The Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 19th century of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987).

QMRThere are also four regular star polyhedra, known as the Kepler-Poinsot polyhedra after their discoverers. The dual of a regular polyhedron is also regular.

QMRA tetrose is a monosaccharide with 4 carbon atoms. They have either an aldehyde functional group in position 1 (aldotetroses) or a ketone functional group in position 2 (ketotetroses).[1][2] D-Erythrose D-Threose D-Erythrulose The aldotetroses have two chiral centers ("asymmetric carbon atoms") and so 4 different stereoisomers are possible. There are two naturally occurring stereoisomers, the enantiomers of erythrose and threose having the D configuration but not the L enantiomers. The ketotetroses have one chiral center and, therefore, two possible stereoisomers: erythrulose (L- and D-form). Again, only the D enantiomer is naturally occurring.

Erythrose is a tetrose carbohydrate with the chemical formula C4H8O4. It has one aldehyde group, and so is part of the aldose family. The natural isomer is D-erythrose. Fischer projections Erythrose was first isolated in 1849 from rhubarb by the French pharmacist Louis Feux Joseph Garot (1798-1869),[2] and was named as such because of its red hue in the presence of alkali metals (ἐρυθρός, "red").[3][4] Erythrose 4-phosphate is an intermediate in the pentose phosphate pathway[5] and the Calvin cycle.[6] Erythrulose, an isomer of erythrose, is non-toxic.[7] Oxidative bacteria can be made to use erythrose as its sole energy source

QMrThe term is most common in biochemistry, where it is a synonym of saccharide, a group that includes sugars, starch, and cellulose. The saccharides are divided into four chemical groups: monosaccharides, disaccharides, oligosaccharides, and polysaccharides. In general, the monosaccharides and disaccharides, which are smaller (lower molecular weight) carbohydrates, are commonly referred to as sugars.[

16 is the squares of the quadrant model

Monosaccharides are classified according to three different characteristics: the placement of its carbonyl group, the number of carbon atoms it contains, and its chiral handedness. If the carbonyl group is an aldehyde, the monosaccharide is an aldose; if the carbonyl group is a ketone, the monosaccharide is a ketose. Monosaccharides with three carbon atoms are called trioses, those with four are called tetroses, five are called pentoses, six are hexoses, and so on.[13] These two systems of classification are often combined. For example, glucose is an aldohexose (a six-carbon aldehyde), ribose is an aldopentose (a five-carbon aldehyde), and fructose is a ketohexose (a six-carbon ketone). Each carbon atom bearing a hydroxyl group (-OH), with the exception of the first and last carbons, are asymmetric, making them stereo centers with two possible configurations each (R or S). Because of this asymmetry, a number of isomers may exist for any given monosaccharide formula. Using Le Bel-van't Hoff rule, the aldohexose D-glucose, for example, has the formula (C·H2O) 6, of which four of its six carbons atoms are stereogenic, making D-glucose one of 24=16 possible stereoisomers. In the case of glyceraldehydes, an aldotriose, there is one pair of possible stereoisomers, which are enantiomers and epimers. 1, 3-dihydroxyacetone, the ketose corresponding to the aldose glyceraldehydes, is a symmetric molecule with no stereo centers. The assignment of D or L is made according to the orientation of the asymmetric carbon furthest from the carbonyl group: in a standard Fischer projection if the hydroxyl group is on the right the molecule is a D sugar, otherwise it is an L sugar. The "D-" and "L-" prefixes should not be confused with "d-" or "l-", which indicate the direction that the sugar rotates plane polarized light. This usage of "d-" and "l-" is no longer followed in carbohydrate chemistry.[14]