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Chessboard Paradox
The chessboard paradoxGreg N. Frederickson: ''Dissections: Plane and Fancy''. Cambridge University Press, 2003, , chapter 23, pp. 268–277 in particular pp. 271–274 Colin Foster: "Slippery Slopes". In: ''Mathematics in School'', vol. 34, no. 3 (May, 2005), pp. 33–34JSTOR or paradox of Loyd and SchlömilchFranz Lemmermeyer: ''Mathematik à la Carte: Elementargeometrie an Quadratwurzeln mit einigen geschichtlichen Bemerkungen''. Springer 2014, , pp95–96(German) is a falsidical paradox based on an optical illusion. A chessboard or a square with a side length of 8 units is cut into four pieces. Those four pieces are used to form a rectangle with side lengths of 13 and 5 units. Hence the combined area of all four pieces is 64 area units in the square but 65 area units in the rectangle, this seeming contradiction is due an optical illusion as the four pieces don't fit exactly in the rectangle, but leave a small barely visible gap around the rectangle's diagonal. The paradox is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similarity (geometry)
In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (geometry), scaling (enlarging or reducing), possibly with additional translation (geometry), translation, rotation (mathematics), rotation and reflection (mathematics), reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruence (geometry), congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. This is because two ellipse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Paradoxes
Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to prove properties of objects, a ''proof'' consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Mathematics
Elementary mathematics, also known as primary or secondary school mathematics, is the study of mathematics topics that are commonly taught at the primary or secondary school levels around the world. It includes a wide range of mathematical concepts and skills, including number sense, algebra, geometry, measurement, and data analysis. These concepts and skills form the foundation for more advanced mathematical study and are essential for success in many fields and everyday life. The study of elementary mathematics is a crucial part of a student's education and lays the foundation for future academic and career success. Strands of elementary mathematics Number sense and numeration Number sense is an understanding of numbers and operations. In the 'Number Sense and Numeration' strand students develop an understanding of numbers by being taught various ways of representing numbers, as well as the relationships among numbers. Properties of the natural numbers such as divisibil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optical Illusions
In visual perception, an optical illusion (also called a visual illusion) is an illusion caused by the visual system and characterized by a visual percept that arguably appears to differ from reality. Illusions come in a wide variety; their categorization is difficult because the underlying cause is often not clear but a classification proposed by Richard Gregory is useful as an orientation. According to that, there are three main classes: physical, physiological, and cognitive illusions, and in each class there are four kinds: Ambiguities, distortions, paradoxes, and fictions. A classical example for a physical distortion would be the apparent bending of a stick half immersed in water; an example for a physiological paradox is the motion aftereffect (where, despite movement, position remains unchanged). An example for a physiological fiction is an afterimage. Three typical cognitive distortions are the Ponzo, Poggendorff, and Müller-Lyer illusion. Physical illusions are cau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Missing Square Puzzle
The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it. Solution The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, nor would either truly be 13x5 if it were, because what appears to be the hypotenuse is bent. In other words, the "hypotenuse" does not maintain a consistent slope, even though it may appear that way to the human eye. A true 13×5 triangle cannot be created from the given component parts. The four figures (the yellow, red, blue and green shapes) total 32 units of area. The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lewis Carroll
Charles Lutwidge Dodgson (27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet, mathematician, photographer and reluctant Anglicanism, Anglican deacon. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ''Through the Looking-Glass'' (1871). He was noted for his facility with word play, logic, and fantasy. His poems ''Jabberwocky'' (1871) and ''The Hunting of the Snark'' (1876) are classified in the genre of literary nonsense. Some of Alice's nonsensical wonderland logic reflects his published work on mathematical logic. Carroll came from a family of high-church Anglicanism, Anglicans, and pursued his clerical training at Christ Church, Oxford, where he lived for most of his life as a scholar, teacher and (necessarily for his academic fellowship at the time) Anglican deacon. Alice Liddell – a daughter of Henry Liddell, the Dean of Christ Church, Oxford, Dean of Christ Church – is wide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Schlegel
Victor Schlegel (4 March 1843 – 22 November 1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams. In the nineteenth century there were various expansions of the traditional field of geometry through the innovations of hyperbolic geometry, non-Euclidean geometry and algebraic geometry. Hermann Grassmann was one of the more advanced innovators with his anticipation of linear algebra and multilinear algebra that he called "Extension theory" (''Ausdehnungslehre''). As recounted by David E. Rowe in 2010: :The most important new convert was Victor Schlegel, Grassmann’s colleague at Stettin Gymnasium from 1866 to 1868. Afterward Schlegel accepted a position as Oberlehrer at the Gymnasium in Waren, a small town in Mecklenburg. In 1872 Schlegel published the first part of his ''System der Raumlehre'' which used Grassmann’s methods to develop plane geometr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmé Gilles Guyot
The name Edmé may refer to: *Edmé Bouchardon (1698–1762), French sculptor * Edmé Boursault (1638–1701), French writer and dramatist * Edme Castaing (1796–1823), French physician * Edmé-Louis Daubenton (1732–1786), French naturalist * Gaston Audiffret-Pasquier (1823–1905), born Edme-Armand-Gaston d'Audiffret-Pasquier, French politician * Edme Étienne Borne Desfourneaux (1767–1849), French general * Edme Gaulle (1762–1841), French sculptor * Edme Henry (1760–1841), Canadian politician * Edme François Jomard (1777–1862), French engineer and cartographer * Edme-Jean Leclaire (1801–1872), French economist *Edme Mariotte (1620–1684), French physicist * Edme Mongin (1668–1746), French bishop and orator * Edmé Samson (1810–1891), French ceramist *Edmé-Martin Vandermaesen Edmé-Martin, comte Vandermaesen (Versailles (city), Versailles, 11 November 1767 – 1 September 1813; also spelled Vander Maesen) was a French general of the French Revolutionary Wars ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hooper's Paradox
Hooper's paradox is a falsidical paradox based on an optical illusion. A geometric shape with an area of 32 units is dissected into four parts, which afterwards get assembled into a rectangle with an area of only 30 units. Explanation Upon close inspection one can notice that the triangles of the dissected shape are not identical to the triangles in the rectangle. The length of the shorter side at the right angle measures 2 units in the original shape but only 1.8 units in the rectangle. This means, the real triangles of the original shape overlap in the rectangle. The overlapping area is a parallelogram, the diagonals and sides of which can be computed via the Pythagorean theorem. : d_1=\sqrt=\sqrt : d_2=\sqrt=\sqrt : s_1=\sqrt=\sqrt : s_2=\sqrt=\sqrt The area of this parallelogram can determined using Heron's formula for triangles. This yields : s=\frac=\frac for the halved circumference of the triangle (half of the parallelogram) and with that for the area of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hooper Paradox
''Hooper'' may refer to: Place names in the United States: * Hooper, Colorado, town in Alamosa County, Colorado * Hooper, Georgia, an unincorporated community * Hooper, Nebraska, town in Dodge County, Nebraska * Hooper, Utah, place in Weber County, Utah * Hooper Bay, Alaska, town in Alaska * Hooper Township, Dodge County, Nebraska Other: * ''Hooper'' (film), 1978 comedy film starring Burt Reynolds * Hooper (mascot), the mascot for the National Basketball Association team, Detroit Pistons * Hooper (coachbuilder), a British coachbuilder fitting bodies to many Rolls-Royce and Daimler cars * USS ''Hooper'' (DE-1026), a destroyer escort in the US Navy * Hooper Ratings, an early audience measurement in early radio and television * Hooper, someone who practices dance form of Hooping * Hooper, an archaic English term for a person who aided a cooper in the building of barrels People with the surname Hooper: * Hooper (surname) See also * Hooper X, a character in Kevin Smith's 1997 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albrecht Beutelspacher
Albrecht Beutelspacher (born 5 June 1950) is a German mathematician and founder of the Mathematikum. He is a professor emeritus at the University of Giessen, where he held the chair for geometry and discrete mathematics from 1988 to 2018. Biography Beutelspacher studied from 1969 to 1973 math, physics and philosophy at the University of Tübingen and received his PhD 1976 from the University of Mainz. His PhD advisor was Judita Cofman. From 1982 to 1985 he was an associate professor at the University of Mainz and from 1985 to 1988 he worked at a research department of Siemens. From 1988 to 2018 he was a tenured professor for geometry and discrete mathematics at the University of Giessen University of Giessen, official name Justus Liebig University Giessen (), is a large public research university in Giessen, Hesse, Germany. It is one of the oldest institutions of higher education in the German-speaking world. It is named afte .... He became a well-known popularizer of math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
