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Snub 24-cell
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular Tetrahedron, tetrahedral and 24 Regular icosahedron, icosahedral cell (mathematics), cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra. Topologically, under its highest symmetry, [3+,4,3], as an alternation of a truncated 24-cell, it contains 24 pyritohedra (an icosahedron with Th symmetry), 24 regular tetrahedra, and 96 triangular pyramids. Semiregular polytope It is one of three Semiregular 4-polytopes#Semiregular polytopes, semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a ''tetricosahedric'' for be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schlegel Half-solid Alternated Cantitruncated 16-cell
Schlegel is a German occupational surname. Notable people with the surname include: * Anthony Schlegel (born 1981), American football player * August Wilhelm Schlegel (1767–1845), German poet, brother of Friedrich * Brad Schlegel (born 1968), Canadian ice hockey player * Bernhard Schlegel (born 1951), German-Canadian chemist * Carmela Schlegel (born 1983), Swiss swimmer * Catharina von Schlegel (1697 – after 1768), German hymn writer * Dorothea von Schlegel (1764–1839), German novelist and translator, wife of Friedrich * Elfi Schlegel (born 1964), Canadian gymnast and sportscaster * Frits Schlegel (1896–1965), Danish architect * Gustaaf Schlegel (1840–1903), Dutch sinologist and field naturalist * Hans Schlegel (born 1951), German astronaut * Helmut Schlegel (born 1943), German Franciscan, priest, author, meditation instructor, songwriter * Hermann Schlegel (1804–1884), German ornithologist and herpetologist * Johan Frederik Schlegel (1817–1896), Danish civil servant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 600-cell
In geometry, the Rectification (geometry), rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cell (mathematics), cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices. Containing the cell Hyperplane#Notes, realms of both the regular 120-cell and the regular 600-cell, it can be considered analogous to the polyhedron icosidodecahedron, which is a rectified icosahedron and rectified dodecahedron. The vertex figure of the rectified 600-cell is a uniform pentagonal prism. Semiregular polytope It is one of three Semiregular 4-polytopes#Semiregular polytopes, semiregular 4-polytopes made of two or more cells which are Platonic solids, discovered by Thorold Gosset in his 1900 paper. He called it a ''octicosahedric'' for being made of octahedron and icosahedron cells. Emanuel Lodewijk Elte, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism. Topologically, under its highest symmetry, ,3,3 there is only one geometrical form, containing 5 regular tetrahedra and 5 rectified tetrahedra (which is geometrically the same as a regular octahedron). It is also topologically identical to a tetrahedron-octahedron segmentochoron. The vertex figure of the ''rectified 5-cell'' is a uniform triangular prism, formed by three octahedra around the sides, and two tetrahedra on the opposite ends. Despite having the same number of vertices as cells (10) and the same number of edges as faces (30), the rectified 5-cell is not self-dual bec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the ( convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , contai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tetrahedron is the simplest of all the ordinary convex polytope, convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean geometry, Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid (geometry), pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such net (polyhedron), nets. For any tetrahedron there exists a sphere (called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and for his generalization of Descartes' theorem on tangent circles to four and higher dimensions. Biography Thorold Gosset was born in Thames Ditton, the son of John Jackson Gosset, a civil servant and statistical officer for HM Customs,UK Census 1871, RG10-863-89-23 and his wife Eleanor Gosset (formerly Thorold). He was admitted to Pembroke College, Cambridge as a pensioner on 1 October 1888, graduated BA in 1891, was called to the bar of the Inner Temple in June 1895, and graduated LLM in 1896. In 1900 he married Emily Florence Wood, and they subsequently had two children, named Kathleen and John.UK Census 1911, RG14-181-9123-19 Mathematics According to H. S. M. Coxeter, after obtaining his law degree in 1896 and having no client ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (identical in shape and size) regular polygons (all angles congruent and all edge (geometry), edges congruent), and the same number of faces meet at each Vertex (geometry), vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the ''Timaeus (dialogue), Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular 4-polytopes
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is isogonal figure, vertex-transitive and has all its facet (geometry), facets being regular polytopes. E.L. Elte compiled a Emanuel_Lodewijk_Elte#Elte's semiregular polytopes of the first kind, longer list in 1912 as ''The Semiregular Polytopes of the Hyperspaces'' which included a wider definition. Gosset's list In three-dimensional space and below, the terms ''semiregular polytope'' and ''uniform polytope'' have identical meanings, because all uniform polygons must be regular polygon, regular. However, since not all uniform polyhedra are regular polyhedra, regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. The three convex semiregular 4-polytopes are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated 24-cell
In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the Truncation (geometry), truncation of the regular 24-cell. There are two degrees of truncations, including a bitruncation. Truncated 24-cell The truncated 24-cell or truncated icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cell (mathematics), cells: 24 cubes, and 24 truncated octahedron, truncated octahedra. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular pyramid vertex figure. Construction The truncated 24-cell can be constructed from polytopes with three symmetry groups: *F4 [3,4,3]: A Truncation (geometry), truncation of the 24-cell. *B4 [3,3,4]: A cantitruncation of the 16-cell, with two families of truncated octahedral cells. *D4 [31,1,1]: An omnitruncation of the demihypercube, demitesseract, with three families of truncated octahedral cells. Zonotope It is also a zonotop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cell (mathematics)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object. For example, a cube has six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that it may have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense. Polygonal face In elementary geometry, a face is a polygon on the boundary of a polyhedron. (Here a "polygon" should be viewed as including the 2-dimensional region inside it.) Other names for a polygonal face include polyhedron side and Euclidean plane ''tile''. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Number of polygonal faces of a polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
