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Eighth Dimension
In mathematics, a sequence of ''n'' real numbers can be understood as a location in ''n''-dimensional space. When ''n'' = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Eight-dimensional Euclidean space is eight-dimensional space equipped with the Euclidean metric. More generally the term may refer to an eight-dimensional vector space over any field, such as an eight-dimensional complex vector space, which has 16 real dimensions. It may also refer to an eight-dimensional manifold such as an 8-sphere, or a variety of other geometric constructions. Geometry 8-polytope A polytope in eight dimensions is called an 8-polytope. The most studied are the regular polytopes, of which there are only three in eight dimensions: the 8-simplex, 8-cube, and 8-orthoplex. A broader family are the uniform 8-polytopes, constructed from fundamental symmetry domains of reflection, each domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 8-polytope
In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope Ridge (geometry), ridge being shared by exactly two 7-polytope Facet (mathematics), facets. A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets. Regular 8-polytopes Regular 8-polytopes can be represented by the Schläfli symbol , with v 7-polytope Facet (mathematics), facets around each Peak (geometry), peak. There are exactly three such List of regular polytopes#Convex 4, convex regular 8-polytopes: # - 8-simplex # - 8-cube # - 8-orthoplex There are no nonconvex regular 8-polytopes. Characteristics The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficient (topology), torsion coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4 21 T0 E8
4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures. In mathematics Four is the smallest composite number, its proper divisors being and . Four is the sum and product of two with itself: 2 + 2 = 4 = 2 x 2, the only number b such that a + a = b = a x a, which also makes four the smallest squared prime number p^. In Knuth's up-arrow notation, , and so forth, for any number of up arrows. By consequence, four is the only square one more than a prime number, specifically three. The sum of the first four prime numbers two + three + five + seven is the only sum of four consecutive prime numbers that yields an odd prime number, seventeen, which is the fourth super-prime. Four lies between the first proper pair of twin primes, three and five, which are the first two Fermat primes, like seventeen, which is the third. On the other hand, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases. Basic description The Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is eight. Therefore, the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the whole group, has order 2357 = . The compact group E8 is unique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-demicube T0 D7
In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8- hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM8 for an 8-dimensional ''half measure'' polytope. Coxeter named this polytope as 151 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol \left\ or . Cartesian coordinates Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube: : (±1,±1,±1,±1,±1,±1,±1,±1) with an odd number of plus signs. Related polytopes and honeycombs This polytope is the vertex figure for the uniform tessellation, 251 with Coxeter-Dynkin diagram: : Images References * H.S.M. Coxeter: ** Coxeter, ''Regular Polytopes'', (3rd edition, 1973), Dover edition, , p. 296, Table I ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-cube T7
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 Vertex (geometry), vertices, 1024 Edge (geometry), edges, 1792 square Face (geometry), faces, 1792 cubic Cell (mathematics), cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces. It is represented by Schläfli symbol , being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the ''4-cube'') and ''oct'' for eight (dimensions) in Greek language, Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-polytope, 8-dimensional polytope constructed from 16 regular Facet (geometry), facets. It is a part of an infinite family of polytopes, called hypercubes. The Dual polytope, dual of an 8-cube can be called an 8-orthoplex and is a part of the infinite family of cross-polytopes. Cartesian coordinates Cartesian coordinates for the vertices of an 8-cube centered at the origin and edge length 2 are : (±1,±1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-cube T0
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 Vertex (geometry), vertices, 1024 Edge (geometry), edges, 1792 square Face (geometry), faces, 1792 cubic Cell (mathematics), cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces. It is represented by Schläfli symbol , being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the ''4-cube'') and ''oct'' for eight (dimensions) in Greek language, Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-polytope, 8-dimensional polytope constructed from 16 regular Facet (geometry), facets. It is a part of an infinite family of polytopes, called hypercubes. The Dual polytope, dual of an 8-cube can be called an 8-orthoplex and is a part of the infinite family of cross-polytopes. Cartesian coordinates Cartesian coordinates for the vertices of an 8-cube centered at the origin and edge length 2 are : (±1,±1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-simplex T0
In geometry, an 8-simplex is a self-dual Regular polytope, regular 8-polytope. It has 9 vertex (geometry), vertices, 36 Edge (geometry), edges, 84 triangle Face (geometry), faces, 126 tetrahedral Cell (mathematics), cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately 82.82°. It can also be called an enneazetton, or ennea-8-tope, as a 9-facet (geometry), facetted polytope in eight-dimensions. The 5-polytope#A note on generality of terms for n-polytopes and elements, name ''enneazetton'' is derived from ''ennea'' for nine Facet (mathematics), facets in Greek language, Greek and Zetta, ''-zetta'' for having seven-dimensional facets, and ''-on''. As a configuration This Regular 4-polytope#As configurations, configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number ''h'' of an irreducible root system. A Coxeter element is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. *The Coxeter number is the order of any Coxeter element;. *The Coxeter number is 2''m''/''n'', where ''n'' is the rank, and ''m'' is the number of reflections. In the crystallographic case, ''m'' is half the number of roots; and ''2m''+''n'' is the dimension of the corresponding semisimple Lie algebra. *If the highest root is Σ''m''iα''i'' for simple roots α''i'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 42 Polytope
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences. The rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241, and the quadrirectified 421. These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: . The projection of 142 to the E8 Coxeter plane (aka. the Petrie projection) with polytope radius 4\sqrt is shown below with 483,840 edges of length 2\sqrt culled 53% on the interior to only 226,444: 142 polytope The 142 is composed of 2400 facets: 240 132 polytopes, and 2160 7-demicubes (141). Its vertex figure is a birectified 7-simplex. This polytope, along ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2 41 Polytope
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 241, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences. The rectified 241 is constructed by points at the mid-edges of the 241. The birectified 241 is constructed by points at the triangle face centers of the 241, and is the same as the rectified 142. These polytopes are part of a family of 255 (28 − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets, defined by all permutations of rings in this Coxeter-Dynkin diagram: . 241 polytope {, class="wikitable" align="right" style="margin-left:10px" width="280" !bgcolor=#e7dcc3 colspan=2, 241 polytope , - , bgcolor=#e7dcc3, Type, , Uniform 8-polytope , - , bgcolor=#e7dcc3, Family, , 2k1 polytope , - , bgcolor=#e7dcc3, Schläfli symbol, , {3,3,34,1} , - , bgcolor=#e7dcc3, Coxeter symbol, , 241 , - , b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4 21 Polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an ''8-ic semi-regular figure''.Gosset, 1900 Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, . The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is constructed by points at the tetrahedral centers of the 421. These polytopes are part of a family of 255 = 28 − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: . 421 polytope The 421 polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the 321 po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
