|
Witting Polytope
In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3333, and Coxeter diagram . It has 240 vertices, 2160 3 edges, 2160 33 faces, and 240 333 cells. It is self-dual. Each vertex belongs to 27 edges, 72 faces, and 27 cells, corresponding to the Hessian polyhedron vertex figure. Symmetry Its symmetry by 3 sub>3 sub>3 sub>3 or , order 155,520. It has 240 copies of , order 648 at each cell. Structure The configuration matrix is: \left begin240&27&72&27\\3&2160&8&8\\8&8&2160&3\\27&72&27&240\end\right /math> The number of vertices, edges, faces, and cells are seen in the diagonal of the matrix. These are computed by the order of the group divided by the order of the subgroup, by removing certain complex reflections, shown with X below. The number of elements of the k-faces are seen in rows below the diagonal. The number of elements in the vertex figure, etc., are given in rows above the digonal. Coordinates Its 240 vertices are give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Regular Complex Polytope
In geometry, a complex polytope is a generalization of a polytope in real coordinate space, real space to an analogous structure in a Complex number, complex Hilbert space, where each real dimension is accompanied by an imaginary number, imaginary one. A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Precise definitions exist only for the #Regular complex polytopes, regular complex polytopes, which are Configuration (polytope), configurations. The regular complex polytopes have been completely characterized, and can be described using a symbolic notation developed by Harold Scott MacDonald Coxeter, Coxeter. Some complex polytopes which are not fully regular have also been described. Definitions and introduction The complex line \mathbb^1 has one dimension with real number, real coordinates and another with imaginary number, imaginary coor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including '' The Fifty-Nine Icosahedra'' (1938) and '' Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
F-vector
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics of polytopes; for instance, they seek inequalities that describe the relations between the numbers of vertices, edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their connectivity and diameter (number of steps needed to reach any vertex from any other vertex). Additionally, many computer scientists use the phrase “polyhedral combinatorics” to describe research into precise descriptions of the faces of certain specific polytopes (especially 0-1 polytopes, whose vertices are subsets of a hypercube) arising from integer pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
5 21 Honeycomb
In geometry, the 521 honeycomb is a Uniform honeycomb, uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.Coxeter, 1973, Chapter 5: The Kaleidoscope By putting spheres at its vertices one obtains the densest-possible packing of spheres in 8 dimensions. This was proven by Maryna Viazovska in 2016 using the theory of Modular form, modular forms. Viazovska was awarded the Fields Medal for this work in 2022. This honeycomb was first studied by Gosset who called it a ''9-ic semi-regular figure'' (Gosset regarded honeycombs in ''n'' dimensions as degenerate ''n''+1 polytopes). Each vertex of the 521 honeycomb is surrounded by 2160 8-orthoplex, 8-orthoplexes and 17280 8-simplex, 8-simplicies. The vertex figure of Gosset's honeycomb is the semiregular 4 21 polytope, 421 polytope. It is the final figure in the Semiregular k 21 polytope, k21 family. This honeycomb is highly regular i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
4 21 Polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 (mathematics), E8 group (mathematics), 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 Facet (geometry), f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Alexander Witting
Carl Johann Adolf Alexander Witting (18 December 1861 – 29 November 1946) was a German mathematician. Family Witting was born in Dresden as the first child of the musician Carl Witting (1823–1907) and the painter Minna Witting, née Japha (1828–1882). Alexander Witting married the pianist Sophie Sebass (1864–1924) in 1889. They had two daughters and a son: Tillyta (1890–1970), Lotte (1894–1971) and the physicist Rudolf Witting (1899–1963). In view of the artistically affected family environment – father musician, mother painter, aunt Louise Japha (1826–1910) pianist, sister Agnes Witting (1863–1937) singer, brother Walther Witting painter – it does not surprise that Alexander Witting also painted sometimes and regularly made music, even beyond the narrow circle of family or colleagues. Education He successfully completed his final grammar school examinations at the Städtische Gymnasium zum Heiligen Kreuz (Kreuzschule) in Dresden at Easter 1880, served ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line. Most commonly, the ambient space is -dimensional Euclidean space, in which case the hyperplanes are the -dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion ( geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces. In non-Euclidean geometry, the ambient space might be the -dimensional sphere or hyperbolic space, or more generally a pseudo-Riemannian space form, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
K-face
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] [Amazon] |
Configuration (polytope)
In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration. Other configurations in geometry are something different. These ''polytope configurations'' may be more accurately called incidence matrices, where like elements are collected together in rows and columns. Regular polytopes will have one row and column per ''k''-face element, while other polytopes will have one row and column for each k-face type by their symmetry classes. A polytope with no symmetry will have one row and column for every element, and the matrix will be filled with 0 if the elements are not connected, and 1 if they are connected. Elements of the same ''k'' will not be connected and will have a "*" table entry. Every polytope, and abstract polytope has a Hasse diagram expressing these connectivities, which can be systematically described with an incidence matrix. Configuration matrix for regular polytopes A configuration for a regular polytope is represented by a matrix w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
