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Runcinated 8-simplex
In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations ( runcination) of the regular 8-simplex. There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The ''triruncinated 8-simplex'' and ''triruncicantitruncated 8-simplex'' have a doubled symmetry, showing 8order reflectional symmetry in the A8 Coxeter plane. Runcinated 8-simplex Alternate names * Runcinated enneazetton * Small prismated enneazetton (Acronym: spene) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of the ''runcinated 8-simplex'' can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 9-orthoplex. Images Biruncinated 8-simplex Alternate names * Biruncinated enneazetton * Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers) Coordinates The Cartesian coordinates of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-simplex T0
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral 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- facetted polytope in eight-dimensions. The name ''enneazetton'' is derived from ''ennea'' for nine facets in Greek and ''-zetta'' for having seven-dimensional facets, and ''-on''. As a configuration This 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 many of each element occur in the whole 8-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation. \begin\begin 9 & 8 & 28 & 56 & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. Uniform truncation In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron, represented as Schläfli symbols r or \begin 5 \\ 3 \end, and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodecahedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8-polytope
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope 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 facets around each peak. There are exactly three such 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 coefficients.Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 8-polytopes, whatever their underlying topology. This inadequa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runcinated 9-orthoplex
In geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets in place of the original face, edge, and vertex centers. It is a higher order truncation operation, following cantellation, and truncation. It is represented by an extended Schläfli symbol t0,3. This operation only exists for 4-polytopes or higher. This operation is dual-symmetric for regular uniform 4-polytopes and 3-space convex uniform honeycombs. For a regular 4-polytope, the original cells remain, but become separated. The gaps at the separated faces become p-gonal prisms. The gaps between the separated edges become r-gonal prisms. The gaps between the separated vertices become cells. The vertex figure for a regular 4-polytope is an ''q''-gonal antiprism (called an ''antipodium'' if ''p'' and ''r'' are different). For regular 4-polytopes/honeycombs, this operation is also called expansion by Alicia Bool ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Facet (geometry)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: * In three-dimensional geometry, a facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a '' face''. To '' facet'' a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to '' stellation'' and may also be applied to higher-dimensional polytopes. * In polyhedral combinatorics and in the general theory of polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...s, a facet (or hyperface) of a polytope of dimension ''n'' is a face that has dimension ''n'' − 1. Facets may also be called (''n'' −&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 . Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tilings or, by extension, to space-filling tessellation with polytope cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. This i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
