41 relations: Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, E8 (mathematics), Edge (geometry), Emanuel Lodewijk Elte, Face (geometry), Facet (geometry), Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, List of E8 polytopes, Petrie polygon, Projection (linear algebra), Rectification (geometry), Rectified 6-simplexes, Rectified 7-simplexes, Regular polygon, Schläfli symbol, Tetrahedron, Triangle, Uniform 2 k1 polytope, Uniform 8-polytope, Uniform polytope, Vertex (geometry), Vertex figure, Wythoff construction, 1 22 polytope, 1 42 polytope, 2 21 polytope, 2 31 polytope, 2 51 honeycomb, 4 21 polytope, 5-cell, 5-orthoplex, 5-simplex, 6-simplex, 7-demicube, 7-simplex.
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.
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In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.
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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).
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In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).
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.
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In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope.
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Emanuel Lodewijk Elte (16 March 1881, Amsterdam – 9 April 1943, Sobibór) at joodsmonument.nl was a Dutch mathematician.
In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.
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In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.
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Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles.
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.
In geometry a hyperplane is a subspace of one dimension less than its ambient space.
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In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry.
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every (n-1) consecutive sides (but no n) belong to one of the facets.
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In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.
In Euclidean geometry, rectification or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
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In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
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In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, three of which meet at each corner or vertex.
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A triangle is a polygon with three edges and three vertices.
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In geometry, 2k1 polytope is a uniform polytope in n dimensions (n.
In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets.
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A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
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In geometry, a vertex (plural vertices) is a special kind of point that describes the corners or intersections of geometric shapes.
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In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.
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In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.
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In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
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In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group.
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In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
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In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation.
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In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.
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In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.
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In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
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In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.
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In geometry, a 6-simplex is a self-dual regular 6-polytope.
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In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices truncated.
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In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.
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