39 relations: Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, E7 (mathematics), Edge (geometry), Emanuel Lodewijk Elte, Face (geometry), Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, List of E7 polytopes, Octadecagon, Petrie polygon, Projection (linear algebra), Rectification (geometry), Rectified 6-simplexes, Schläfli symbol, Simple Lie group, Tetrahedron, Triangle, Uniform 2 k1 polytope, Uniform 7-polytope, Uniform honeycomb, Uniform polytope, Vertex (geometry), Vertex figure, Wythoff construction, 1 32 polytope, 2 21 polytope, 2 31 polytope, 3 21 polytope, 3 31 honeycomb, 5-cell, 5-orthoplex, 5-simplex, 6-demicube, 6-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).

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In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.

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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.

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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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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.

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Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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In geometry a hyperplane is a subspace of one dimension less than its ambient space.

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In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry.

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An octadecagon (or octakaidecagon) is a polygon with 18 sides and 18 vertices.

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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.

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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.

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In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

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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 group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.

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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.

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In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets.

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In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform 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.

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In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 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 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group.

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In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

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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-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices truncated.

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In geometry, a 6-simplex is a self-dual regular 6-polytope.

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## Redirects here:

Gosset 2 31 polytope, Rectified 2 31 polytope.