41 relations: Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, E8 lattice, Face (geometry), Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, Kissing number problem, Messenger of Mathematics, Norman Johnson (mathematician), Regular polytope, Regular Polytopes (book), Schläfli symbol, Semiregular polytope, Simplex, Sphere packing, Tetrahedron, Thorold Gosset, Triangle, Uniform honeycomb, Uniform k 21 polytope, Vertex arrangement, Vertex figure, Wythoff construction, 1 22 polytope, 1 52 honeycomb, 2 21 polytope, 2 51 honeycomb, 3 21 polytope, 4 21 polytope, 5-cell, 5-demicube, 5-simplex, 6-simplex, 7-simplex, 8-demicubic honeycomb, 8-orthoplex, 8-simplex, 8-simplex honeycomb.

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 geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions.

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In mathematics, the E8 lattice is a special lattice in R8.

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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 geometry, a kissing number is defined as the number of non-overlapping unit spheres that can be arranged such that they each touch another given unit sphere.

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The Messenger of Mathematics is a defunct mathematics journal.

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Norman W. Johnson (born November 12, 1930) is a mathematician, previously at Wheaton College, Norton, Massachusetts.

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In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry.

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Regular Polytopes is a mathematical geometry book written by Canadian mathematician H.S.M. Coxeter.

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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, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes.

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In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.

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In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space.

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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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John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.

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A triangle is a polygon with three edges and three vertices.

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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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In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the ''E''''n'' Coxeter group, and having only regular polytope facets.

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In geometry, a vertex arrangement is a set of points in space described by their relative positions.

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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 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.

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In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.

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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 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation.

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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 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 demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices truncated.

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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 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.

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The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space.

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In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

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In geometry, an 8-simplex is a self-dual regular 8-polytope.

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In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb).

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E8 honeycomb, Gosset 5 21 honeycomb, Gosset 5 21 lattice.