There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C10 or 4,38 symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D10 or 37,1,1 symmetry group.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the Coxeter group.
n-cube Coxeter plane projections in the Bk Coxeter groups project into k-cube graphs, with power of two vertices overlapping in the projective graphs.
There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or 4,3,3,3 Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or 32,1,1 Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.
The regular 5-simplex is one of 19 uniform polytera based on the 3,3,3,3 Coxeter group, all shown here in A5 Coxeter plane orthographic projections.
There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or 4,3,3,3,3 Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or 33,1,1 Coxeter group.
The regular 6-simplex is one of 35 uniform 6-polytopes based on the 3,3,3,3,3 Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or 4,3,3,3,3,3 symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or 34,1,1 symmetry group.
There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C8 or 4,3,3,3,3,3,3 symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D8 or 35,1,1 symmetry group.A lowest symmetry construction is based on a dual of a 8-orthotope, called a 8-fusil.
There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C9 or 4,37 symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D9 or 36,1,1 symmetry group.
The cantellated 5-simplex is one of 19 uniform polytera based on the 3,3,3,3 Coxeter group, all shown here in A5 Coxeter plane orthographic projections.
There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the BC6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
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There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the BC6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
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There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the BC6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
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There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the BC6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
The truncated 6-simplex is one of 35 uniform 6-polytopes based on the 3,3,3,3,3 Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
This polyhedron is also a part of a sequence of truncated rhombic polyhedra and tilings with n,3 Coxeter group symmetry.
The Coxeter groups of type Dn, E6, E7, and E8 are the symmetry groups of certain semiregular polytopes.
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Anders Björner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol.
The pentellated 6-simplex is one of 35 uniform 6-polytopes based on the 3,3,3,3,3 Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, h, is Coxeter number of the Coxeter group.
There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C10 or 4,38 Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D10 or 37,1,1 Coxeter group.
There are two Coxeter groups associated with the rectified pentacross, one with the C5 or 4,3,3,3 Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D5 or 32,1,1 Coxeter group.
It is also one of 19 uniform polytera based on the 3,3,3,3 Coxeter group, all shown here in A5 Coxeter plane orthographic projections.
There are two Coxeter groups associated with the rectified hexacross, one with the C6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or 33,1,1 Coxeter group.
These polytopes are a part of 35 uniform 6-polytopes based on the 3,3,3,3,3 Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
There are two Coxeter groups associated with the rectified heptacross, one with the C7 or 4,3,3,3,3,3 Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D7 or 34,1,1 Coxeter group.
There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or 4,36 Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or 35,1,1 Coxeter group.
There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C9 or 4,37 Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or 36,1,1 Coxeter group.
This polyhedron is a part of a sequence of rhombic polyhedra and tilings with n,3 Coxeter group symmetry.
These polytopes are in a set of 19 uniform polytera based on the 3,3,3,3 Coxeter group, all shown here in A5 Coxeter plane orthographic projections.
A Schläfli symbol is closely related to reflection symmetry groups, also called Coxeter groups, given with the same indices, but square brackets instead p,q,r,....
These polytopes are a part of 19 uniform polytera based on the 3,3,3,3 Coxeter group, all shown here in A5 Coxeter plane orthographic projections.
The truncated 5-simplex is one of 19 uniform polytera based on the 3,3,3,3 Coxeter group, all shown here in A5 Coxeter plane orthographic projections.
There are two Coxeter groups associated with the truncated hexacross, one with the C6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C7 or 4,35 Coxeter group, and a lower symmetry with the D7 or 34,1,1 Coxeter group.
There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C8 or 4,3,3,3,3,3,3 Coxeter group, and a lower symmetry with the D8 or 35,1,1 Coxeter group.
In geometry, 2k1 polytope is a uniform polytope in n dimensions (n = k+4) constructed from the Coxeter group.
The orthoplex faces are constructed from the Coxeter group Dn−1 and have a Schlafli symbol of {31,n−1,1} rather than the regular {3n−2,4}.
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In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the Coxeter group, and having only regular polytope facets.
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In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a one-parameter deformation of the group algebra of a Coxeter group.
As A. J. Coleman says, "He exhibited the characteristic equation of the Weyl group when Weyl was 3 years old and listed the orders of the Coxeter transformation 19 years before Coxeter was born."