Coxeter called it the Witting polytope, after Alexander Witting.
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.
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.
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 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.
In the mid 1970s he contacted the Canadian geometer Coxeter as he felt his art work was of some mathematical interest as well.
J. Richard Gott in 1967 published a larger set of seven infinite skew polyhedra which he called regular pseudopolyhedrons, including the three from Coxeter as {4,6}, {6,4}, and {6,6} and four new ones: {5,5}, {4,5}, {3,8}, {3,10}.
In combinatorial mathematics, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by Coxeter and Frucht, for the representation of cubic graphs that are Hamiltonian.
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.
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.
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.
Coxeter's manuscript collections were largely used in Theophilus Cibber's Lives of the Poets and in Thomas Warton's History of English Poetry.
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.
The family was named by Coxeter as k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.
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."
The Coxeter–Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.