X-Nico

unusual facts about Dual polytope



10-orthoplex

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.

5-orthoplex

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.

6-orthoplex

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.

7-orthoplex

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.

8-orthoplex

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.

9-orthoplex

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.


see also