Hexicated 7-orthoplexes explained

In seven-dimensional geometry, a hexicated 7-orthoplex (also hexicated 7-cube) is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-orthoplex.

There are 32 hexications for the 7-orthoplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 12 are represented here, while 20 are more easily constructed from the 7-cube.

Hexitruncated 7-orthoplex

Hexitruncated 7-orthoplex
Type	Uniform 7-polytope
Schläfli symbol	t0,1,6 Alternate names Petitruncated heptacross Images Hexicantellated 7-orthoplex Hexicantellated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,2,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 94080 Vertices 13440 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Petirhombated heptacross Images Hexicantitruncated 7-orthoplex Hexicantitruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,2,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 134400 Vertices 26880 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Petigreatorhombated heptacross Images Hexiruncitruncated 7-orthoplex Hexiruncitruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,3,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 322560 Vertices 53760 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Petiprismatotruncated heptacross Images Hexiruncicantellated 7-orthoplex Hexiruncicantellated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,2,3,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 268800 Vertices 53760 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex In seven-dimensional geometry, a hexiruncicantellated 7-orthoplex is a uniform 7-polytope. Alternate names Petiprismatorhombated heptacross Images Hexisteritruncated 7-orthoplex hexisteritruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,4,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 322560 Vertices 53760 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Peticellitruncated heptacross Images Hexiruncicantitruncated 7-orthoplex Hexiruncicantitruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,2,3,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 483840 Vertices 107520 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Petigreatoprismated heptacross Images Hexistericantitruncated 7-orthoplex Hexistericantitruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,2,4,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 806400 Vertices 161280 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Peticelligreatorhombated heptacross Images Hexisteriruncitruncated 7-orthoplex Hexisteriruncitruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,3,4,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 725760 Vertices 161280 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Peticelliprismatotruncated heptacross Images Hexipenticantitruncated 7-orthoplex hexipenticantitruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,2,5,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 483840 Vertices 107520 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Petiterigreatorhombated heptacross Images Hexisteriruncicantitruncated 7-orthoplex Hexisteriruncicantitruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,2,3,4,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 1290240 Vertices 322560 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Great petacellated heptacross Images Hexipentiruncicantitruncated 7-orthoplex Hexipentiruncicantitruncated 7-orthoplex Type uniform 7-polytope Schläfli symbol t0,1,2,3,5,6 Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 1290240 Vertices 322560 Vertex figure Coxeter groups B7, [4,3<sup>5</sup>] Properties convex Alternate names Petiterigreatoprismated heptacross Images References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Norman Johnson Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, PhD (1966) External links Polytopes of Various Dimensions Multi-dimensional Glossary