Hexicated 7-orthoplexes

Orthogonal projections in B4 Coxeter plane
7-orthoplex	Hexicated 7-orthoplex Hexicated 7-cube	Hexi-truncated 7-orthoplex	Hexi-cantellated 7-orthoplex	Hexicanti-truncated 7-orthoplex
Hexirunci-truncated 7-orthoplex	Hexirunci-cantellated 7-orthoplex	Hexisteri-truncated 7-orthoplex	Hexiruncicanti-truncated 7-orthoplex	Hexistericanti-truncated 7-orthoplex
Hexisterirunci-truncated 7-orthoplex	Hexipenticanti-truncated 7-orthoplex	Hexisteriruncicanti-truncated 7-orthoplex	Hexipentiruncicanti-truncated 7-orthoplex

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{35,4
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	29568
Vertices	5376
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petitruncated heptacross

Hexicantellated 7-orthoplex

Hexicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	94080
Vertices	13440
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petirhombated heptacross

Hexicantitruncated 7-orthoplex

Hexicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	134400
Vertices	26880
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petigreatorhombated heptacross

Hexiruncitruncated 7-orthoplex

Hexiruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	322560
Vertices	53760
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petiprismatotruncated heptacross

Hexiruncicantellated 7-orthoplex

Hexiruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	268800
Vertices	53760
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

In seven-dimensional geometry, a hexiruncicantellated 7-orthoplex is a uniform 7-polytope.

Alternate names

• Petiprismatorhombated heptacross

Hexisteritruncated 7-orthoplex

hexisteritruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	322560
Vertices	53760
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Peticellitruncated heptacross

Hexiruncicantitruncated 7-orthoplex

Hexiruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	483840
Vertices	107520
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petigreatoprismated heptacross

Hexistericantitruncated 7-orthoplex

Hexistericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	806400
Vertices	161280
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Peticelligreatorhombated heptacross

Hexisteriruncitruncated 7-orthoplex

Hexisteriruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,3,4,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	725760
Vertices	161280
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Peticelliprismatotruncated heptacross

Hexipenticantitruncated 7-orthoplex

hexipenticantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,5,6{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	483840
Vertices	107520
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petiterigreatorhombated heptacross

Hexisteriruncicantitruncated 7-orthoplex

Hexisteriruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,4,6{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	1290240
Vertices	322560
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Great petacellated heptacross

Hexipentiruncicantitruncated 7-orthoplex

Hexipentiruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t0,1,2,3,5,6{35,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	1290240
Vertices	322560
Vertex figure
Coxeter groups	B7, [4,35]
Properties	convex

Alternate names

• Petiterigreatoprismated heptacross

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, ISBN:978-0-471-01003-6 [1]
    • (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)
• Klitzing, Richard. "7D uniform polytopes (polyexa)". https://bendwavy.org/klitzing/dimensions/polyexa.htm. 

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary