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	t_0,1,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexicantellated 7-orthoplex

Hexicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexicantitruncated 7-orthoplex

Hexicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexiruncitruncated 7-orthoplex

Hexiruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexiruncicantellated 7-orthoplex

Hexiruncicantellated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexisteritruncated 7-orthoplex

hexisteritruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,4,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexiruncicantitruncated 7-orthoplex

Hexiruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexistericantitruncated 7-orthoplex

Hexistericantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,4,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexisteriruncitruncated 7-orthoplex

Hexisteriruncitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,4,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph	too complex
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipenticantitruncated 7-orthoplex

hexipenticantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,5,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexisteriruncicantitruncated 7-orthoplex

Hexisteriruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,4,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph	too complex
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipentiruncicantitruncated 7-orthoplex

Hexipentiruncicantitruncated 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,5,6{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

Images

orthographic projections
Coxeter plane	B₇ / A₆	B₆ / D₇	B₅ / D₆ / A₄
Graph	too complex
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B₄ / D₅	B₃ / D₄ / A₂	B₂ / D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Notes

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)".

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds