Hexic 7-cubes

Orthogonal projections in D₇ Coxeter plane
7-demicube	Hexic 7-cube	Hexicantic 7-cube	Hexiruncic 7-cube	Hexiruncicantic 7-cube
Hexisteric 7-cube	Hexistericantic 7-cube	Hexisteriruncic 7-cube	Hexisteriruncicantic 7-cube	Hexipentic 7-cube
Hexipenticantic 7-cube	Hexipentiruncic 7-cube	Hexipentiruncicantic 7-cube	Hexipentisteric 7-cube	Hexipentistericantic 7-cube
Hexipentisteriruncic 7-cube	Hexipentisteriruncicantic 7-cube

In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms.

Hexic 7-cube

Hexic 7-cube
Type	uniform 7-polytope
Schläfli symbol	t_0,5{3,3^4,1} h₆{4,3⁵}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	4704
Vertices	448
Vertex figure
Coxeter groups	D₇, [3^4,1,1]
Properties	convex

Cartesian coordinates

The Cartesian coordinates for the vertices of a hexic 7-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexicantic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexiruncic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexisteric 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipentic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexiruncicantic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexistericantic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipenticantic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexisteriruncic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipentiruncic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipentisteric 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexisteriruncicantic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipentiruncicantic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipentisteriruncic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipentistericantic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Hexipentisteriruncicantic 7-cube

Images

orthographic projections
Coxeter plane	B₇	D₇	D₆
Graph
Dihedral symmetry	[14/2]	[12]	[10]
Coxeter plane	D₅	D₄	D₃
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

This polytope is based on the 7-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 95 uniform polytopes with D₇ symmetry, 63 are shared by the BC₇ symmetry, and 32 are unique:

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, Ph.D.
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

D7 polytopes
t₀(1₄₁)	t_0,1(1₄₁)	t_0,2(1₄₁)	t_0,3(1₄₁)	t_0,4(1₄₁)	t_0,5(1₄₁)	t_0,1,2(1₄₁)	t_0,1,3(1₄₁)
t_0,1,4(1₄₁)	t_0,1,5(1₄₁)	t_0,2,3(1₄₁)	t_0,2,4(1₄₁)	t_0,2,5(1₄₁)	t_0,3,4(1₄₁)	t_0,3,5(1₄₁)	t_0,4,5(1₄₁)
t_0,1,2,3(1₄₁)	t_0,1,2,4(1₄₁)	t_0,1,2,5(1₄₁)	t_0,1,3,4(1₄₁)	t_0,1,3,5(1₄₁)	t_0,1,4,5(1₄₁)	t_0,2,3,4(1₄₁)	t_0,2,3,5(1₄₁)
t_0,2,4,5(1₄₁)	t_0,3,4,5(1₄₁)	t_0,1,2,3,4(1₄₁)	t_0,1,2,3,5(1₄₁)	t_0,1,2,4,5(1₄₁)	t_0,1,3,4,5(1₄₁)	t_0,2,3,4,5(1₄₁)	t_0,1,2,3,4,5(1₄₁)

Hexic 7-cubes

Contents

Hexic 7-cube

Cartesian coordinates

Images

Hexicantic 7-cube

Images

Hexiruncic 7-cube

Images

Hexisteric 7-cube

Images

Hexipentic 7-cube

Images

Hexiruncicantic 7-cube

Images

Hexistericantic 7-cube

Images

Hexipenticantic 7-cube

Images

Hexisteriruncic 7-cube

Images

Hexipentiruncic 7-cube

Images

Hexipentisteric 7-cube

Images

Hexisteriruncicantic 7-cube

Images

Hexipentiruncicantic 7-cube

Images

Hexipentisteriruncic 7-cube

Images

Hexipentistericantic 7-cube

Images

Hexipentisteriruncicantic 7-cube

Images

Related polytopes

Notes

References

External links

Navigation menu