Pentic 7-cubes

Orthogonal projections in D₇ Coxeter plane
7-demicube (half 7-cube, h{4,3⁵})	Pentic 7-cube h₅{4,3⁵}	Penticantic 7-cube h_2,5{4,3⁵}
Pentiruncic 7-cube h_3,5{4,3⁵}	Pentiruncicantic 7-cube h_2,3,5{4,3⁵}	Pentisteric 7-cube h_4,5{4,3⁵}
Pentistericantic 7-cube h_2,4,5{4,3⁵}	Pentisteriruncic 7-cube h_3,4,5{4,3⁵}	Penticsteriruncicantic 7-cube h_2,3,4,5{4,3⁵}

In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms.

Pentic 7-cube

Pentic 7-cube
Type	uniform 7-polytope
Schläfli symbol	t_0,4{3,3^4,1} h₅{4,3⁵}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	13440
Vertices	1344
Vertex figure
Coxeter groups	D₇, [3^4,1,1]
Properties	convex

Cartesian coordinates

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

(±1,±1,±1,±1,±1,±3,±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]

Related polytopes

Dimensional family of pentic n-cubes
n	6	7
[1⁺,4,3^n-2] = [3,3^n-3,1]	[1⁺,4,3⁴] = [3,3^3,1]	[1⁺,4,3⁵] = [3,3^4,1]
Cantic figure
Coxeter	=	=
Schläfli	h₅{4,3⁴}	h₅{4,3⁵}

Penticantic 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]

Pentiruncic 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]

Pentiruncicantic 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]

Pentisteric 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]

Pentistericantic 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]

Pentisterirunic 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]

Pentisteriruncicantic 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:

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)

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
  - (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.
Richard Klitzing, 7D, uniform polytopes (polyexa)

External links

Weisstein, Eric W., "Hypercube", MathWorld.
Olshevsky, George, Measure polytope at Glossary for Hyperspace.
Polytopes of Various Dimensions
Multi-dimensional Glossary

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 4-polytope	5-cell	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

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