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
6-faces
5-faces
4-faces
Cells
Faces
Edges	13440
Vertices	1344
Vertex figure
Coxeter groups	D₇, [3^4,1,1]
Properties	convex

Alternate names

Small cellated demihepteract (acronym: sochesa)^[1]

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	8
[1⁺,4,3^n-2] = [3,3^n-3,1]	[1⁺,4,3⁴] = [3,3^3,1]	[1⁺,4,3⁵] = [3,3^4,1]	[1⁺,4,3⁶] = [3,3^5,1]
Cantic figure
Coxeter	=	=	=
Schläfli	h₅{4,3⁴}	h₅{4,3⁵}	h₅{4,3⁶}

Penticantic 7-cube

Alternate names

Cellitruncated demihepteract (acronym: cothesa)^[2]

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

Alternate names

Cellirhombated demihepteract (acronym: crohesa)^[3]

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

Alternate names

Celligreatorhombated demihepteract (acronym: cagrohesa)^[4]

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

Alternate names

Celliprismated demihepteract (acronym: caphesa)^[5]

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

Alternate names

Celliprismatotruncated demihepteract (acronym: capthesa)^[6]

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]

Pentisteriruncic 7-cube

Alternate names

Celliprismatorhombated demihepteract (acronym: coprahesa)^[7]

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

Alternate names

Great cellated demihepteract (acronym: gochesa)^[8]

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

These polytopes are based on the 7-demicube, a member 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

↑ Klitzing, (x3o3o *b3o3o3x3o - sochesa)
↑ Klitzing, (x3x3o *b3o3o3x3o - cothesa)
↑ Klitzing, (x3o3o *b3x3o3x3o - crohesa)
↑ Klitzing, (x3x3o *b3x3o3x3o - cagrohesa)
↑ Klitzing, (x3o3o *b3o3x3x3o - caphesa)
↑ Klitzing, (x3x3o *b3o3x3x3o - capthesa)
↑ Klitzing, (x3o3o *b3x3x3x3o - coprahesa)
↑ Klitzing, (x3x3o *b3x3x3x3o - gochesa)

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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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.
Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms".

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
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 • Polytope operations

[1] Klitzing, (x3o3o *b3o3o3x3o - sochesa)

[2] Klitzing, (x3x3o *b3o3o3x3o - cothesa)

[3] Klitzing, (x3o3o *b3x3o3x3o - crohesa)

[4] Klitzing, (x3x3o *b3x3o3x3o - cagrohesa)

[5] Klitzing, (x3o3o *b3o3x3x3o - caphesa)

[6] Klitzing, (x3x3o *b3o3x3x3o - capthesa)

[7] Klitzing, (x3o3o *b3x3x3x3o - coprahesa)

[8] Klitzing, (x3x3o *b3x3x3x3o - gochesa)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

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)