Steric 6-cubes

Orthogonal projections in D₅ Coxeter plane
6-demicube =	Steric 6-cube =	Stericantic 6-cube =
	Steriruncic 6-cube =	Steriruncicantic 6-cube =

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

Steric 6-cube

Steric 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,3{3,3^3,1} h₄{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	3360
Vertices	480
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

Alternate names

Runcinated demihexeract
Runcinated 6-demicube
Small prismated hemihexeract (Acronym: sophax) (Jonathan Bowers)^[1]

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

Dimensional family of steric n-cubes
n	5	6	7	8
[1⁺,4,3^n-2] = [3,3^n-3,1]	[1⁺,4,3³] = [3,3^2,1]	[1⁺,4,3⁴] = [3,3^3,1]	[1⁺,4,3⁵] = [3,3^4,1]	[1⁺,4,3⁶] = [3,3^5,1]
Steric figure
Coxeter	=	=	=	=
Schläfli	h₄{4,3³}	h₄{4,3⁴}	h₄{4,3⁵}	h₄{4,3⁶}

Stericantic 6-cube

Stericantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,1,3{3,3^3,1} h_2,4{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	12960
Vertices	2880
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

Alternate names

Runcitruncated demihexeract
Runcitruncated 6-demicube
Prismatotruncated hemihexeract (Acronym: pithax) (Jonathan Bowers)^[2]

Cartesian coordinates

The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Steriruncic 6-cube

Steriruncic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,2,3{3,3^3,1} h_3,4{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	7680
Vertices	1920
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

Alternate names

Runcicantellated demihexeract
Runcicantellated 6-demicube
Prismatorhombated hemihexeract (Acronym: prohax) (Jonathan Bowers)^[3]

Cartesian coordinates

The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Steriruncicantic 6-cube

Steriruncicantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,1,2,3{3,3^2,1} h_2,3,4{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	17280
Vertices	5760
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

Alternate names

Runcicantitruncated demihexeract
Runcicantitruncated 6-demicube
Great prismated hemihexeract (Acronym: gophax) (Jonathan Bowers)^[4]

Cartesian coordinates

The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

There are 47 uniform polytopes with D₆ symmetry, 31 are shared by the B₆ symmetry, and 16 are unique:

D6 polytopes
h{4,3⁴}	h₂{4,3⁴}	h₃{4,3⁴}	h₄{4,3⁴}	h₅{4,3⁴}	h_2,3{4,3⁴}	h_2,4{4,3⁴}	h_2,5{4,3⁴}
h_3,4{4,3⁴}	h_3,5{4,3⁴}	h_4,5{4,3⁴}	h_2,3,4{4,3⁴}	h_2,3,5{4,3⁴}	h_2,4,5{4,3⁴}	h_3,4,5{4,3⁴}	h_2,3,4,5{4,3⁴}

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, 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. "6D uniform polytopes (polypeta) with acronyms". x3o3o *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax

External links

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