Runcinated 6-cubes

Orthogonal projections in B₆ Coxeter plane
6-cube	Runcinated 6-cube	Biruncinated 6-cube	Runcinated 6-orthoplex	6-orthoplex
Runcitruncated 6-cube	Biruncitruncated 6-cube	Runcicantellated 6-orthoplex	Runcicantellated 6-cube	Biruncitruncated 6-orthoplex
Runcitruncated 6-orthoplex	Runcicanti-truncated 6-cube	Biruncicanti-truncated 6-cube	Runcicanti-truncated 6-orthoplex

In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.

There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. 5 are expressed relative to the dual 6-orthoplex.

Runcinated 6-cube

Runcinated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t_0,3{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	7680
Vertices	1280
Vertex figure
Coxeter group	B₆ [4,3,3,3,3]
Properties	convex

Alternate names

Small prismated hexeract (spox) (Jonathan Bowers)^[1]

Images

Orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Biruncinated 6-cube

Biruncinated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t_1,4{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	11520
Vertices	1920
Vertex figure
Coxeter group	B₆ [4,3,3,3,3]
Properties	convex

Alternate names

Small biprismated hexeractihexacontatetrapeton (sobpoxog) (Jonathan Bowers)^[2]

Images

Orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Runcitruncated 6-cube

Runcitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t_0,1,3{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	17280
Vertices	3840
Vertex figure
Coxeter group	B₆ [4,3,3,3,3]
Properties	convex

Alternate names

Prismatotruncated hexeract (potax) (Jonathan Bowers)^[3]

Images

Orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Biruncitruncated 6-cube

Biruncitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t_1,2,4{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	23040
Vertices	5760
Vertex figure
Coxeter group	B₆ [4,3,3,3,3]
Properties	convex

Alternate names

Biprismatotruncated hexeract (boprag) (Jonathan Bowers)^[4]

Images

Orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Runcicantellated 6-cube

Runcicantellated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t_0,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	13440
Vertices	3840
Vertex figure
Coxeter group	B₆ [4,3,3,3,3]
Properties	convex

Alternate names

Prismatorhombated hexeract (prox) (Jonathan Bowers)^[5]

Images

Orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Runcicantitruncated 6-cube

Runcicantitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t_0,1,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	23040
Vertices	7680
Vertex figure
Coxeter group	B₆ [4,3,3,3,3]
Properties	convex

Alternate names

Great prismated hexeract (gippox) (Jonathan Bowers)^[6]

Images

Orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Biruncicantitruncated 6-cube

Biruncicantitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t_1,2,3,4{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	23040
Vertices	5760
Vertex figure
Coxeter group	B₆ [4,3,3,3,3]
Properties	convex

Alternate names

Great biprismated hexeractihexacontatetrapeton (gobpoxog) (Jonathan Bowers)^[7]

Images

Orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

These polytopes are from a set of 63 uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube and 6-orthoplex.

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 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. "6D uniform polytopes (polypeta) with acronyms". o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - gobpoxog

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

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsspoxhtm_(o3o3x3o3o4x_-_spox)]-1] Klitzing, (o3o3x3o3o4x - spox).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatssobpoxoghtm_(o3x3o3o3x4o_-_sobpoxog)]-2] Klitzing, (o3x3o3o3x4o - sobpoxog).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatspotaxhtm_(o3o3x3o3x4x_-_potax)]-3] Klitzing, (o3o3x3o3x4x - potax).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsbopraghtm_(o3x3o3x3x4o_-_boprag)]-4] Klitzing, (o3x3o3x3x4o - boprag).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsproxhtm_(o3o3x3x3o4x_-_prox)]-5] Klitzing, (o3o3x3x3o4x - prox).

[6] Klitzing, (o3o3x3x3x4x - gippox)

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsgobpoxoghtm_(o3x3x3x3x4o_-_gobpoxog)]-7] Klitzing, (o3x3x3x3x4o - gobpoxog).

[1]

[2]

[3]

[4]

[5]

[6]

[7]

B6 polytopes
β₆	t₁β₆	t₂β₆	t₂γ₆	t₁γ₆	γ₆	t_0,1β₆	t_0,2β₆
t_1,2β₆	t_0,3β₆	t_1,3β₆	t_2,3γ₆	t_0,4β₆	t_1,4γ₆	t_1,3γ₆	t_1,2γ₆
t_0,5γ₆	t_0,4γ₆	t_0,3γ₆	t_0,2γ₆	t_0,1γ₆	t_0,1,2β₆	t_0,1,3β₆	t_0,2,3β₆
t_1,2,3β₆	t_0,1,4β₆	t_0,2,4β₆	t_1,2,4β₆	t_0,3,4β₆	t_1,2,4γ₆	t_1,2,3γ₆	t_0,1,5β₆
t_0,2,5β₆	t_0,3,4γ₆	t_0,2,5γ₆	t_0,2,4γ₆	t_0,2,3γ₆	t_0,1,5γ₆	t_0,1,4γ₆	t_0,1,3γ₆
t_0,1,2γ₆	t_0,1,2,3β₆	t_0,1,2,4β₆	t_0,1,3,4β₆	t_0,2,3,4β₆	t_1,2,3,4γ₆	t_0,1,2,5β₆	t_0,1,3,5β₆
t_0,2,3,5γ₆	t_0,2,3,4γ₆	t_0,1,4,5γ₆	t_0,1,3,5γ₆	t_0,1,3,4γ₆	t_0,1,2,5γ₆	t_0,1,2,4γ₆	t_0,1,2,3γ₆
t_0,1,2,3,4β₆	t_0,1,2,3,5β₆	t_0,1,2,4,5β₆	t_0,1,2,4,5γ₆	t_0,1,2,3,5γ₆	t_0,1,2,3,4γ₆	t_0,1,2,3,4,5γ₆