Runcinated 6-cube

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	Runcicantitruncated 6-cube	Biruncicantitruncated 6-cube	Runcicantitruncated 6-orthoplex
Orthogonal projections in BC6 Coxeter plane

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. Half are expressed relative to the dual 6-orthoplex.

Contents 1 Runcinated 6-cube 1.1 Alternate names 1.2 Images 2 Biruncinated 6-cube 2.1 Alternate names 2.2 Images 3 Runcitruncated 6-cube 3.1 Alternate names 3.2 Images 4 Biruncitruncated 6-cube 4.1 Alternate names 4.2 Images 5 Runcicantellated 6-cube 5.1 Alternate names 5.2 Images 6 Runcicantitruncated 6-cube 6.1 Alternate names 6.2 Images 7 Biruncitruncated 6-cube 7.1 Alternate names 7.2 Images 8 Related polytopes 9 Notes 10 References 11 External links

Runcinated 6-cube

Runcinated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t0,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	7680
Vertices	1280
Vertex figure
Coxeter group	BC6 [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	t1,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	11520
Vertices	1920
Vertex figure
Coxeter group	BC6 [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	t0,1,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	17280
Vertices	3840
Vertex figure
Coxeter group	BC6 [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	t1,2,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	23040
Vertices	5760
Vertex figure
Coxeter group	BC6 [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	t0,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	13440
Vertices	3840
Vertex figure
Coxeter group	BC6 [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	t0,1,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	23040
Vertices	7680
Vertex figure
Coxeter group	BC6 [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]

Biruncitruncated 6-cube

Biruncitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t1,2,3,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	23040
Vertices	5760
Vertex figure
Coxeter group	BC6 [4,3,3,3,3]
Properties	convex

Alternate names

• Biprismatotruncated hexeract (boprag) (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 polypeta generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.

β6	t1β6	t2β6	t2γ6	t1γ6	γ6	t0,1β6	t0,2β6
t1,2β6	t0,3β6	t1,3β6	t2,3γ6	t0,4β6	t1,4γ6	t1,3γ6	t1,2γ6
t0,5γ6	t0,4γ6	t0,3γ6	t0,2γ6	t0,1γ6	t0,1,2β6	t0,1,3β6	t0,2,3β6
t1,2,3β6	t0,1,4β6	t0,2,4β6	t1,2,4β6	t0,3,4β6	t1,2,4γ6	t1,2,3γ6	t0,1,5β6
t0,2,5β6	t0,3,4γ6	t0,2,5γ6	t0,2,4γ6	t0,2,3γ6	t0,1,5γ6	t0,1,4γ6	t0,1,3γ6
t0,1,2γ6	t0,1,2,3β6	t0,1,2,4β6	t0,1,3,4β6	t0,2,3,4β6	t1,2,3,4γ6	t0,1,2,5β6	t0,1,3,5β6
t0,2,3,5γ6	t0,2,3,4γ6	t0,1,4,5γ6	t0,1,3,5γ6	t0,1,3,4γ6	t0,1,2,5γ6	t0,1,2,4γ6	t0,1,2,3γ6
t0,1,2,3,4β6	t0,1,2,3,5β6	t0,1,2,4,5β6	t0,1,2,4,5γ6	t0,1,2,3,5γ6	t0,1,2,3,4γ6	t0,1,2,3,4,5γ6

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, editied 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.
• Richard Klitzing, 6D, uniform polytopes (polypeta) o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - boprag

External links

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

Family	An	BCn	Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes