Runcinated 6-orthoplexes

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-orthplex is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-orthoplex.

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

Runcinated 6-orthoplex

Alternate names

• Small prismatohexacontatetrapeton (spog) (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]

Runcicantellated 6-orthoplex

Alternate names

• Prismatorhombated hexacontatetrapeton (prog) (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-orthoplex

Alternate names

• Prismatotruncated hexacontatetrapeton (potag) (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]

Biruncicantellated 6-cube

Alternate names

• Great biprismated hexeractihexacontatetrapeton (gobpoxog) (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]

Related polytopes

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

B6 polytopes 

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

1. Klitzing, (x3o3o3x3o4o - spog)
2. Klitzing, (x3o3x3x3o4o - prog)
3. Klitzing, (x3x3o3x3o4o - potag)
4. Klitzing, (o3x3x3x3x4o - gobpoxog)

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, 6D, uniform polytopes (polypeta) x3o3o3x3o4o - spog, x3o3x3x3o4o - prog, x3x3o3x3o4o - potag, o3x3x3x3x4o - gobpoxog

External links

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