Truncated 6-orthoplexes

6-orthoplex	Truncated 6-orthoplex	Bitruncated 6-orthoplex	Tritruncated 6-cube
6-cube	Truncated 6-cube	Bitruncated 6-cube
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.

There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.

Truncated 6-orthoplex

Truncated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	t{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces	76
4-faces	576
Cells	1200
Faces	1120
Edges	540
Vertices	120
Vertex figure	( )v{3,4}
Coxeter groups	B6, [3,3,3,3,4] D6, [33,1,1]
Properties	convex

Alternate names

• Truncated hexacross
• Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)

Construction

There are two Coxeter groups associated with the truncated hexacross, one with the C₆ or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D₆ or [3^3,1,1] Coxeter group.

Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of

(±2,±1,0,0,0,0)

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]

Bitruncated 6-orthoplex

Bitruncated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	2t{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{ }v{3,4}
Coxeter groups	B6, [3,3,3,3,4] D6, [33,1,1]
Properties	convex

Alternate names

• Bitruncated hexacross
• Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)

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

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 [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.
• Klitzing, Richard. "6D uniform polytopes (polypeta)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.  x3x3o3o3o4o - tag, o3x3x3o3o4o - botag

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary