Truncated 6-cube

6-cube	Truncated 6-cube	Bitruncated 6-cube	Tritruncated 6-cube
6-orthoplex	Truncated 6-orthoplex	Bitruncated 6-orthoplex
Orthogonal projections in BC6 Coxeter plane

In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.

There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.

Contents 1 Truncated 6-cube 1.1 Alternate names 1.2 Construction and coordinates 1.3 Images 2 Bitruncated 6-cube 2.1 Alternate names 2.2 Construction and coordinates 2.3 Images 3 Tritruncated 6-cube 3.1 Alternate names 3.2 Construction and coordinates 3.3 Images 4 Related polytopes 5 Notes 6 References 7 References 8 External links

Truncated 6-cube

Truncated 6-cube
Type	uniform polypeton
Schläfli symbol	t0,1{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	76
4-faces	464
Cells	1120
Faces	1520
Edges	1152
Vertices	384
Vertex figure	Elongated 5-cell pyramid
Coxeter groups	BC6, [3,3,3,3,4]
Properties	convex

Alternate names

• Truncated hexeract (Acronym: tox) (Jonathan Bowers)^[1]

Construction and coordinates

The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at of the edge length. A regular 5-simplex replaces each original vertex.

The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:

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-cube

Bitruncated 6-cube
Type	uniform polypeton
Schläfli symbol	t1,2{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	BC6, [3,3,3,3,4]
Properties	convex

Alternate names

• Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)^[2]

Construction and coordinates

The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:

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]

Tritruncated 6-cube

Tritruncated 6-cube
Type	uniform polypeton
Schläfli symbol	t2,3{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	BC6, [3,3,3,3,4]
Properties	convex

Alternate names

• Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)^[3]

Construction and coordinates

The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:

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

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) o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog

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