Truncated 8-cubes

In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.


8-cube
Truncated 8-cube
Bitruncated 8-cube

Quadritruncated 8-cube
Tritruncated 8-cube
Tritruncated 8-orthoplex

Bitruncated 8-orthoplex
Truncated 8-orthoplex
8-orthoplex
Orthogonal projections in B8 Coxeter plane

There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.

Truncated 8-cube
Truncated 8-cube
Typeuniform 8-polytope
Schläfli symbolt{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure( )v{3,3,3,3,3}
Coxeter groupsB8, [3,3,3,3,3,3,4]
Propertiesconvex

Alternate names
  • Truncated octeract (acronym tocto) (Jonathan Bowers)[1]

Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of

(±2,±2,±2,±2,±2,±2,±1,0)

Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

The truncated 8-cube, is seventh in a sequence of truncated hypercubes:

Truncated hypercubes
Image ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v{ }
( )v{3}
( )v{3,3}		 ( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

Bitruncated 8-cube
Bitruncated 8-cube
Typeuniform 8-polytope
Schläfli symbol2t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{ }v{3,3,3,3}
Coxeter groupsB8, [3,3,3,3,3,3,4]
Propertiesconvex

Alternate names
  • Bitruncated octeract (acronym bato) (Jonathan Bowers)[2]

Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

(±2,±2,±2,±2,±2,±1,0,0)

Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
Image ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter
Vertex figure
( )v{ }
{ }v{ }
{ }v{3}
{ }v{3,3}		 { }v{3,3,3} { }v{3,3,3,3}

Tritruncated 8-cube
Tritruncated 8-cube
Typeuniform 8-polytope
Schläfli symbol3t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{4}v{3,3,3}
Coxeter groupsB8, [3,3,3,3,3,3,4]
Propertiesconvex

Alternate names
  • Tritruncated octeract (acronym tato) (Jonathan Bowers)[3]

Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

(±2,±2,±2,±2,±1,0,0,0)

Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Quadritruncated 8-cube
Quadritruncated 8-cube
Typeuniform 8-polytope
Schläfli symbol4t{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams

6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{3,4}v{3,3}
Coxeter groupsB8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex

Alternate names
  • Quadritruncated octeract (acronym oke) (Jonathan Bowers)[4]

Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±1,0,0,0)

Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]
2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8 n
Name t{4} r{4,3} 2t{4,3,3} 2r{4,3,3,3} 3t{4,3,3,3,3} 3r{4,3,3,3,3,3} 4t{4,3,3,3,3,3,3} ...
Coxeter
diagram
Images
Facets {3}
{4} 		t{3,3}
t{3,4} 		r{3,3,3}
r{3,3,4} 		2t{3,3,3,3}
2t{3,3,3,4} 		2r{3,3,3,3,3}
2r{3,3,3,3,4} 		3t{3,3,3,3,3,3}
3t{3,3,3,3,3,4}
Vertex
figure		 ( )v( )
{ }×{ }
{ }v{ }
{3}×{4}
{3}v{4}		 {3,3}×{3,4} {3,3}v{3,4}

Notes
  1. Klitizing, (o3o3o3o3o3o3x4x – tocto)
  2. Klitizing, (o3o3o3o3o3x3x4o – bato)
  3. Klitizing, (o3o3o3o3x3x3o4o – tato)
  4. Klitizing, (o3o3o3x3x3o3o4o – oke)

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.
  • Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
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 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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