Truncated 8-cubes

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File:8-cube t0.svg
8-cube
File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:8-cube t01.svg
Truncated 8-cube
File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:8-cube t12.svg
Bitruncated 8-cube
File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:8-cube t34.svg
Quadritruncated 8-cube
File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png 		File:8-cube t23.svg
Tritruncated 8-cube
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png 		File:8-cube t45.svg
Tritruncated 8-orthoplex
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:8-cube t56.svg
Bitruncated 8-orthoplex
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png 		File:8-cube t67.svg
Truncated 8-orthoplex
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png 		File:8-cube t7.svg
8-orthoplex
File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
Orthogonal projections in B8 Coxeter plane

In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube. 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
Type uniform 8-polytope
Schläfli symbol t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure ( )v{3,3,3,3,3}
Coxeter groups B8, [3,3,3,3,3,3,4]
Properties convex

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
File:8-cube t01.svg File:8-cube t01 B7.svg
[16] [14]
B6 B5
File:8-cube t01 B6.svg File:8-cube t01 B5.svg
[12] [10]
B4 B3 B2
File:8-cube t01 B4.svg File:8-cube t01 B3.svg File:8-cube t01 B2.svg
[8] [6] [4]
A7 A5 A3
File:8-cube t01 A7.svg File:8-cube t01 A5.svg File:8-cube t01 A3.svg
[8] [6] [4]

Related polytopes

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

Truncated hypercubes
Image File:Regular polygon 8 annotated.svg File:3-cube t01.svgFile:Truncated hexahedron.png File:4-cube t01.svgFile:Schlegel half-solid truncated tesseract.png File:5-cube t01.svgFile:5-cube t01 A3.svg File:6-cube t01.svgFile:6-cube t01 A5.svg File:7-cube t01.svgFile:7-cube t01 A5.svg File:8-cube t01.svgFile:8-cube t01 A7.svg ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Vertex figure ( )v( ) File:Truncated cube vertfig.svg
( )v{ } 		File:Truncated 8-cell verf.png
( )v{3} 		File:Truncated 5-cube verf.png
( )v{3,3} 		( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

Bitruncated 8-cube

Bitruncated 8-cube
Type uniform 8-polytope
Schläfli symbol 2t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure { }v{3,3,3,3}
Coxeter groups B8, [3,3,3,3,3,3,4]
Properties convex

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
File:8-cube t12.svg File:8-cube t12 B7.svg
[16] [14]
B6 B5
File:8-cube t12 B6.svg File:8-cube t12 B5.svg
[12] [10]
B4 B3 B2
File:8-cube t12 B4.svg File:8-cube t12 B3.svg File:8-cube t12 B2.svg
[8] [6] [4]
A7 A5 A3
File:8-cube t12 A7.svg File:8-cube t12 A5.svg File:8-cube t12 A3.svg
[8] [6] [4]

Related polytopes

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

Bitruncated hypercubes
Image File:3-cube t12.svgFile:Truncated octahedron.png File:4-cube t12.svgFile:Schlegel half-solid bitruncated 8-cell.png File:5-cube t12.svgFile:5-cube t12 A3.svg File:6-cube t12.svgFile:6-cube t12 A5.svg File:7-cube t12.svgFile:7-cube t12 A5.svg File:8-cube t12.svgFile:8-cube t12 A7.svg ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Vertex figure File:Truncated octahedron vertfig.png
( )v{ } 		File:Bitruncated 8-cell verf.png
{ }v{ } 		File:Bitruncated penteract verf.png
{ }v{3} 		File:Bitruncated 6-cube verf.png
{ }v{3,3} 		{ }v{3,3,3} { }v{3,3,3,3}

Tritruncated 8-cube

Tritruncated 8-cube
Type uniform 8-polytope
Schläfli symbol 3t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {4}v{3,3,3}
Coxeter groups B8, [3,3,3,3,3,3,4]
Properties convex

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
File:8-cube t23.svg File:8-cube t23 B7.svg
[16] [14]
B6 B5
File:8-cube t23 B6.svg File:8-cube t23 B5.svg
[12] [10]
B4 B3 B2
File:8-cube t23 B4.svg File:8-cube t23 B3.svg File:8-cube t23 B2.svg
[8] [6] [4]
A7 A5 A3
File:8-cube t23 A7.svg File:8-cube t23 A5.svg File:8-cube t23 A3.svg
[8] [6] [4]

Quadritruncated 8-cube

Quadritruncated 8-cube
Type uniform 8-polytope
Schläfli symbol 4t{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png

File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png

6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,4}v{3,3}
Coxeter groups B8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

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
File:8-cube t34.svg File:8-cube t34 B7.svg
[16] [14]
B6 B5
File:8-cube t34 B6.svg File:8-cube t34 B5.svg
[12] [10]
B4 B3 B2
File:8-cube t34 B4.svg File:8-cube t34 B3.svg File:8-cube t34 B2.svg
[8] [6] [4]
A7 A5 A3
File:8-cube t34 A7.svg File:8-cube t34 A5.svg File:8-cube t34 A3.svg
[8] [6] [4]

Related polytopes

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 		File:CDel label4.pngFile:CDel branch 11.png File:CDel node 1.pngFile:CDel split1-43.pngFile:CDel nodes.png File:CDel branch 11.pngFile:CDel 4a3b.pngFile:CDel nodes.png File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 4a3b.pngFile:CDel nodes.png
Images File:Truncated square.png File:3-cube t1.svgFile:Cuboctahedron.png File:4-cube t12.svgFile:Schlegel half-solid bitruncated 8-cell.png File:5-cube t2.svgFile:5-cube t2 A3.svg File:6-cube t23.svgFile:6-cube t23 A5.svg File:7-cube t3.svgFile:7-cube t3 A5.svg File:8-cube t34.svgFile:8-cube t34 A7.svg
Facets {3} File:Regular polygon 3 annotated.svg
{4} File:Regular polygon 4 annotated.svg 		t{3,3} File:Uniform polyhedron-33-t01.png
t{3,4} File:Uniform polyhedron-43-t12.png 		r{3,3,3} File:Schlegel half-solid rectified 5-cell.png
r{3,3,4} File:Schlegel wireframe 24-cell.png 		2t{3,3,3,3} File:5-simplex t12.svg
2t{3,3,3,4} File:5-cube t23.svg 		2r{3,3,3,3,3} File:6-simplex t2.svg
2r{3,3,3,3,4} File:6-cube t4.svg 		3t{3,3,3,3,3,3} File:7-simplex t23.svg
3t{3,3,3,3,3,4} File:7-cube t45.svg
Vertex
figure 		( )v( ) File:Cuboctahedron vertfig.png
{ }×{ } 		File:Bitruncated 8-cell verf.png
{ }v{ } 		File:Birectified penteract verf.png
{3}×{4} 		File:Tritruncated 6-cube verf.png
{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 [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. "8D uniform polytopes (polyzetta)". o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke

External links

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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