Truncated 8-cubes

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8-cube t0.svg
8-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t01.svg
Truncated 8-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t12.svg
Bitruncated 8-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t34.svg
Quadritruncated 8-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t23.svg
Tritruncated 8-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t45.svg
Tritruncated 8-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t56.svg
Bitruncated 8-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
8-cube t67.svg
Truncated 8-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
8-cube t7.svg
8-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 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[edit]

Truncated 8-cube
Type uniform 8-polytope
Schläfli symbol t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure Elongated 6-simplex pyramid
Coxeter groups B8, [3,3,3,3,3,3,4]
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
B8 B7
8-cube t01.svg 8-cube t01 B7.svg
[16] [14]
B6 B5
8-cube t01 B6.svg 8-cube t01 B5.svg
[12] [10]
B4 B3 B2
8-cube t01 B4.svg 8-cube t01 B3.svg 8-cube t01 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t01 A7.svg 8-cube t01 A5.svg 8-cube t01 A3.svg
[8] [6] [4]

Related polytopes[edit]

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

Truncated hypercubes
Regular polygon 8 annotated.svg 3-cube t01.svgTruncated hexahedron.png 4-cube t01.svgSchlegel half-solid truncated tesseract.png 5-cube t01.svg5-cube t01 A3.svg 6-cube t01.svg6-cube t01 A5.svg 7-cube t01.svg7-cube t01 A5.svg 8-cube t01.svg8-cube t01 A7.svg ...
Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
CDel node 1.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Bitruncated 8-cube[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
B8 B7
8-cube t12.svg 8-cube t12 B7.svg
[16] [14]
B6 B5
8-cube t12 B6.svg 8-cube t12 B5.svg
[12] [10]
B4 B3 B2
8-cube t12 B4.svg 8-cube t12 B3.svg 8-cube t12 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t12 A7.svg 8-cube t12 A5.svg 8-cube t12 A3.svg
[8] [6] [4]

Related polytopes[edit]

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

Bitruncated hypercubes
3-cube t12.svgTruncated octahedron.png 4-cube t12.svgSchlegel half-solid bitruncated 8-cell.png 5-cube t12.svg5-cube t12 A3.svg 6-cube t12.svg6-cube t12 A5.svg 7-cube t12.svg7-cube t12 A5.svg 8-cube t12.svg8-cube t12 A7.svg ...
Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Tritruncated 8-cube[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
B8 B7
8-cube t23.svg 8-cube t23 B7.svg
[16] [14]
B6 B5
8-cube t23 B6.svg 8-cube t23 B5.svg
[12] [10]
B4 B3 B2
8-cube t23 B4.svg 8-cube t23 B3.svg 8-cube t23 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t23 A7.svg 8-cube t23 A5.svg 8-cube t23 A3.svg
[8] [6] [4]

Quadritruncated 8-cube[edit]

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

CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
B8 B7
8-cube t34.svg 8-cube t34 B7.svg
[16] [14]
B6 B5
8-cube t34 B6.svg 8-cube t34 B5.svg
[12] [10]
B4 B3 B2
8-cube t34 B4.svg 8-cube t34 B3.svg 8-cube t34 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t34 A7.svg 8-cube t34 A5.svg 8-cube t34 A3.svg
[8] [6] [4]

Related polytopes[edit]

2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8
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
CDel label4.pngCDel branch 11.png CDel node 1.pngCDel split1-43.pngCDel nodes.png CDel branch 11.pngCDel 4a3b.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png
Images Truncated square.png 3-cube t1.svgCuboctahedron.png 4-cube t12.svgSchlegel half-solid bitruncated 8-cell.png 5-cube t2.svg5-cube t2 A3.svg 6-cube t23.svg6-cube t23 A5.svg 7-cube t3.svg7-cube t3 A5.svg 8-cube t34.svg8-cube t34 A7.svg ...
Facets {3} Regular polygon 3 annotated.svg
{4} Regular polygon 4 annotated.svg
t{3,3} Uniform polyhedron-33-t01.png
t{3,4} Uniform polyhedron-43-t12.png
r{3,3,3} Schlegel half-solid rectified 5-cell.png
r{3,3,4} Schlegel wireframe 24-cell.png
2t{3,3,3,3} 5-simplex t12.svg
2t{3,3,3,4} 5-cube t23.svg
2r{3,3,3,3,3} 6-simplex t2.svg
2r{3,3,3,3,4} 6-cube t4.svg
3t{3,3,3,3,3,3} 7-simplex t23.svg
3t{3,3,3,3,3,4} 7-cube t45.svg
Vertex
figure
Cuboctahedron vertfig.png
Rectangle
Bitruncated 8-cell verf.png
Disphenoid
Birectified penteract verf.png
{3}×{4} duoprism
{3,3}×{3,4} duoprism

Notes[edit]

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

References[edit]

  • 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[edit]

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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds