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Truncated 5-cubes

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5-cube

Truncated 5-cube

Bitruncated 5-cube

5-orthoplex

Truncated 5-orthoplex

Bitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.

There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the square faces of the 5-cube. The third and fourth truncations are more easily constructed as second and first truncations of the 5-orthoplex.

Truncated 5-cube

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Truncated 5-cube
Type uniform 5-polytope
Schläfli symbol t{4,3,3,3}
Coxeter-Dynkin diagram
4-faces 42 10
32
Cells 200 40
160
Faces 400 80
320
Edges 400 80
320
Vertices 160
Vertex figure
( )v{3,3}
Coxeter group B5, [3,3,3,4], order 3840
Properties convex

Alternate names

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  • Truncated penteract (Acronym: tan) (Jonathan Bowers)

Construction and coordinates

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The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at of the edge length. A regular 5-cell is formed at each truncated vertex.

The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:

Images

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The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
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The truncated 5-cube, is fourth 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 5-cube

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Bitruncated 5-cube
Type uniform 5-polytope
Schläfli symbol 2t{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces 42 10
32
Cells 280 40
160
80
Faces 720 80
320
320
Edges 800 320
480
Vertices 320
Vertex figure
{ }v{3}
Coxeter groups B5, [3,3,3,4], order 3840
Properties convex

Alternate names

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  • Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)

Construction and coordinates

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The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at of the edge length.

The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:

Images

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orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
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The bitruncated 5-cube is third 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}
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This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.

B5 polytopes

β5

t1β5

t2γ5

t1γ5

γ5

t0,1β5

t0,2β5

t1,2β5

t0,3β5

t1,3γ5

t1,2γ5

t0,4γ5

t0,3γ5

t0,2γ5

t0,1γ5

t0,1,2β5

t0,1,3β5

t0,2,3β5

t1,2,3γ5

t0,1,4β5

t0,2,4γ5

t0,2,3γ5

t0,1,4γ5

t0,1,3γ5

t0,1,2γ5

t0,1,2,3β5

t0,1,2,4β5

t0,1,3,4γ5

t0,1,2,4γ5

t0,1,2,3γ5

t0,1,2,3,4γ5

Notes

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References

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  • 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. "5D uniform polytopes (polytera)". o3o3o3x4x - tan, o3o3x3x4o - bittin
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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 polychoron Pentachoron 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