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Cantic 5-cube

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Uniform 5-polytope
Truncated 5-demicube
Cantic 5-cube

D5 Coxeter plane projection
Type uniform 5-polytope
Schläfli symbol h2{4,3,3,3}
t{3,3}
Coxeter-Dynkin diagram =
4-faces 42 total:
16 r{3,3,3}
16 t{3,3,3}
10 t{3,3,4}
Cells 280 total:
80 {3,3}
120 t{3,3}
80 {3,4}
Faces 640 total:
480 {3}
160 {6}
Edges 560
Vertices 160
Vertex figure
( )v{ }×{3}
Coxeter groups D5,
Properties convex

In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.

Cartesian coordinates

The Cartesian coordinates for the 160 vertices of a cantic 5-cube centered at the origin and edge length 6√2 are coordinate permutations:

(±1,±1,±3,±3,±3)

with an odd number of plus signs.

Alternate names

  • Cantic penteract, truncated demipenteract
  • Truncated hemipenteract (thin) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B5
Graph
Dihedral symmetry
Coxeter plane D5 D4
Graph
Dihedral symmetry
Coxeter plane D3 A3
Graph
Dihedral symmetry

Related polytopes

It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections:


Cantic 5-cube
Cantellated 5-cube

This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

Dimensional family of cantic n-cubes
n 3 4 5 6 7 8
Symmetry
=
=
=
=
=
=
Cantic
figure
Coxeter
=
=
=
=
=
=
Schläfli h2{4,3} h2{4,3} h2{4,3} h2{4,3} h2{4,3} h2{4,3}

There are 23 uniform 5-polytope that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.

D5 polytopes

h{4,3,3,3}
h2{4,3,3,3}
h3{4,3,3,3}
h4{4,3,3,3}
h2,3{4,3,3,3}
h2,4{4,3,3,3}
h3,4{4,3,3,3}
h2,3,4{4,3,3,3}

Notes

  1. Klitzing, (x3x3o *b3o3o - thin)

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,
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • Klitzing, Richard. "5D uniform polytopes (polytera) x3x3o *b3o3o - thin".

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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