Cantic 5-cube

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Uniform 5-polytope

title: "Cantic 5-cube" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["5-polytopes"] description: "Uniform 5-polytope" topic_path: "general/5-polytopes" source: "https://en.wikipedia.org/wiki/Cantic_5-cube" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Uniform 5-polytope ::

::data[format=table]

Truncated 5-demicubeCantic 5-cube
[[File:Truncated 5-demicube D5.svg
Type
Schläfli symbol
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties
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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 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

Related polytopes

It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections: ::data[format=table] | [[File:5-demicube t01 B5.svg|240px]]Cantic 5-cube | [[File:5-cube t02.svg|240px]]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.

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.

Notes

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
      • (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.

References

  1. Klitzing, (x3x3o *b3o3o - thin)

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