5-cube
Runcic 5-cube
=

5-demicube
=
Runcicantic 5-cube
=
Orthogonal projections in B5 Coxeter plane

In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, runcihalf 5-cube) is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

Runcic 5-cube

Runcic 5-cube
Typeuniform 5-polytope
Schläfli symbolh3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces42
Cells360
Faces880
Edges720
Vertices160
Vertex figure
Coxeter groupsD5, [32,1,1]
Propertiesconvex

Alternate names

  • Cantellated 5-demicube/demipenteract
  • Small rhombated hemipenteract (sirhin) (Jonathan Bowers)[1]

Cartesian coordinates

The Cartesian coordinates for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B5
Graph
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph
Dihedral symmetry [4] [4]

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


Runcic 5-cube
Runcinated 5-cube
Runcic n-cubes
n45678
[1+,4,3n-2]
= [3,3n-3,1]		 [1+,4,32]
= [3,31,1]		 [1+,4,33]
= [3,32,1]		 [1+,4,34]
= [3,33,1]		 [1+,4,35]
= [3,34,1]		 [1+,4,36]
= [3,35,1]
Runcic
figure
Coxeter
=
=
=
=
=
Schläfli h3{4,32} h3{4,33} h3{4,34} h3{4,35} h3{4,36}

Runcicantic 5-cube

Runcicantic 5-cube
Typeuniform 5-polytope
Schläfli symbolt0,1,2{3,32,1}
h3{4,33}
Coxeter-Dynkin diagram
4-faces42
Cells360
Faces1040
Edges1200
Vertices480
Vertex figure
Coxeter groupsD5, [32,1,1]
Propertiesconvex

Alternate names

  • Cantitruncated 5-demicube/demipenteract
  • Great rhombated hemipenteract (girhin) (Jonathan Bowers)[2]

Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±5,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane B5
Graph
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph
Dihedral symmetry [4] [4]

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


Runcicantic 5-cube
Runcicantellated 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-polytopes 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, (x3o3o *b3x3o - sirhin)
  2. ↑ Klitzing, (x3x3o *b3x3o - girhin)

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, [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)". x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin
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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