Misplaced Pages

Runcic 5-cubes

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Concept in geometry

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
Type uniform 5-polytope
Schläfli symbol h3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces 42
Cells 360
Faces 880
Edges 720
Vertices 160
Vertex figure
Coxeter groups D5,
Properties convex

Alternate names

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

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
Coxeter plane D5 D4
Graph
Dihedral symmetry
Coxeter plane D3 A3
Graph
Dihedral symmetry

Related polytopes

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
n 4 5 6 7 8

=
=
=
=
=
=
Runcic
figure
Coxeter
=
=
=
=
=
Schläfli h3{4,3} h3{4,3} h3{4,3} h3{4,3} h3{4,3}

Runcicantic 5-cube

Runcicantic 5-cube
Type uniform 5-polytope
Schläfli symbol t0,1,2{3,3}
h3{4,3}
Coxeter-Dynkin diagram
4-faces 42
Cells 360
Faces 1040
Edges 1200
Vertices 480
Vertex figure
Coxeter groups D5,
Properties convex

Alternate names

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

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
Coxeter plane D5 D4
Graph
Dihedral symmetry
Coxeter plane D3 A3
Graph
Dihedral symmetry

Related polytopes

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

Related polytopes

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,
      • (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)". x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin

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
Category: