Runcic 5-cubes

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5-cube t0.svg
5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t02 B5.svg
Runcic 5-cube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t0 B5.svg
5-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-demicube t012 B5.svg
Runcicantic 5-cube
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

Runcic 5-cube
Type uniform 5-polytope
Schläfli symbol h3{4,3,3,3}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 42
Cells 360
Faces 880
Edges 720
Vertices 160
Vertex figure
Coxeter groups D5, [32,1,1]
Properties convex

Alternate names[edit]

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

Cartesian coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B5
Graph 5-demicube t02 B5.svg
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph 5-demicube t02 D5.svg 5-demicube t02 D4.svg
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph 5-demicube t02 D3.svg 5-demicube t02 A3.svg
Dihedral symmetry [4] [4]

Related polytopes[edit]

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

5-demicube t02 B5.svg
Runcic 5-cube
5-cube t03.svg
Runcinated 5-cube

Runcicantic 5-cube[edit]

Runcicantic 5-cube
Type uniform 5-polytope
Schläfli symbol t0,1,2{3,32,1}
h3{4,33}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 42
Cells 360
Faces 1040
Edges 1200
Vertices 480
Vertex figure
Coxeter groups D5, [32,1,1]
Properties convex

Alternate names[edit]

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

Cartesian coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B5
Graph 5-demicube t0123 B5.svg
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph 5-demicube t0123 D5.svg 5-demicube t0123 D4.svg
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph 5-demicube t0123 D3.svg 5-demicube t0123 A3.svg
Dihedral symmetry [4] [4]

Related polytopes[edit]

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

5-demicube t012 B5.svg
Runcicantic 5-cube
5-cube t023.svg
Runcicantellated 5-cube

Related polytopes[edit]

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.

Notes[edit]

  1. ^ Klitzing, (x3o3o *b3x3o - sirhin)
  2. ^ Klitzing, (x3x3o *b3x3o - girhin)

References[edit]

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

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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