Runcinated 5-cubes

In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.


5-cube

Runcinated 5-cube

Runcinated 5-orthoplex

Runcitruncated 5-cube

Runcicantellated 5-cube

Runcicantitruncated 5-cube

Runcitruncated 5-orthoplex

Runcicantellated 5-orthoplex

Runcicantitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane

There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.

Runcinated 5-cube

Runcinated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,3{4,3,3,3}
Coxeter diagram
4-faces 202 10
80
80
32
Cells 1240 40
240
320
160
320
160
Faces 2160 240
960
640
320
Edges 1440 480+960
Vertices 320
Vertex figure
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

  • Small prismated penteract (Acronym: span) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]




Runcitruncated 5-cube

Runcitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces 202 10
80
80
32
Cells 1560 40
240
320
320
160
320
160
Faces 3760 240
960
320
960
640
640
Edges 3360 480+960+1920
Vertices 960
Vertex figure
Coxeter group B5, [3,3,3,4]
Properties convex

Alternate names

  • Runcitruncated penteract
  • Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a runcitruncated 5-cube having edge length 2 are all permutations of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]





Runcicantellated 5-cube

Runcicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,2,3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces 202 10
80
80
32
Cells 1240 40
240
320
320
160
160
Faces 2960 240
480
960
320
640
320
Edges 2880 960+960+960
Vertices 960
Vertex figure
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

  • Runcicantellated penteract
  • Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]





Runcicantitruncated 5-cube

Runcicantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{4,3,3,3}
Coxeter-Dynkin
diagram
4-faces202
Cells1560
Faces4240
Edges4800
Vertices1920
Vertex figure
Irregular 5-cell
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

  • Runcicantitruncated penteract
  • Biruncicantitruncated pentacross
  • great prismated penteract (gippin) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

These polytopes are a part of a set of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

B5 polytopes

β5

t1β5

t2γ5

t1γ5

γ5

t0,1β5

t0,2β5

t1,2β5

t0,3β5

t1,3γ5

t1,2γ5

t0,4γ5

t0,3γ5

t0,2γ5

t0,1γ5

t0,1,2β5

t0,1,3β5

t0,2,3β5

t1,2,3γ5

t0,1,4β5

t0,2,4γ5

t0,2,3γ5

t0,1,4γ5

t0,1,3γ5

t0,1,2γ5

t0,1,2,3β5

t0,1,2,4β5

t0,1,3,4γ5

t0,1,2,4γ5

t0,1,2,3γ5

t0,1,2,3,4γ5

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)". o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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
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