Runcinated 5-orthoplexes

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5-cube
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Runcinated 5-cube
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Runcinated 5-orthoplex
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Runcitruncated 5-cube
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Runcicantellated 5-cube
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Runcicantitruncated 5-cube
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Runcitruncated 5-orthoplex
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Runcicantellated 5-orthoplex
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Runcicantitruncated 5-orthoplex
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Orthogonal projections in BC5 Coxeter plane

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.

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

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Runcinated 5-cube

Runcinated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,3{4,3,3,3}
Coxeter-Dynkin diagram Template:CDD
4-faces 202
Cells 1240
Faces 2160
Edges 1440
Vertices 320
Vertex figure
Coxeter group BC5 [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:

(±1,±1,±1,±(1+2),±(1+2))

Images

Template:5-cube Coxeter plane graphs

Runcitruncated 5-cube

Runcitruncated 5-cube
Type uniform polyteron
Schläfli symbol t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams Template:CDD
4-faces 202
Cells 1560
Faces 3760
Edges 3360
Vertices 960
Vertex figure
Coxeter groups BC5, [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:

(±1,±(1+2),±(1+2),±(1+22),±(1+22))

Images

Template:5-cube Coxeter plane graphs

Runcicantellated 5-cube

Runcicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,2,3{4,3,3,3}
Coxeter-Dynkin diagram Template:CDD
4-faces 202
Cells 1240
Faces 2960
Edges 2880
Vertices 960
Vertex figure
Coxeter group BC5 [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:

(±1,±1,±(1+2),±(1+22),±(1+22))

Images

Template:5-cube Coxeter plane graphs

Runcicantitruncated 5-cube

Runcicantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{4,3,3,3}
Coxeter-Dynkin
diagram
Template:CDD
4-faces 202
Cells 1560
Faces 4240
Edges 4800
Vertices 1920
Vertex figure
Irregular 5-cell
Coxeter group BC5 [4,3,3,3]
Properties convex, isogonal

Alternate names

  • Runcicantitruncated penteract
  • Biruncicantitruncated 16-cell / Biruncicantitruncated pentacross
  • great prismated penteract (gippin) (Jonathan Bowers)

Coordinates

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

(1,1+2,1+22,1+32,1+32)

Images

Template:5-cube Coxeter plane graphs

Related polytopes

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

Template:Penteract family

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, editied 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.
  • Template:KlitzingPolytopes o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin

External links

Template:Polytopes