Runcinated 5-orthoplexes

5-cube Template:CDD	Runcinated 5-cube Template:CDD	Runcinated 5-orthoplex Template:CDD
Runcitruncated 5-cube Template:CDD	Runcicantellated 5-cube Template:CDD	Runcicantitruncated 5-cube Template:CDD
Runcitruncated 5-orthoplex Template:CDD	Runcicantellated 5-orthoplex Template:CDD	Runcicantitruncated 5-orthoplex Template:CDD
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)

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

$(\pm 1,\pm 1,\pm 1,\pm (1+\sqrt{2}),\pm (1+\sqrt{2}))$

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:

$(\pm 1,\pm (1+\sqrt{2}),\pm (1+\sqrt{2}),\pm (1+2\sqrt{2}),\pm (1+2\sqrt{2}))$

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:

$(\pm 1,\pm 1,\pm (1+\sqrt{2}),\pm (1+2\sqrt{2}),\pm (1+2\sqrt{2}))$

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:

$(1,1+\sqrt{2},1+2\sqrt{2},1+3\sqrt{2},1+3\sqrt{2})$

Related polytopes

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

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:PolyCell
• Polytopes of Various Dimensions, Jonathan Bowers
  • Runcinated uniform polytera (spid), Jonathan Bowers
• Multi-dimensional Glossary