Cantellated 5-orthoplexes

5-orthoplex	Cantellated 5-orthoplex	Bicantellated 5-cube	Cantellated 5-cube
5-cube	Cantitruncated 5-orthoplex	Bicantitruncated 5-cube	Cantitruncated 5-cube
Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.

There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube.

Cantellated 5-orthoplex

Cantellated 5-orthoplex
Type	Uniform 5-polytope
Schläfli symbol	rr{3,3,3,4} rr{3,3,31,1}
Coxeter-Dynkin diagrams
4-faces	82	10 40 36px|link=Octahedral prism 32
Cells	640	80 160 25px|link=Octahedron 320 25px|link=Triangular prism 80
Faces	1520	640 25px|link=Triangle 320 25px|link=Triangle 480 25px|link=Square 80 25px|link=Triangle
Edges	1200	960 240
Vertices	240
Vertex figure	Square pyramidal prism
Coxeter group	B5, [4,3,3,3], order 3840 D5, [32,1,1], order 1920
Properties	convex

Alternate names

• Cantellated 5-orthoplex
• Bicantellated 5-demicube
• Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)

The vertices of the can be made in 5-space, as permutations and sign combinations of:

(0,0,1,1,2)

Images

The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.

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]

Cantitruncated 5-orthoplex

Cantitruncated 5-orthoplex
Type	uniform 5-polytope
Schläfli symbol	tr{3,3,3,4} tr{3,31,1}
Coxeter-Dynkin diagrams
4-faces	82	10 40 36px|link=Octahedral prism 32
Cells	640	80 160 25px|link=Truncated tetrahedron 320 25px|link=Triangular prism 80
Faces	1520	640 320 25px|link=Hexagon 480 25px|link=Square 80 25px|link=Hexagon
Edges	1440	960 240 240
Vertices	480
Vertex figure	Square pyramidal pyramid
Coxeter groups	B5, [3,3,3,4], order 3840 D5, [32,1,1], order 1920
Properties	convex

Alternate names

• Cantitruncated pentacross
• Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±3,±2,±1,0,0)

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]

Related polytopes

These polytopes are from a set of 31 uniform 5-polytopes 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, 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)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  x3o3x3o4o - sart, x3x3x3o4o - gart

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary