Cantellated 5-orthoplexes

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           32
Cells	640	80         160         320         80
Faces	1520	640       320       480       80
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

• Cantellated 5-orthoplex
• Bicantellated 5-demicube
• Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)^[1]

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

(0,0,1,1,2)

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
Type	uniform 5-polytope
Schläfli symbol	tr{3,3,3,4} tr{3,31,1}
Coxeter-Dynkin diagrams
4-faces	82	10           40           32
Cells	640	80         160         320         80
Faces	1520	640       320       480       80
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

• Cantitruncated pentacross
• Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)^[2]

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)

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 from a set of 31 uniform 5-polytopes 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

• 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)". x3o3x3o4o - sart, x3x3x3o4o - gart

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