Cantellated 5-orthoplexes

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

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

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 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

Alternate names

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

Coordinates

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 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

Alternate names

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

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.

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

Notes

1. Klitizing, (x3o3x3o4o - sart)
2. Klitizing, (x3x3x3o4o - gart)

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)". x3o3x3o4o - sart, x3x3x3o4o - gart

External links

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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds