Cantellated 5-simplexes

Orthogonal projections in A₅ Coxeter plane
File:5-simplex t0.svg 5-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:5-simplex t02.svg Cantellated 5-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:5-simplex t13.svg Bicantellated 5-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
File:5-simplex t2.svg Birectified 5-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:5-simplex t012.svg Cantitruncated 5-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:5-simplex t123.svg Bicantitruncated 5-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png

In five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex. There are unique 4 degrees of cantellation for the 5-simplex, including truncations.

Cantellated 5-simplex

Cantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	rr{3,3,3,3} = $r {\begin{cases} 3, 3, 3 \\ 3 \end{cases}}$
Coxeter-Dynkin diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
4-faces	27	6 r{3,3,3}File:Schlegel half-solid rectified 5-cell.png 6 rr{3,3,3}File:Schlegel half-solid cantellated 5-cell.png 15 {}x{3,3}File:Tetrahedral prism.png
Cells	135	30 {3,3}File:Tetrahedron.png 30 r{3,3}File:Uniform polyhedron-33-t1.svg 15 rr{3,3}File:Cantellated tetrahedron.png 60 {}x{3}File:Triangular prism.png
Faces	290	200 {3} 90 {4}
Edges	240
Vertices	60
Vertex figure	File:Cantellated hexateron verf.png Tetrahedral prism
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex

The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).

Alternate names

Cantellated hexateron
Small rhombated hexateron (Acronym: sarx) (Jonathan Bowers)^[1]

Coordinates

The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,1,2) or of (0,1,1,2,2,2). These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively.

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph	File:5-simplex t02.svg	File:5-simplex t02 A4.svg
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph	File:5-simplex t02 A3.svg	File:5-simplex t02 A2.svg
Dihedral symmetry	[4]	[3]

Bicantellated 5-simplex

Bicantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	2rr{3,3,3,3} = $r {\begin{cases} 3, 3 \\ 3, 3 \end{cases}}$
Coxeter-Dynkin diagram	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel 3ab.pngFile:CDel nodes.png
4-faces	32	12 t02{3,3,3} 20 {3}x{3}
Cells	180	30 t1{3,3} 120 {}x{3} 30 t02{3,3}
Faces	420	240 {3} 180 {4}
Edges	360
Vertices	90
Vertex figure	File:Bicantellated 5-simplex verf.png
Coxeter group	A₅×2, [[3,3,3,3]], order 1440
Properties	convex, isogonal

Alternate names

Bicantellated hexateron
Small birhombated dodecateron (Acronym: sibrid) (Jonathan Bowers)^[2]

Coordinates

The coordinates can be made in 6-space, as 90 permutations of:

(0,0,1,1,2,2)

This construction exists as one of 64 orthant facets of the bicantellated 6-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph	File:5-simplex t13.svg	File:5-simplex t13 A4.svg
Dihedral symmetry	[6]	[[5]]=[10]
A_k Coxeter plane	A₃	A₂
Graph	File:5-simplex t13 A3.svg	File:5-simplex t13 A2.svg
Dihedral symmetry	[4]	[[3]]=[6]

Cantitruncated 5-simplex

cantitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	tr{3,3,3,3} = $t {\begin{cases} 3, 3, 3 \\ 3 \end{cases}}$
Coxeter-Dynkin diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
4-faces	27	6 t012{3,3,3}File:4-simplex t012.svg 6 t{3,3,3}File:4-simplex t01.svg 15 {}x{3,3}
Cells	135	15 t012{3,3} File:3-simplex t012.svg 30 t{3,3}File:3-simplex t01.svg 60 {}x{3} 30 {3,3}File:3-simplex t0.svg
Faces	290	120 {3}File:2-simplex t0.svg 80 {6}File:2-simplex t01.svg 90 {}x{}File:2-cube.svg
Edges	300
Vertices	120
Vertex figure	File:Canitruncated 5-simplex verf.png Irr. 5-cell
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex

Alternate names

Cantitruncated hexateron
Great rhombated hexateron (Acronym: garx) (Jonathan Bowers)^[3]

Coordinates

The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 6-cube respectively.

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph	File:5-simplex t012.svg	File:5-simplex t012 A4.svg
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph	File:5-simplex t012 A3.svg	File:5-simplex t012 A2.svg
Dihedral symmetry	[4]	[3]

Bicantitruncated 5-simplex

Bicantitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	2tr{3,3,3,3} = $t {\begin{cases} 3, 3 \\ 3, 3 \end{cases}}$
Coxeter-Dynkin diagram	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel 3ab.pngFile:CDel nodes.png
4-faces	32	12 tr{3,3,3} 20 {3}x{3}
Cells	180	30 t{3,3} 120 {}x{3} 30 t{3,4}
Faces	420	240 {3} 180 {4}
Edges	450
Vertices	180
Vertex figure	File:Bicanitruncated 5-simplex verf.png
Coxeter group	A₅×2, [[3,3,3,3]], order 1440
Properties	convex, isogonal

Alternate names

Bicantitruncated hexateron
Great birhombated dodecateron (Acronym: gibrid) (Jonathan Bowers)^[4]

Coordinates

The coordinates can be made in 6-space, as 180 permutations of:

(0,0,1,2,3,3)

This construction exists as one of 64 orthant facets of the bicantitruncated 6-orthoplex.

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph	File:5-simplex t123.svg	File:5-simplex t123 A4.svg
Dihedral symmetry	[6]	[[5]]=[10]
A_k Coxeter plane	A₃	A₂
Graph	File:5-simplex t123 A3.svg	File:5-simplex t123 A2.svg
Dihedral symmetry	[4]	[[3]]=[6]

Related uniform 5-polytopes

The cantellated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A₅ Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

A5 polytopes
File:5-simplex t0.svg t₀	File:5-simplex t1.svg t₁	File:5-simplex t2.svg t₂	File:5-simplex t01.svg t_0,1	File:5-simplex t02.svg t_0,2	File:5-simplex t12.svg t_1,2	File:5-simplex t03.svg t_0,3
File:5-simplex t13.svg t_1,3	File:5-simplex t04.svg t_0,4	File:5-simplex t012.svg t_0,1,2	File:5-simplex t013.svg t_0,1,3	File:5-simplex t023.svg t_0,2,3	File:5-simplex t123.svg t_1,2,3	File:5-simplex t014.svg t_0,1,4
File:5-simplex t024.svg t_0,2,4	File:5-simplex t0123.svg t_0,1,2,3	File:5-simplex t0124.svg t_0,1,2,4	File:5-simplex t0134.svg t_0,1,3,4	File:5-simplex t01234.svg t_0,1,2,3,4

Notes

↑ Klitizing, (x3o3x3o3o - sarx)
↑ Klitizing, (o3x3o3x3o - sibrid)
↑ Klitizing, (x3x3x3o3o - garx)
↑ Klitizing, (o3x3x3x3o - gibrid)

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)". x3o3x3o3o - sarx, o3x3o3x3o - sibrid, x3x3x3o3o - garx, o3x3x3x3o - gibrid

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	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] Klitizing, (x3o3x3o3o - sarx)

[2] Klitizing, (o3x3o3x3o - sibrid)

[3] Klitizing, (x3x3x3o3o - garx)

[4] Klitizing, (o3x3x3x3o - gibrid)

[1]

[2]

[3]

[4]

Cantellated 5-simplexes

Contents