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Orthogonal projections in A₅ Coxeter plane
5-simplex	Runcinated 5-simplex	Runcitruncated 5-simplex
Birectified 5-simplex	Runcicantellated 5-simplex	Runcicantitruncated 5-simplex

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.

Runcinated 5-simplex

Runcinated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t_0,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 t_0,3{3,3,3} 20 {3}×{3} 15 { }×r{3,3} 6 r{3,3,3}
Cells	255	45 {3,3} 180 { }×{3} 30 r{3,3}
Faces	420	240 {3} 180 {4}
Edges	270
Vertices	60
Vertex figure
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex

Alternate names

Runcinated hexateron
Small prismated hexateron (Acronym: spix) (Jonathan Bowers)^[1]

Coordinates

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

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Runcitruncated 5-simplex

Runcitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t_0,1,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 t_0,1,3{3,3,3} 20 {3}×{6} 15 { }×r{3,3} 6 rr{3,3,3}
Cells	315
Faces	720
Edges	630
Vertices	180
Vertex figure
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex, isogonal

Alternate names

Runcitruncated hexateron
Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)^[2]

Coordinates

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

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

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

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Runcicantellated 5-simplex

Runcicantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t_0,2,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47
Cells	255
Faces	570
Edges	540
Vertices	180
Vertex figure
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex, isogonal

Alternate names

Runcicantellated hexateron
Biruncitruncated 5-simplex/hexateron
Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)^[3]

Coordinates

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

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

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

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Runcicantitruncated 5-simplex

Runcicantitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t_0,1,2,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 t_0,1,2,3{3,3,3} 20 {3}×{6} 15 {}×t{3,3} 6 tr{3,3,3}
Cells	315	45 t_0,1,2{3,3} 120 { }×{3} 120 { }×{6} 30 t{3,3}
Faces	810	120 {3} 450 {4} 240 {6}
Edges	900
Vertices	360
Vertex figure	Irregular 5-cell
Coxeter group	A₅ [3,3,3,3], order 720
Properties	convex, isogonal

Alternate names

Runcicantitruncated hexateron
Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)^[4]

Coordinates

The coordinates can be made in 6-space, as 360 permutations of:

(0,0,1,2,3,4)

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

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph
Dihedral symmetry	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[3]

Related uniform 5-polytopes

These polytopes are in a set 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
t₀	t₁	t₂	t_0,1	t_0,2	t_1,2	t_0,3
t_1,3	t_0,4	t_0,1,2	t_0,1,3	t_0,2,3	t_1,2,3	t_0,1,4
t_0,2,4	t_0,1,2,3	t_0,1,2,4	t_0,1,3,4	t_0,1,2,3,4

Notes

^ Klitizing, (x3o3o3x3o - spidtix)
^ Klitizing, (x3x3o3x3o - pattix)
^ Klitizing, (x3o3x3x3o - pirx)
^ Klitizing, (x3x3x3x3o - gippix)

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)". x3o3o3x3o - spidtix, x3x3o3x3o - pattix, x3o3x3x3o - pirx, x3x3x3x3o - gippix

External links

Glossary for hyperspace, George Olshevsky.
Polytopes of Various Dimensions, Jonathan Bowers
- Runcinated uniform polytera (spid), 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