Runcinated 5-simplexes

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

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
Type	Uniform 5-polytope
Schläfli symbol	t0,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 t0,3{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

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

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.

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Runcitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t0,1,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 t0,1,3{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

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

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.

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Runcicantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t0,2,3{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

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

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.

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Runcicantitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t0,1,2,3{3,3,3,3}
Coxeter-Dynkin diagram
4-faces	47	6 t0,1,2,3{3,3,3} 20 {3}×{6} 15 {}×t{3,3} 6 tr{3,3,3}
Cells	315	45 t0,1,2{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

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

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.

orthographic projections

Ak Coxeter plane	A5	A4
Graph
Dihedral symmetry	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

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	t0,1	t0,2	t1,2	t0,3
t1,3	t0,4	t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4
t0,2,4	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,1,2,3,4

• 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

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