# Runcinated 5-simplexes

(Redirected from Runcinated 5-simplex)
 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 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

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

## Runcitruncated 5-simplex

 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

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

## Runcicantellated 5-simplex

 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

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

## Runcicantitruncated 5-simplex

 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

### 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
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [5]
Ak
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)

## Notes

1. ^ Klitizing, (x3o3o3x3o - spidtix)
2. ^ Klitizing, (x3x3o3x3o - pattix)
3. ^ Klitizing, (x3o3x3x3o - pirx)
4. ^ 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