# Stericated 5-simplexes

(Redirected from Stericated hexateron)
 Orthogonal projections in A5 and A4 Coxeter planes 5-simplex Stericated 5-simplex Steritruncated 5-simplex Stericantellated 5-simplex Stericantitruncated 5-simplex Steriruncitruncated 5-simplex Steriruncicantitruncated 5-simplex (Omnitruncated 5-simplex)

In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.

There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an expanded 5-simplex, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-simplex. The highest form, the steriruncicantitruncated 5-simplex is more simply called an omnitruncated 5-simplex with all of the nodes ringed.

## Stericated 5-simplex

 Stericated 5-simplex Type Uniform 5-polytope Schläfli symbol 2r2r{3,3,3,3} 2r{32,2} = ${\displaystyle 2r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$ Coxeter-Dynkin diagram or 4-faces 62 6+6 {3,3,3} 15+15 {}×{3,3} 20 {3}×{3} Cells 180 60 {3,3} 120 {}×{3} Faces 210 120 {3} 90 {4} Edges 120 Vertices 30 Vertex figure Tetrahedral antiprism Coxeter group A5×2, [[3,3,3,3]], order 1440 Properties convex, isogonal, isotoxal

A stericated 5-simplex can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an expanded 5-simplex. It has 30 vertices, 120 edges, 210 faces (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duoprisms).

### Alternate names

• Expanded 5-simplex
• Stericated hexateron
• Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)[1]

### Cross-sections

The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6 5-cells, 15 tetrahedral prisms and 10 3-3 duoprisms each.

### Coordinates

The vertices of the stericated 5-simplex can be constructed on a hyperplane in 6-space as permutations of (0,1,1,1,1,2). This represents the positive orthant facet of the stericated 6-orthoplex.

A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0)

The Cartesian coordinates in 5-space for the normalized vertices of an origin-centered stericated hexateron are:

${\displaystyle \left(\pm 1,\ 0,\ 0,\ 0,\ 0\right)}$
${\displaystyle \left(0,\ \pm 1,\ 0,\ 0,\ 0\right)}$
${\displaystyle \left(0,\ 0,\ \pm 1,\ 0,\ 0\right)}$
${\displaystyle \left(\pm 1/2,\ 0,\ \pm 1/2,\ -{\sqrt {1/8}},\ -{\sqrt {3/8}}\right)}$
${\displaystyle \left(\pm 1/2,\ 0,\ \pm 1/2,\ {\sqrt {1/8}},\ {\sqrt {3/8}}\right)}$
${\displaystyle \left(0,\ \pm 1/2,\ \pm 1/2,\ -{\sqrt {1/8}},\ {\sqrt {3/8}}\right)}$
${\displaystyle \left(0,\ \pm 1/2,\ \pm 1/2,\ {\sqrt {1/8}},\ -{\sqrt {3/8}}\right)}$
${\displaystyle \left(\pm 1/2,\ \pm 1/2,\ 0,\ \pm {\sqrt {1/2}},\ 0\right)}$

### Root system

Its 30 vertices represent the root vectors of the simple Lie group A5. It is also the vertex figure of the 5-simplex honeycomb.

### Images

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [[3]]=[6]
 orthogonal projection with [6] symmetry

## Steritruncated 5-simplex

 Steritruncated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,1,4{3,3,3,3} Coxeter-Dynkin diagram 4-faces 62 6 t{3,3,3} 15 {}×t{3,3} 20 {3}×{6} 15 {}×{3,3} 6 t0,3{3,3,3} Cells 330 Faces 570 Edges 420 Vertices 120 Vertex figure Coxeter group A5 [3,3,3,3], order 720 Properties convex, isogonal

### Alternate names

• Steritruncated hexateron
• Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)[2]

### Coordinates

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

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

This construction exists as one of 64 orthant facets of the steritruncated 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]

## Stericantellated 5-simplex

 Stericantellated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,2,4{3,3,3,3} Coxeter-Dynkin diagram or 4-faces 62 12 rr{3,3,3} 30 rr{3,3}x{} 20 {3}×{3} Cells 420 60 rr{3,3} 240 {}×{3} 90 {}×{}×{} 30 r{3,3} Faces 900 360 {3} 540 {4} Edges 720 Vertices 180 Vertex figure Coxeter group A5×2, [[3,3,3,3]], order 1440 Properties convex, isogonal

### Alternate names

• Stericantellated hexateron
• Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)[3]

### Coordinates

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

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

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

### Images

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [[3]]=[6]

## Stericantitruncated 5-simplex

 Stericantitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,1,2,4{3,3,3,3} Coxeter-Dynkin diagram 4-faces 62 Cells 480 Faces 1140 Edges 1080 Vertices 360 Vertex figure Coxeter group A5 [3,3,3,3], order 720 Properties convex, isogonal

### Alternate names

• Stericantitruncated hexateron
• Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)[4]

### Coordinates

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

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

This construction exists as one of 64 orthant facets of the stericantitruncated 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]

## Steriruncitruncated 5-simplex

 Steriruncitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,1,3,4{3,3,3,3} 2t{32,2} Coxeter-Dynkin diagram or 4-faces 62 12 t0,1,3{3,3,3} 30 {}×t{3,3} 20 {6}×{6} Cells 450 Faces 1110 Edges 1080 Vertices 360 Vertex figure Coxeter group A5×2, [[3,3,3,3]], order 1440 Properties convex, isogonal

### Alternate names

• Steriruncitruncated hexateron
• Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)[5]

### Coordinates

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

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

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

### Images

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [[3]]=[6]

## Omnitruncated 5-simplex

 Omnitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,1,2,3,4{3,3,3,3} 2tr{32,2} Coxeter-Dynkin diagram or 4-faces 62 12 t0,1,2,3{3,3,3} 30 {}×tr{3,3} 20 {6}×{6} Cells 540 360 t{3,4} 90 {4,3} 90 {}×{6} Faces 1560 480 {6} 1080 {4} Edges 1800 Vertices 720 Vertex figure Irregular 5-cell Coxeter group A5×2, [[3,3,3,3]], order 1440 Properties convex, isogonal, zonotope

The omnitruncated 5-simplex has 720 vertices, 1800 edges, 1560 faces (480 hexagons and 1080 squares), 540 cells (360 truncated octahedrons, 90 cubes, and 90 hexagonal prisms), and 62 4-faces (12 omnitruncated 5-cells, 30 truncated octahedral prisms, and 20 6-6 duoprisms).

### Alternate names

• Steriruncicantitruncated 5-simplex (Full description of omnitruncation for 5-polytopes by Johnson)
• Omnitruncated hexateron
• Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers)[6]

### Coordinates

The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive orthant facet of the steriruncicantitruncated 6-orthoplex, t0,1,2,3,4{34,4}, .

### Images

orthographic projections
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [[3]]=[6]

### Permutohedron

The omnitruncated 5-simplex is the permutohedron of order 6. It is also a zonotope, the Minkowski sum of six line segments parallel to the six lines through the origin and the six vertices of the 5-simplex.

 Orthogonal projection, vertices labeled as a permutohedron.

### Related honeycomb

The omnitruncated 5-simplex honeycomb is constructed by omnitruncated 5-simplex facets with 3 facets around each ridge. It has Coxeter-Dynkin diagram of .

Coxeter group ${\displaystyle {\tilde {I}}_{1}}$ ${\displaystyle {\tilde {A}}_{2}}$ ${\displaystyle {\tilde {A}}_{3}}$ ${\displaystyle {\tilde {A}}_{4}}$ ${\displaystyle {\tilde {A}}_{5}}$
Coxeter-Dynkin
Picture
Name Apeirogon Hextille Omnitruncated
3-simplex
honeycomb
Omnitruncated
4-simplex
honeycomb
Omnitruncated
5-simplex
honeycomb
Facets

## Related uniform polytopes

These polytopes are a part 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, (x3o3o3o3x - scad)
2. ^ Klitizing, (x3x3o3o3x - cappix)
3. ^ Klitizing, (x3o3x3o3x - card)
4. ^ Klitizing, (x3x3x3o3x - cograx)
5. ^ Klitizing, (x3x3o3x3x - captid)
6. ^ Klitizing, (x3x3x3x3x - gocad)

## 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)". x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad