Stericated 5-simplexes

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
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)

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:


 * $$\left(\pm1,\ 0,\ 0,\ 0,\ 0\right)$$
 * $$\left(0,\ \pm1,\ 0,\ 0,\ 0\right)$$
 * $$\left(0,\ 0,\ \pm1,\ 0,\ 0\right)$$
 * $$\left(\pm1/2,\     0,\ \pm1/2,\ -\sqrt{1/8},\ -\sqrt{3/8}\right)$$
 * $$\left(\pm1/2,\     0,\ \pm1/2,\  \sqrt{1/8},\  \sqrt{3/8}\right)$$
 * $$\left(    0,\ \pm1/2,\ \pm1/2,\ -\sqrt{1/8},\  \sqrt{3/8}\right)$$
 * $$\left(    0,\ \pm1/2,\ \pm1/2,\  \sqrt{1/8},\ -\sqrt{3/8}\right)$$
 * $$\left(\pm1/2,\ \pm1/2,\ 0,\ \pm\sqrt{1/2},\ 0\right)$$

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.

Alternate names

 * Steritruncated hexateron
 * Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)

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.

Alternate names

 * Stericantellated hexateron
 * Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers)

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.

Alternate names

 * Stericantitruncated hexateron
 * Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)

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.

Alternate names

 * Steriruncitruncated hexateron
 * Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)

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.

Omnitruncated 5-simplex
The omnitruncated 5-simplex has 720 vertices, 1800 edges, 1560 faces (480 hexagons and 1080 squares), 540 cells (360 truncated octahedra, 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)

Coordinates
The vertices of the omnitruncated 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},.

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.

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.

Full snub 5-simplex
The full snub 5-simplex or omnisnub 5-simplex, defined as an alternation of the omnitruncated 5-simplex is not uniform, but it can be given Coxeter diagram and symmetry [3,3,3,3]+, and constructed from 12 snub 5-cells, 30 snub tetrahedral antiprisms, 20 3-3 duoantiprisms, and 360 irregular 5-cells filling the gaps at the deleted vertices.

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)