Runcinated 5-simplexes

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.

Alternate names

 * Runcinated hexateron
 * Small prismated hexateron (Acronym: spix) (Jonathan Bowers)

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.

Alternate names

 * Runcitruncated hexateron
 * Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)

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.

Alternate names

 * Runcicantellated hexateron
 * Biruncitruncated 5-simplex/hexateron
 * Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)

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.

Alternate names

 * Runcicantitruncated hexateron
 * Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)

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.

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)