User:Tomruen/Cantellated 5-simplexes

In five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex.

There are unique 4 degrees of cantellation for the 5-simplex, including truncations.

Cantellated 5-simplex
The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).

Alternate names

 * Cantellated hexateron
 * Small rhombated hexateron (Acronym: sarx) (Jonathan Bowers)

Coordinates
The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,1,2) or of (0,1,1,2,2,2). These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively.

Alternate names

 * Bicantellated hexateron
 * Small birhombated dodecateron (Acronym: sibrid) (Jonathan Bowers)

Coordinates
The coordinates can be made in 6-space, as 90 permutations of:
 * (0,0,1,1,2,2)

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

Alternate names

 * Cantitruncated hexateron
 * Great rhombated hexateron (Acronym: garx) (Jonathan Bowers)

Coordinates
The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 6-cube respectively.

Alternate names

 * Bicantitruncated hexateron
 * Great birhombated dodecateron (Acronym: gibrid) (Jonathan Bowers)

Coordinates
The coordinates can be made in 6-space, as 180 permutations of:
 * (0,0,1,2,3,3)

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

Related uniform 5-polytopes
The cantellated 5-simplex is one 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)