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). 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. Bicantellated 5-simplex. Coordinates. The coordinates can be made in 6-space, as 90 permutations of: This construction exists as one of 64 orthant facets of the bicantellated 6-orthoplex. Cantitruncated 5-simplex. 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 . Bicantitruncated 5-simplex. Coordinates. The coordinates can be made in 6-space, as 180 permutations of: 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 A Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)