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Concept
Cantellated tesseract
Summary
In four-dimensional geometry, a cantellated tesseract is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation) of the regular tesseract.
There are four degrees of cantellations of the tesseract including with permutations truncations. Two are also derived from the 24-cell family.
The cantellated tesseract, bicantellated 16-cell, or small rhombated tesseract is a convex uniform 4-polytope or 4-dimensional polytope bounded by 56 cells: 8 small rhombicuboctahedra, 16 octahedra, and 32 triangular prisms.
In the process of cantellation, a polytope's 2-faces are effectively shrunk. The rhombicuboctahedron can be called a cantellated cube, since if its six faces are shrunk in their respective planes, each vertex will separate into the three vertices of the rhombicuboctahedron's triangles, and each edge will separate into two of the opposite edges of the rhombicuboctahedrons twelve non-axial squares.
When the same process is applied to the tesseract, each of the eight cubes becomes a rhombicuboctahedron in the described way. In addition however, since each cube's edge was previously shared with two other cubes, the separating edges form the three parallel edges of a triangular prism—32 triangular prisms, since there were 32 edges. Further, since each vertex was previously shared with three other cubes, the vertex would split into 12 rather than three new vertices. However, since some of the shrunken faces continues to be shared, certain pairs of these 12 potential vertices are identical to each other, and therefore only 6 new vertices are created from each original vertex (hence the cantellated tesseract's 96 vertices compared to the tesseract's 16). These six new vertices form the vertices of an octahedron—16 octahedra, since the tesseract had 16 vertices.
The Cartesian coordinates of the vertices of a cantellated tesseract with edge length 2 is given by all permutations of:
The 8 small rhombicuboctahedral cells are joined to each other via their axial square faces.
Official source
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