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. Cantellated tesseract 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. Construction 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. Cartesian coordinates The Cartesian coordinates of the vertices of a cantellated tesseract with edge length 2 is given by all permutations of: Structure The 8 small rhombicuboctahedral cells are joined to each other via their axial square faces. Their non-axial square faces, which correspond with the edges of a cube, are connected to the triangular prisms. The triangular faces of the small rhombicuboctahedra and the triangular prisms are connected to the 16 octahedra. Its structure can be imagined by means of the tesseract itself: the rhombicuboctahedra are analogous to the tesseract's cells, the triangular prisms are analogous to the tesseract's edges, and the octahedra are analogous to the tesseract's vertices. Images Projections The following is the layout of the cantellated tesseract's cells under the parallel projection