into 3-dimensional space, small rhombicuboctahedron first: The projection envelope is a truncated cube. The nearest and farthest small rhombicuboctahedral cells from the 4D viewpoint project to the volume of the same shape inscribed in the projection envelope. The axial squares of this central small rhombicuboctahedron touches the centers of the 6 octagons of the envelope. The octagons are the image of the other 6 small rhombicuboctahedral cells. The 12 wedge-shaped volumes connecting the non-axial square faces of the central small rhombicuboctahedron to the neighbouring octagons are the images of 24 of the triangular prisms. The remaining 8 triangular prisms project onto the triangular faces of the envelope. Between the triangular faces of the envelope and the triangular faces of the central small rhombicuboctahedron are 8 octahedral volumes, which are the images of the 16 octahedral cells. This layout of cells in projection is analogous to the layout of faces in the projection of the truncated cube into 2 dimensions. Hence, the cantellated tesseract may be thought of as an analogue of the truncated cube in 4 dimensions. (It is not the only possible analogue; another close candidate is the truncated tesseract.) Another uniform 4-polytope with a similar layout of cells is the runcitruncated 16-cell. Cantitruncated tesseract In geometry, the cantitruncated tesseract or great rhombated tesseract is a uniform 4-polytope (or uniform 4-dimensional polytope) that is bounded by 56 cells: 8 truncated cuboctahedra, 16 truncated tetrahedra, and 32 triangular prisms. Construction The cantitruncated tesseract is constructed by the cantitruncation of the tesseract. Cantitruncation is often thought of as rectification followed by truncation. However, the result of this construction would be a polytope which, while its structure would be very similar to that given by cantitruncation, not all of its faces would be uniform. Alternatively, a uniform cantitruncated tesseract may be constructed by placing 8 uniform truncated cuboctahedra in the hyperplanes of a tesseract's cells, displaced along the coordinate axes such that their octagonal faces coincide. For an edge length of 2, this construction gives the Cartesian coordinates of its vertices as all permutations of: Structure The 8 truncated cuboctahedra are joined to each other via their octagonal faces, in an arrangement corresponding to the 8 cubical cells of the tesseract. They are joined to the 16 truncated tetrahedra via their hexagonal faces, and their square faces are joined to the square faces of the 32 triangular prisms. The triangular faces of