Truncated tesseract
In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract. There are three truncations, including a bitruncation, and a tritruncation, which creates the "truncated 16-cell". Truncated tesseract. The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra. Construction. The truncated tesseract may be constructed by truncating the vertices of the tesseract at formula_1 of the edge length. A regular tetrahedron is formed at each truncated vertex. The Cartesian coordinates of the vertices of a truncated tesseract having edge length 2 is given by all permutations  of: Projections. In the truncated cube first parallel projection of the truncated tesseract into 3-dimensional space, the image is laid out as follows: Related polytopes. The "truncated tesseract", is third in a sequence of truncated hypercubes: Bitruncated tesseract. The bitruncated tesseract, bitruncated 16-cell, or tesseractihexadecachoron is constructed by a bitruncation operation applied to the tesseract. It can also be called a runcicantic tesseract with half the vertices of a runcicantellated tesseract with a construction. Construction. A tesseract is bitruncated by truncating its cells beyond their midpoints, turning the eight cubes into eight truncated octahedra. These still share their  square faces, but the hexagonal faces form truncated tetrahedra which share their triangular faces with each other. The Cartesian coordinates of the vertices of a bitruncated tesseract having edge length 2 is given by all permutations of: Structure. The truncated octahedra are connected to each other via their square faces, and to the truncated tetrahedra via their hexagonal faces. The truncated tetrahedra are connected to each other via their triangular faces. Stereographic projections. The truncated-octahedron-first projection of the bitruncated tesseract into 3D space has a truncated cubical envelope. Two of the truncated octahedral cells project onto a truncated  octahedron inscribed in this envelope, with the square faces touching the centers of the octahedral faces. The 6 octahedral faces are the images of the remaining 6 truncated octahedral cells. The remaining gap between the inscribed truncated octahedron and the envelope are filled by 8 flattened truncated tetrahedra, each of which is the image of a pair of truncated tetrahedral cells. Related polytopes. The "bitruncated tesseract" is second in a sequence of bitruncated hypercubes: Truncated 16-cell. The truncated 16-cell, truncated hexadecachoron, cantic tesseract which is bounded by 24 cells: 8 regular octahedra, and 16 truncated tetrahedra. It has  half the vertices of a cantellated tesseract with construction . It is related to, but not to be confused with, the 24-cell, which is a regular 4-polytope bounded by 24 regular octahedra. Construction. The truncated 16-cell may be constructed from the 16-cell by truncating its vertices at 1/3 of the edge length. This results in the 16 truncated tetrahedral cells, and introduces the 8 octahedra (vertex figures). The Cartesian coordinates of the vertices of a truncated 16-cell having edge length √2 are given by all permutations, and sign combinations of An alternate construction begins with a  demitesseract with vertex coordinates (±3,±3,±3,±3), having an even number of each sign, and truncates it to obtain the permutations of Structure. The truncated tetrahedra are joined to each other via their hexagonal faces. The octahedra are joined to the truncated tetrahedra via their triangular faces. Projections. Centered on octahedron. The octahedron-first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure: This layout of cells in projection is analogous to the layout of faces in the projection of the truncated octahedron into 2-dimensional space. Hence, the truncated 16-cell may be thought of  as the 4-dimensional analogue of the truncated octahedron. Centered on truncated tetrahedron. The truncated tetrahedron first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure: Related polytopes. A truncated 16-cell, as a cantic 4-cube, is related to the dimensional family of cantic n-cubes: