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. The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra. Truncated tesseract (Norman W. Johnson) Truncated tesseract (Acronym tat) (George Olshevsky, and Jonathan Bowers) The truncated tesseract may be constructed by truncating the vertices of the tesseract at 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: In the truncated cube first parallel projection of the truncated tesseract into 3-dimensional space, the image is laid out as follows: The projection envelope is a cube. Two of the truncated cube cells project onto a truncated cube inscribed in the cubical envelope. The other 6 truncated cubes project onto the square faces of the envelope. The 8 tetrahedral volumes between the envelope and the triangular faces of the central truncated cube are the images of the 16 tetrahedra, a pair of cells to each image. The truncated tesseract, is third in a sequence of truncated hypercubes: 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. Bitruncated tesseract/Runcicantic tesseract (Norman W. Johnson) Tesseractihexadecachoron (Acronym tah) (George Olshevsky, and Jonathan Bowers) 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.

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Runcinated tesseracts

In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract. There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations. The runcinated tesseract or (small) disprismatotesseractihexadecachoron has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.

Uniform polytope

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges). This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions.

Cantellated tesseract

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.

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