The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. It has 12 regular square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron.
Note that the name truncated cuboctahedron may be a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get an actual regular truncated cuboctahedron: some of the faces will be irregular polygons. However, the resulting figure is topologically equivalent to truncated cuboctahedron and can always be deformed until the faces are regular. The alternative name great rhombicuboctahedron refers to the fact that the 12 square faces lie in the same planes as the 12 faces of the rhombic dodecahedron which is dual to the cuboctahedron. Compare to small rhombicuboctahedron.
Canonical coordinates for the vertices of a truncated cuboctahedron centered at the origin are all permutations of (±1, ±(1+√2), ±(1+√8)).
In geometry, the omnitruncated tesseract is a uniform polychoron (or uniform 4-dimensional polytope) bounded by 80 cells: 8 great rhombicuboctahedra, 16 truncated octahedra, 24 octagonal prisms, and 32 hexagonal prisms.
Finally, the 8 volumes between the hexagonal faces of the projection envelope and the hexagonal faces of the central greatrhombicuboctahedron are the images of the 16 truncated octahedra, two cells to each image.
This layout of cells in projection is similar to that of the runcitruncated 16-cell, which is analogous to the layout of faces in the octagon-first projection of the greatrhombicuboctahedron into 2 dimensions.
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