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The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron G IV (1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a ...
There are many relations among the uniform polyhedra. [1] [2] [3] Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron.Others share the same vertices and edges as other polyhedron.
truncated triangular dihedron (Half of the "edges" count as degenerate digon faces. The other half are normal edges.) triangular prism: truncated tetrahedron: truncated octahedron: truncated cube: truncated icosahedron: truncated dodecahedron: truncated triangular tiling: truncated hexagonal tiling: Truncated order-7 triangular tiling ...
In the geometry of hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal tiling, and trihexagonal tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
A bitruncated cube is a truncated octahedron. A bitruncated cubic honeycomb - Cubic cells become orange truncated octahedra, and vertices are replaced by blue truncated octahedra. In geometry, a bitruncation is an operation on regular polytopes. The original edges are lost completely and the original faces remain as smaller copies of themselves.
The truncated order-4 hexagonal tiling honeycomb, t 0,1 {6,3,4}, has octahedron and truncated hexagonal tiling facets, with a square pyramid vertex figure. It is similar to the 2D hyperbolic truncated order-4 apeirogonal tiling , t{∞,4}, with apeirogonal and square faces:
The snub octahedron represents the pseudoicosahedron, a regular icosahedron with pyritohedral symmetry. The snub tetratetrahedron, as {}, and , is the alternation of the truncated tetrahedral symmetry form, {}, and .
The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout: The projection envelope is a truncated cuboctahedron. Two of the truncated octahedra project onto a truncated octahedron lying in the center of the envelope.