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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.
Example: a truncated octahedron is a bitruncated cube: t{3,4} = 2t{4,3}. A complete bitruncation, called a birectification, reduces original faces to points. For polyhedra, this becomes the dual polyhedron. Example: an octahedron is a birectification of a cube: {3,4} = 2r{4,3}.
The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive.
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 ...
The cantic cubic honeycomb, cantic cubic cellulation or truncated half cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated octahedra, cuboctahedra and truncated tetrahedra in a ratio of 1:1:2. Its vertex figure is a rectangular pyramid.
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:
For example, tC represents a truncated cube, and taC, parsed as t(aC), is (topologically) a truncated cuboctahedron. The simplest operator dual swaps vertex and face elements; e.g., a dual cube is an octahedron: dC = O. Applied in a series, these operators allow many higher order polyhedra to be generated.
There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron. Each parallelohedron is a zonohedron and a plesiohedron. It has point reflection symmetry, as do its faces.