Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol. John Horton Conway called this honeycomb a cubille.
Description
The cubic honeycomb is a space-filling or three-dimensional tessellation consisting of many cubes that attach each other to the faces; the cube is known as cell of a honeycomb. The parallelepiped is the member of a parallelohedron, generated from three line segments that are not all parallel to a common plane. The cube is the special case of a parallelepiped for having the most symmetric form, generated by three perpendicular unit-length line segments. In three-dimensional space, the cubic honeycomb is the only proper regular space-filling tessellation. It is self-dual.Related honeycombs
The rectified cubic honeycomb or rectified cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1, with a square prism vertex figure. John Horton Conway calls this honeycomb a cuboctahedrille, and its dual an oblate octahedrille.The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1, with an isosceles square pyramid vertex figure. John Horton Conway calls this honeycomb a truncated cubille, and its dual pyramidille.
The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has four truncated octahedra around each vertex, in a tetragonal disphenoid vertex figure. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs. John Horton Conway calls this honeycomb a truncated octahedrille in his Architectonic and catoptric tessellation list, with its dual called an oblate tetrahedrille, also called a tetrahedral honeycomb">tetrahedron">tetrahedral honeycomb. Although a regular tetrahedron can not tessellate space alone, this dual has identical disphenoid tetrahedron cells with isosceles triangle faces.
The alternated bitruncated cubic honeycomb or bisnub cubic honeycomb is non-uniform, with the highest symmetry construction reflecting an alternation of the uniform bitruncated cubic honeycomb. A lower-symmetry construction involves regular icosahedra paired with golden icosahedra. There are three constructions from three related Coxeter diagrams:,, and. These have symmetry, and ]+ respectively. The first and last symmetry can be doubled as
The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space. It is composed of rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3, with a wedge vertex figure. John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.
[Image:Cantitruncated cubic tiling.png|thumb|Cantitruncated cubic tiling]
The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3, with a mirrored sphenoid vertex figure. John Horton Conway calls this honeycomb a n-tCO-trille, and its dual triangular pyramidille. Its dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram,. These honeycomb cells represent the fundamental domains of symmetry. A cell can be as 1/24 of a translational cube with vertices positioned: taking two corners, ne face center, and the cube center. The edge colors and labels specify how many cells exist around the edge.
The alternated cantitruncated cubic honeycomb or snub rectified cubic honeycomb contains three types of cells: snub cubes, icosahedra, tetrahedra, and new tetrahedral cells created at the gaps.
Although it is not uniform, constructionally it can be given as Coxeter diagrams or. Despite being non-uniform, there is a near-miss version with two edge lengths shown below, one of which is around 4.3% greater than the other. The snub cubes in this case are uniform, but the rest of the cells are not.
The cantic snub cubic honeycomb is constructed by snubbing the truncated octahedra in a way that leaves only rectangles from the cubes. It is not uniform but it can be represented as Coxeter diagram. It has rhombicuboctahedra, icosahedra, and triangular prisms filling the gaps.
The runcitruncated cubic honeycomb or runcitruncated cubic cellulation is a uniform space-filling tessellation in Euclidean 3-space. It is composed of rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3, with an isosceles-trapezoidal pyramid vertex figure. Its name is derived from its Coxeter diagram, with three ringed nodes representing 3 active mirrors in the Wythoff construction from its relation to the regular cubic honeycomb. John Horton Conway calls this honeycomb a 1-RCO-trille, and its dual square quarter pyramidille. Its dual is square quarter pyramidille, with Coxeter diagram. Faces exist in 3 of 4 hyperplanes of the, Coxeter group. Cells are irregular pyramids and can be seen as 1/24 of a cube, using one corner, one mid-edge point, two face centers, and the cube center.
An alternated omnitruncated cubic honeycomb or omnisnub cubic honeycomb can be constructed by alternation of the omnitruncated cubic honeycomb, although it can not be made uniform, but it can be given Coxeter diagram: and has symmetry +. It makes square antiprisms from the octagonal prisms, tetrahedra from the cubes, and two tetrahedra from the triangular bipyramids.