Текше кубтық ұяшықтардың кеңістіктік толықтыруы

Only regular space filling tessellation of the cube
Cubic honeycombTypeRegular honeycombFamilyHypercube honeycombIndexingJ11,15, A1W1, G22Schläfli symbol {4,3,4}Coxeter diagramCell type{4,3} Face typesquare {4}Vertex figureoctahedronSpace groupFibrifold notationPmm (221)4−:2Coxeter group, [4,3,4]Dualself dualCell: PropertiesVertex transitive, regular
The cubic honeycomb or cubic cellulation is the only proper regular space filling tessellation (or honeycomb) 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 {4,3,4}. John Horton Conway called this honeycomb a cubille.

It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3, ,3,4}, starting with the square tiling, {4,4} in the plane. It is one of 28 uniform honeycombs using convex uniform polyhedral cells.

It is related to the regular 4 polytope tesseract, Schläfli symbol {4,3,3}, which exists in 4 space, and only has 3 cubes around each edge. It's also related to the order 5 cubic honeycomb, Schläfli symbol {4,3,5}, of hyperbolic space with 5 cubes around each edge. It is in a sequence of polychora and honeycombs with octahedral vertex figures. It in a sequence of regular polytopes and honeycombs with cubic cells.

The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated cubic honeycomb) is geometrically identical to the cubic honeycomb. The [4,31,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb. This honeycomb is one of five distinct uniform honeycombs constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Nonuniform variants with [4,3,4] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with truncated octahedra and hexagonal prisms (as ditrigonal trapezoprisms). Its vertex figure is a C2v symmetric triangular bipyramid. This honeycomb can then be alternated to produce another nonuniform honeycomb with pyritohedral icosahedra, octahedra (as triangular antiprisms), and tetrahedra (as sphenoids). Its vertex figure has C2v symmetry and consists of 2 pentagons, 4 rectangles, 4 isosceles triangles (divided into two sets of 2), and 4 scalene triangles.

It is closely related to the perovskite structure, shown here with cubic symmetry, with atoms placed at the center of the cells of this honeycomb.

A double symmetry construction can be made by placing cuboctahedra on the rhombicuboctahedra, which results in the rectified cubic honeycomb, by taking the triangular antiprism gaps as regular octahedra, square antiprism pairs and zero height tetragonal disphenoids as components of the cuboctahedron. Other variants result in cuboctahedra, square antiprisms, octahedra (as triangular antipodiums), and tetrahedra (as tetragonal disphenoids), with a vertex figure topologically equivalent to a cube with a triangular prism attached to one of its square faces.

The dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram , containing faces from two of four hyperplanes of the cubic [4,3,4] fundamental domain. It has irregular triangle bipyramid cells which can be seen as 1/12 of a cube, made from the cube center, 2 face centers, and 2 vertices.

The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram, This honeycomb cells represents the fundamental domains of symmetry. A cell can be as 1/24 of a translational cube with vertices positioned: taking two corner, ne face center, and the cube center. The edge colors and labels specify how many cells exist around the edge.

Two related uniform skew apeirohedrons exists with the same vertex arrangement, seen as boundary cells from a subset of cells. One has triangles and squares, and the other triangles, squares, and octagons.

The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram Faces exist in 3 of 4 hyperplanes of the [4,3,4], 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.

Two related uniform skew apeirohedron exist with the same vertex arrangement. The first has octagons removed, and vertex configuration 4.4.4.6. It can be seen as truncated cuboctahedra and octagonal prisms augmented together. The second can be seen as augmented octagonal prisms, vertex configuration 4.8.4.8. 4.4.4.64.8.4.8