Cubic honeycomb
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.
These include: The cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb.
(1), 8, 9 The [4,31,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.
John Horton Conway calls this honeycomb a cuboctahedrille, and its dual an oblate octahedrille.
The rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
This honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra, creating two triangular cupolae.
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 truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
The bitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.
The cantellated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
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.
The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) 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.
Four cells exist around each vertex: The cantitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
The bifurcating diagram form can be drawn with two types (colors) of truncated cuboctahedron cells alternating.
The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram, .
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.
The runcitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram .
A double symmetry construction can be made by placing rhombicuboctahedra on the truncated cubes, resulting in a nonuniform honeycomb with rhombicuboctahedra, octahedra (as triangular antiprisms), cubes (as square prisms), two kinds of triangular prisms (both C2v-symmetric wedges), and tetrahedra (as digonal disphenoids).
It is composed of truncated cuboctahedra and octagonal prisms in a ratio of 1:3, with a phyllic disphenoid vertex figure.
John Horton Conway calls this honeycomb a b-tCO-trille, and its dual eighth pyramidille.
The omnitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
The symmetry can be doubled by relating the first and last branches of the Coxeter diagram, which can be shown with one color for all the truncated cuboctahedral and octagonal prism cells.
Vertex figure Dual cell This honeycomb can then be alternated to produce another nonuniform honeycomb with snub cubes, square antiprisms, octahedra (as triangular antiprisms), and three kinds of tetrahedra (as tetragonal disphenoids, phyllic disphenoids, and irregular tetrahedra).
It makes snub cubes from the truncated cuboctahedra, square antiprisms from the octagonal prisms, and creates new tetrahedral cells from the gaps.
The biorthosnub cubic honeycomb is constructed by removing alternating long rectangles from the octagons orthogonally and is not uniform, but it can be represented as Coxeter diagram .
