Бүршіктелген кубтық ұяшық құрылымы

Bitruncated cubic honeycomb TypeUniform honeycombSchläfli symbol2t{4,3,4}t1,2{4,3,4}Coxeter Dynkin diagramCell type(4.6.6)Face typessquare {4}hexagon {6}Edge figureisosceles triangle {3}Vertex figure(tetragonal disphenoid)Space groupFibrifold notationCoxeter notationImm (229)8o:2[[4,3,4]]Coxeter group, [4,3,4]DualOblate tetrahedrilleDisphenoid tetrahedral honeycombCell: Propertiesisogonal, isotoxal, isochoric

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. 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 disphenoid tetrahedral honeycomb. Although a regular tetrahedron can not tessellate space alone, this dual has identical disphenoid tetrahedron cells with isosceles triangle faces.

It can be realized as the Voronoi tessellation of the body centred cubic lattice. Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) was the optimal soap bubble foam. However, a number of less symmetrical structures have later been found to be more efficient foams of soap bubbles, among which the Weaire–Phelan structure appears to be the best. The honeycomb represents the permutohedron tessellation for 3 space. The coordinates of the vertices for one octahedron represent a hyperplane of integers in 4 space, specifically permutations of (1,2,3,4). The tessellation is formed by translated copies within the hyperplane. The tessellation is the highest tessellation of parallelohedrons in 3 space.

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 tesseractic 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.