Bitruncated cubic honeycomb
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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.
Bitruncated cubic honeycomb | |
---|---|
Type | Uniform honeycomb |
Schläfli symbol | 2t{4,3,4} t1,2{4,3,4} |
Coxeter-Dynkin diagram | |
Cell type | (4.6.6) |
Face types | square {4} hexagon {6} |
Edge figure | isosceles triangle {3} |
Vertex figure | (tetragonal disphenoid) |
Space group Fibrifold notation Coxeter notation | Im3m (229) 8o:2 [[4,3,4]] |
Coxeter group | , [4,3,4] |
Dual | Oblate tetrahedrille Disphenoid tetrahedral honeycomb Cell: |
Properties | isogonal, isotoxal, isochoric |
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.
Projections
The bitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a nonuniform rhombitrihexagonal tiling. A square symmetry projection forms two overlapping truncated square tiling, which combine together as a chamfered square tiling.
The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the Coxeter group. This honeycomb has four uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions. These uniform symmetries can be represented by coloring differently the cells in each construction.
Space group | Im3m (229) | Pm3m (221) | Fm3m (225) | F43m (216) | Fd3m (227) |
---|---|---|---|---|---|
Fibrifold | 8o:2 | 4−:2 | 2−:2 | 1o:2 | 2+:2 |
Coxeter group | ×2 [[4,3,4]] =[4[3[4]]] = |
[4,3,4] =[2[3[4]]] = |
[4,31,1] =<[3[4]]> = |
[3[4]] |
×2 [[3[4]]] =[[3[4]]] |
Coxeter diagram | |||||
truncated octahedra | 1 |
1:1 : |
2:1:1 :: |
1:1:1:1 ::: |
1:1 : |
Vertex figure | |||||
Vertex figure symmetry |
[2+,4] (order 8) |
[2] (order 4) |
[ ] (order 2) |
[ ]+ (order 1) |
[2]+ (order 2) |
Image Colored by cell |