Bitruncated cubic honeycomb

Bitruncated cubic honeycomb
Bitruncated cubic tiling.png HC-A4.png
Type Uniform honeycomb
Schläfli symbol 2t{4,3,4}
Coxeter-Dynkin diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cell type (4.6.6)
Face types square {4}
hexagon {6}
Edge figure isosceles triangle {3}
Vertex figure Bitruncated cubic honeycomb verf2.png
(tetragonal disphenoid)
Space group
Fibrifold notation
Coxeter notation
Im3m (229)
Coxeter group , [4,3,4]
Dual Oblate tetrahedrille
Disphenoid tetrahedral honeycomb
Cell: Oblate tetrahedrille cell.png
Properties isogonal, isotoxal, isochoric
The bitruncated cubic honeycomb shown here in relation to a cubic honeycomb

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) is the optimal soap bubble foam. However, the Weaire–Phelan structure is a less symmetrical, but more efficient, foam of soap bubbles.

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 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.

Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)


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.

Five uniform colorings by cell
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
Coxeter diagram                            
truncated octahedra 1
 : : 
 : : : 
Vertex figure          
(order 8)
(order 4)
[ ]
(order 2)
[ ]+
(order 1)
(order 2)
Colored by

Related polyhedra and honeycombsEdit

The regular skew apeirohedron {6,4|4} contains the hexagons of this honeycomb.

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[1] constructed by the   Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Alternated formEdit

Alternated bitruncated cubic honeycomb
Type Convex honeycomb
Schläfli symbol 2s{4,3,4}
Coxeter diagrams        
Cells tetrahedron
Vertex figure  
Coxeter group [[4,3+,4]],  
Dual Ten-of-diamonds honeycomb
Properties vertex-transitive

This honeycomb can be alternated, creating pyritohedral icosahedra from the truncated octahedra with disphenoid tetrahedral cells created in the gaps. There are three constructions from three related Coxeter-Dynkin diagrams:        ,      , and      . These have symmetry [4,3+,4], [4,(31,1)+] and [3[4]]+ respectively. The first and last symmetry can be doubled as [[4,3+,4]] and [[3[4]]]+.

The dual honeycomb is made of cells called ten-of-diamonds decahedra.

Five uniform colorings
Space group I3 (204) Pm3 (200) Fm3 (202) Fd3 (203) F23 (196)
Fibrifold 8−o 4 2 2o+ 1o
Coxeter group [[4,3+,4]] [4,3+,4] [4,(31,1)+] [[3[4]]]+ [3[4]]+
Coxeter diagram                            
Order double full half quarter
colored by cells

This honeycomb is represented in the boron atoms of the α-rhombihedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.[2]


Related polytopesEdit

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.

See alsoEdit


  1. ^ [1], A000029 6-1 cases, skipping one with zero marks
  2. ^ Williams, 1979, p 199, Figure 5-38.


  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  • Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
  • A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
  • Klitzing, Richard. "3D Euclidean Honeycombs o4x3x4o - batch - O16".
  • Uniform Honeycombs in 3-Space: 05-Batch
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.

External linksEdit