Bitruncated cubic honeycomb

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Bitruncated cubic honeycomb
Bitruncated cubic tiling.png HC-A4.png
Type Uniform honeycomb
Schläfli symbol 2t{4,3,4}
t1,2{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)
8o:2
[[4,3,4]]
Coxeter group {\tilde{C}}_3, [4,3,4]
Dual Oblate tetrahedrille
Disphenoid tetrahedral honeycomb
Properties isogonal, 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) is the optimal soap bubble foam. However, the Weaire–Phelan structure is a less symmetrical, but more efficient, foam of soap bubbles.

Symmetry[edit]

The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the {\tilde{A}}_3 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 {\tilde{C}}_3×2
[[4,3,4]]
=[4[3[4]]]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.png = CDel branch c1.pngCDel 3ab.pngCDel branch c1.png
{\tilde{C}}_3
[4,3,4]
=[2[3[4]]]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png = CDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.png
{\tilde{B}}_3
[4,31,1]
=<[3[4]]>
CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 4.pngCDel node.png = CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png
{\tilde{A}}_3
[3[4]]
 
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c4.png
{\tilde{A}}_3×2
[[3[4]]]
=[[3[4]]]
CDel branch c1.pngCDel 3ab.pngCDel branch c2.png
Coxeter diagram CDel branch 11.pngCDel 4a4b.pngCDel nodes.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png CDel branch 11.pngCDel 3ab.pngCDel branch 11.png
truncated octahedra 1
Uniform polyhedron-43-t12.svg
1:1
Uniform polyhedron-43-t12.svg:Uniform polyhedron-43-t12.svg
2:1:1
Uniform polyhedron-43-t12.svg:Uniform polyhedron-43-t12.svg:Uniform polyhedron-33-t012.png
1:1:1:1
Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png
1:1
Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png
Vertex figure Bitruncated cubic honeycomb verf2.png Bitruncated cubic honeycomb verf.png Cantitruncated alternate cubic honeycomb verf.png Omnitruncated 3-simplex honeycomb verf.png Omnitruncated 3-simplex honeycomb verf2.png
Vertex
figure
symmetry
[2+,4]
(order 8)
[2]
(order 4)
[ ]
(order 2)
[ ]+
(order 1)
[2]+
(order 2)
Image
Colored by
cell
Bitruncated Cubic Honeycomb1.svg Bitruncated Cubic Honeycomb.svg Bitruncated cubic honeycomb3.png Bitruncated cubic honeycomb2.png Bitruncated Cubic Honeycomb1.svg

Related polyhedra and honeycombs[edit]

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

The [4,3,4], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, 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.

Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Pm3m
(221)
4:2 [4,3,4] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node c4.png ×1 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 1, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 2, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 3, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 4,
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 5, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 6
Fm3m
(225)
2:2 [1+,4,3,4]
↔ [4,31,1]
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node c2.png
CDel nodes 10ru.pngCDel split2.pngCDel node c1.pngCDel 4.pngCDel node c2.png
Half CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 7, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 11, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 12, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png 13
I43m
(217)
4o:2 [[(4,3,4,2+)]] CDel branch.pngCDel 4a4b.pngCDel nodes hh.png Half × 2 CDel branch.pngCDel 4a4b.pngCDel nodes hh.png (7),
Fd3m
(227)
2+:2 [[1+,4,3,4,1+]]
↔ [[3[4]]]
CDel branch.pngCDel 4a4b.pngCDel nodes h1h1.png
CDel branch 11.pngCDel 3ab.pngCDel branch.png
Quarter × 2 CDel branch.pngCDel 4a4b.pngCDel nodes h1h1.png 10,
Im3m
(229)
8o:2 [[4,3,4]] CDel branch c2.pngCDel 4a4b.pngCDel nodeab c1.png ×2

CDel branch.pngCDel 4a4b.pngCDel nodes 11.png (1), CDel branch 11.pngCDel 4a4b.pngCDel nodes.png 8, CDel branch 11.pngCDel 4a4b.pngCDel nodes 11.png 9

The [4,31,1], CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png, Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.

Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Fm3m
(225)
2:2 [4,31,1]
↔ [4,3,4,1+]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel split1.pngCDel nodes 10lu.png
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
×1 CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 1, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 2, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 3, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 4
Fm3m
(225)
2:2 <[1+,4,31,1]>
↔ <[3[4]]>
CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodeab c1.png
CDel node 1.pngCDel split1.pngCDel nodeab c1.pngCDel split2.pngCDel node.png
×2 CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png (1), CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png (3)
Pm3m
(221)
4:2 <[4,31,1]> CDel node c3.pngCDel 4.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png ×2

CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png 5, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png 6, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png 7, CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png (6), CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png 9, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 10, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 11

This honeycomb is one of five distinct uniform honeycombs[1] constructed by the {\tilde{A}}_3 Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Space
group
Fibrifold Square
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
F43m
(216)
1o:2 a1 [3[4]] CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png ×1 (None)
Fd3m
(227)
2+:2 p2 [[3[4]]] CDel branch 11.pngCDel 3ab.pngCDel branch.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
×2 CDel branch 11.pngCDel 3ab.pngCDel branch.png 3
Fm3m
(225)
2:2 d2 <[3[4]]>
↔ [4,3,31,1]
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png
CDel node.pngCDel 4.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png
×2 CDel node.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node.png 1,CDel node 1.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node 1.png 2
Pm3m
(221)
4:2 d4 [2[3[4]]]
↔ [4,3,4]
CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png
×4 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node.png 4
Im3m
(229)
8o:2 r8 [4[3[4]]]
↔ [[4,3,4]]
CDel branch c1.pngCDel 3ab.pngCDel branch c1.png
CDel branch c1.pngCDel 4a4b.pngCDel nodes.png
×8 CDel branch 11.pngCDel 3ab.pngCDel branch 11.png 5, CDel branch hh.pngCDel 3ab.pngCDel branch hh.png (*)

Alternated form[edit]

Vertex figure for alternated bitruncated cubic honeycomb, with 4 tetrahedral and 4 icosahedral cells. All edges represent triangles in the honeycomb, but edge-lengths can't be made equal.

This honeycomb can be alternated, creating regular icosahedron from the truncated octahedra with irregular tetrahedral cells created in the gaps. There are three constructions from three related Coxeter-Dynkin diagrams: CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png, CDel node.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.png, and CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel split2.pngCDel node h.png. 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]]]+.

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]

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 CDel branch hh.pngCDel 4a4b.pngCDel nodes.png CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.png CDel branch hh.pngCDel 3ab.pngCDel branch hh.png CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel split2.pngCDel node h.png
Order double full half quarter
double
quarter
Image
colored by cells
Alternated bitruncated cubic honeycomb1.png Alternated bitruncated cubic honeycomb2.png Alternated bitruncated cubic honeycomb3.png Alternated bitruncated cubic honeycomb1.png Alternated bitruncated cubic honeycomb4.png

Projection by folding[edit]

The bitruncated cubic honeycomb can be orthogonally projected into the planar truncated square tiling by a geometric folding operation that maps two pairs of mirrors into each other. The projection of the bitruncated cubic honeycomb creating two offset copies of the truncated square tiling vertex arrangement of the plane:

Coxeter
group
{\tilde{A}}_3 {\tilde{C}}_2
Coxeter
diagram
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Graph Bitruncated Cubic Honeycomb flat.png
Bitruncated cubic honeycomb
Uniform tiling 44-t012.png
Truncated square tiling

See also[edit]

Notes[edit]

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

References[edit]

  • 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.
  • Richard Klitzing, 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 links[edit]