Tetrahedraloctahedral honeycomb
The tetrahedraloctahedral honeycomb, alternated cubic honeycomb is a quasiregular spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
Alternated cubic honeycomb  

Type  Uniform honeycomb 
Family  Alternated hypercubic honeycomb Simplectic honeycomb 
Indexing[1]  J_{21,31,51}, A_{2} W_{9}, G_{1} 
Schläfli symbols  h{4,3,4} {3^{[4]}} ht_{0,3}{4,3,4} h{4,4}h{∞} ht_{0,2}{4,4}h{∞} h{∞}h{∞}h{∞} s{∞}s{∞}s{∞} 
Coxeter diagrams  
Cell types  {3,3}, {3,4} 
Face types  triangle {3} 
Edge figure  [{3,3}.{3,4}]^{2} (rectangle) 
Vertex figure  (cuboctahedron) 
Symmetry group  Fm3m (225) 
Coxeter group  , [4,3^{1,1}] 
Dual  Dodecahedrille rhombic dodecahedral honeycomb Cell: 
Properties  vertextransitive, edgetransitive, facetransitive, quasiregular honeycomb 
Other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation. John Horton Conway calls this honeycomb a tetroctahedrille, and its dual a dodecahedrille.
It is vertextransitive with 8 tetrahedra and 6 octahedra around each vertex. It is edgetransitive with 2 tetrahedra and 2 octahedra alternating on each edge.
A geometric honeycomb is a spacefilling of polyhedral or higherdimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in nonEuclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
It is part of an infinite family of uniform honeycombs called alternated hypercubic honeycombs, formed as an alternation of a hypercubic honeycomb and being composed of demihypercube and crosspolytope facets. It is also part of another infinite family of uniform honeycombs called simplectic honeycombs.
In this case of 3space, the cubic honeycomb is alternated, reducing the cubic cells to tetrahedra, and the deleted vertices create octahedral voids. As such it can be represented by an extended Schläfli symbol h{4,3,4} as containing half the vertices of the {4,3,4} cubic honeycomb.
There is a similar honeycomb called gyrated tetrahedraloctahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra.
The tetrahedraloctahedral honeycomb can have its symmetry doubled by placing tetrahedra on the octahedral cells, creating a nonuniform honeycomb consisting of tetrahedra and octahedra (as triangular antiprisms). Its vertex figure is an order3 truncated triakis tetrahedron. This honeycomb is the dual of the triakis truncated tetrahedral honeycomb, with triakis truncated tetrahedral cells.
Cartesian coordinates
For an alternated cubic honeycomb, with edges parallel to the axes and with an edge length of 1, the Cartesian coordinates of the vertices are: (For all integral values: i,j,k with i+j+k even)
 (i, j, k)
Symmetry
There are two reflective constructions and many alternated cubic honeycomb ones; examples:
Symmetry  , [4,3^{1,1}] = ½, [1^{+},4,3,4] 
, [3^{[4]}] = ½, [1^{+},4,3^{1,1}] 
[[(4,3,4,2^{+})]]  [(4,3,4,2^{+})] 

Space group  Fm3m (225)  F43m (216)  I43m (217)  P43m (215) 
Image  
Types of tetrahedra  1  2  3  4 
Coxeter diagram 
Alternated cubic honeycomb slices
The alternated cubic honeycomb can be sliced into sections, where new square faces are created from inside of the octahedron. Each slice will contain up and downward facing square pyramids and tetrahedra sitting on their edges. A second slice direction needs no new faces and includes alternating tetrahedral and octahedral. This slab honeycomb is a scaliform honeycomb rather than uniform because it has nonuniform cells.
Projection by folding
The alternated cubic honeycomb can be orthogonally projected into the planar square tiling by a geometric folding operation that maps one pairs of mirrors into each other. The projection of the alternated cubic honeycomb creates two offset copies of the square tiling vertex arrangement of the plane:
Coxeter group 


Coxeter diagram 

Image  
Name  alternated cubic honeycomb  square tiling 
A3/D3 lattice
Its vertex arrangement represents an A_{3} lattice or D_{3} lattice.[2][3] This lattice is known as the facecentered cubic lattice in crystallography and is also referred to as the cubic close packed lattice as its vertices are the centers of a closepacking with equal spheres that achieves the highest possible average density. The tetrahedraloctahedral honeycomb is the 3dimensional case of a simplectic honeycomb. Its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tetoct honeycomb.
The D^{+}
_{3} packing can be constructed by the union of two D_{3} (or A_{3}) lattices. The D^{+}
_{n} packing is only a lattice for even dimensions. The kissing number is 2^{2}=4, (2^{n1} for n<8, 240 for n=8, and 2n(n1) for n>8).[4]
∪
The A^{*}
_{3} or D^{*}
_{3} lattice (also called A^{4}
_{3} or D^{4}
_{3}) can be constructed by the union of all four A_{3} lattices, and is identical to the vertex arrangement of the disphenoid tetrahedral honeycomb, dual honeycomb of the uniform bitruncated cubic honeycomb:[5] It is also the body centered cubic, the union of two cubic honeycombs in dual positions.
∪ ∪ ∪ = dual of = ∪ .
The kissing number of the D^{*}
_{3} lattice is 8[6] and its Voronoi tessellation is a bitruncated cubic honeycomb,
Related honeycombs
C3 honeycombs
The [4,3,4],
C3 honeycombs  

Space group 
Fibrifold  Extended symmetry 
Extended diagram 
Order  Honeycombs 
Pm3m (221) 
4^{−}:2  [4,3,4]  ×1  
Fm3m (225) 
2^{−}:2  [1^{+},4,3,4] ↔ [4,3^{1,1}] 
↔ 
Half  
I43m (217) 
4^{o}:2  [[(4,3,4,2^{+})]]  Half × 2  
Fd3m (227) 
2^{+}:2  [[1^{+},4,3,4,1^{+}]] ↔ [[3^{[4]}]] 
↔ 
Quarter × 2  
Im3m (229) 
8^{o}:2  [[4,3,4]]  ×2 
B3 honeycombs
The [4,3^{1,1}],
B3 honeycombs  

Space group 
Fibrifold  Extended symmetry 
Extended diagram 
Order  Honeycombs 
Fm3m (225) 
2^{−}:2  [4,3^{1,1}] ↔ [4,3,4,1^{+}] 
↔ 
×1  
Fm3m (225) 
2^{−}:2  <[1^{+},4,3^{1,1}]> ↔ <[3^{[4]}]> 
↔ 
×2  
Pm3m (221) 
4^{−}:2  <[4,3^{1,1}]>  ×2 
A3 honeycombs
This honeycomb is one of five distinct uniform honeycombs[8] constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:
A3 honeycombs  

Space group 
Fibrifold  Square symmetry 
Extended symmetry 
Extended diagram 
Extended group 
Honeycomb diagrams 
F43m (216) 
1^{o}:2  a1 
[3^{[4]}]  (None)  
Fm3m (225) 
2^{−}:2  d2 
<[3^{[4]}]> ↔ [4,3^{1,1}] 
↔ 
×2_{1} ↔ 

Fd3m (227) 
2^{+}:2  g2 
[[3^{[4]}]] or [2^{+}[3^{[4]}]] 
↔ 
×2_{2}  
Pm3m (221) 
4^{−}:2  d4 
<2[3^{[4]}]> ↔ [4,3,4] 
↔ 
×4_{1} ↔ 

I3 (204) 
8^{−o}  r8 
[4[3^{[4]}]]^{+} ↔ [[4,3<sup>+</sup>,4]] 
↔ 
½×8 ↔ ½×2 

Im3m (229) 
8^{o}:2  [4[3^{[4]}]] ↔ [[4,3,4]] 
×8 ↔ ×2 
Quasiregular honeycombs
Quasiregular polychora and honeycombs: h{4,p,q}  

Space  Finite  Affine  Compact  Paracompact  
Schläfli symbol 
h{4,3,3}  h{4,3,4}  h{4,3,5}  h{4,3,6}  h{4,4,3}  h{4,4,4}  
Coxeter diagram 

Image  
Vertex figure r{p,3} 
Cantic cubic honeycomb
Cantic cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  h_{2}{4,3,4} 
Coxeter diagrams  
Cells  t{3,4} r{4,3} t{3,3} 
Vertex figure  
Coxeter groups  [4,3^{1,1}], [3^{[4]}], 
Symmetry group  Fm3m (225) 
Dual  half oblate octahedrille Cell: 
Properties  vertextransitive 
The cantic cubic honeycomb, cantic cubic cellulation or truncated half cubic honeycomb is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of truncated octahedra, cuboctahedra and truncated tetrahedra in a ratio of 1:1:2. Its vertex figure is a rectangular pyramid.
John Horton Conway calls this honeycomb a truncated tetraoctahedrille, and its dual half oblate octahedrille.
Symmetry
It has two different uniform constructions. The construction can be seen with alternately colored truncated tetrahedra.
Symmetry  [4,3^{1,1}], =<[3^{[4]}]> 
[3^{[4]}], 

Space group  Fm3m (225)  F43m (216) 
Coloring  
Coxeter  
Vertex figure 
Related honeycombs
It is related to the cantellated cubic honeycomb. Rhombicuboctahedra are reduced to truncated octahedra, and cubes are reduced to truncated tetrahedra.
cantellated cubic 
Cantic cubic 
rr{4,3}, r{4,3}, {4,3} 
t{3,4}, r{4,3}, t{3,3} 
Runcic cubic honeycomb
Runcic cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  h_{3}{4,3,4} 
Coxeter diagrams  
Face  rr{4,3} {4,3} {3,3} 
Vertex figure  Tapered triangular prism 
Coxeter group  , [4,3^{1,1}] 
Symmetry group  Fm3m (225) 
Dual  quarter cubille Cell: 
Properties  vertextransitive 
The runcic cubic honeycomb or runcicantic cubic cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of rhombicuboctahedra, cubes, and tetrahedra in a ratio of 1:1:2. Its vertex figure is a triangular prism, with a tetrahedron on one end, cube on the opposite end, and three rhombicuboctahedra around the trapezoidal sides.
John Horton Conway calls this honeycomb a 3RCOtrille, and its dual quarter cubille.
Quarter cubille
The dual of a runcic cubic honeycomb is called a quarter cubille, with Coxeter diagram
Cells can be seen as 1/4 of dissected cube, using 4 vertices and the center. Four cells exist around 6 edges, and 3 cells around 3 edges.
Related honeycombs
It is related to the runcinated cubic honeycomb, with quarter of the cubes alternated into tetrahedra, and half expanded into rhombicuboctahedra.
Runcinated cubic 
Runcic cubic 
{4,3}, {4,3}, {4,3}, {4,3} 
h{4,3}, rr{4,3}, {4,3} 
This honeycomb can be divided on truncated square tiling planes, using the octagons centers of the rhombicuboctahedra, creating square cupolae. This scaliform honeycomb is represented by Coxeter diagram
.
Runcicantic cubic honeycomb
Runcicantic cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  h_{2,3}{4,3,4} 
Coxeter diagrams  
Coxeter group  , [4,3^{1,1}] 
Vertex figure  
Symmetry group  Fm3m (225) 
Dual  half pyramidille Cell: 
Properties  vertextransitive 
The runcicantic cubic honeycomb or runcicantic cubic cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of truncated cuboctahedra, truncated cubes and truncated tetrahedra in a ratio of 1:1:2. It is related to the runcicantellated cubic honeycomb.
John Horton Conway calls this honeycomb a ftCOtrille, and its dual half pyramidille.
Half pyramidille
The dual to the runcitruncated cubic honeycomb is called a half pyramidille, with Coxeter diagram
Cells are irregular pyramids and can be seen as 1/12 of a cube, or 1/24 of a rhombic dodecahedron, each defined with three corner and the cube center.
Related skew apeirohedra
A related uniform skew apeirohedron exists with the same vertex arrangement, but triangles and square removed. It can be seen as truncated tetrahedra and truncated cubes augmented together.
Related honeycombs
Runcicantic cubic 
Runcicantellated cubic 
Gyrated tetrahedraloctahedral honeycomb
Gyrated tetrahedraloctahedral honeycomb  

Type  convex uniform honeycomb 
Coxeter diagram  
Schläfli symbol  h{4,3,4}:g h{6,3}h{∞} s{3,6}h{∞} s{3^{[3]}}h{∞} 
Cell types  {3,3}, {3,4} 
Vertex figure  Triangular orthobicupola G3.4.3.4 
Space group  P6_{3}/mmc (194) [3,6,2^{+},∞] 
Dual  trapezorhombic dodecahedral honeycomb 
Properties  vertextransitive 
The gyrated tetrahedraloctahedral honeycomb or gyrated alternated cubic honeycomb is a spacefilling tessellation (or honeycomb) in Euclidean 3space made up of octahedra and tetrahedra in a ratio of 1:2.
It is vertexuniform with 8 tetrahedra and 6 octahedra around each vertex.
It is not edgeuniform. All edges have 2 tetrahedra and 2 octahedra, but some are alternating, and some are paired.
It can be seen as reflective layers of this layer honeycomb:
Construction by gyration
This is a less symmetric version of another honeycomb, tetrahedraloctahedral honeycomb, in which each edge is surrounded by alternating tetrahedra and octahedra. Both can be considered as consisting of layers one cell thick, within which the two kinds of cell strictly alternate. Because the faces on the planes separating these layers form a regular pattern of triangles, adjacent layers can be placed so that each octahedron in one layer meets a tetrahedron in the next layer, or so that each cell meets a cell of its own kind (the layer boundary thus becomes a reflection plane). The latter form is called gyrated.
The vertex figure is called a triangular orthobicupola, compared to the tetrahedraloctahedral honeycomb whose vertex figure cuboctahedron in a lower symmetry is called a triangular gyrobicupola, so the gyro prefix is reversed in usage.
Honeycomb  Gyrated tetoct  Reflective tetoct 

Image  
Name  triangular orthobicupola  triangular gyrobicupola 
Vertex figure  
Symmetry  D_{3h}, order 12 
D_{3d}, order 12 (O_{h}, order 48) 
Construction by alternation
The geometry can also be constructed with an alternation operation applied to a hexagonal prismatic honeycomb. The hexagonal prism cells become octahedra and the voids create triangular bipyramids which can be divided into pairs of tetrahedra of this honeycomb. This honeycomb with bipyramids is called a ditetrahedraloctahedral honeycomb. There are 3 CoxeterDynkin diagrams, which can be seen as 1, 2, or 3 colors of octahedra:
Gyroelongated alternated cubic honeycomb
Gyroelongated alternated cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  h{4,3,4}:ge {3,6}h_{1}{∞} 
Coxeter diagram  
Cell types  {3,3}, {3,4}, (3.4.4) 
Face types  {3}, {4} 
Vertex figure  
Space group  P6_{3}/mmc (194) [3,6,2^{+},∞] 
Properties  vertexuniform 
The gyroelongated alternated cubic honeycomb or elongated triangular antiprismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2.
It is vertexuniform with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex.
It is one of 28 convex uniform honeycombs.
The elongated alternated cubic honeycomb has the same arrangement of cells at each vertex, but the overall arrangement differs. In the elongated form, each prism meets a tetrahedron at one of its triangular faces and an octahedron at the other; in the gyroelongated form, the prism meets the same kind of deltahedron at each end.
Elongated alternated cubic honeycomb
Elongated alternated cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  h{4,3,4}:e {3,6}g_{1}{∞} 
Cell types  {3,3}, {3,4}, (3.4.4) 
Vertex figure  triangular cupola joined to isosceles hexagonal pyramid 
Space group  [6,(3,2^{+},∞,2^{+})] ? 
Properties  vertextransitive 
The elongated alternated cubic honeycomb or elongated triangular gyroprismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2.
It is vertexuniform with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex. Each prism meets an octahedron at one end and a tetrahedron at the other.
It is one of 28 convex uniform honeycombs.
It has a gyrated form called the gyroelongated alternated cubic honeycomb with the same arrangement of cells at each vertex.
Notes
 For crossreferencing, they are given with list indices from Andreini (122), Williams(12,919), Johnson (1119, 2125, 3134, 4149, 5152, 6165), and Grünbaum(128).
 http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/D3.html
 http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/A3.html
 Conway (1998), p. 119
 http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/Ds3.html
 Conway (1998), p. 120
 Conway (1998), p. 466
 , OEIS sequence A000029 61 cases, skipping one with zero marks
References
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292298, 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 3space. Geombinatorics 4(1994), 49  56.
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X.
 Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0500340331.
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (1.9 Uniform spacefillings)
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 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.
 D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
 Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0387985859.
External links
Wikimedia Commons has media related to Tetrahedraloctahedral honeycomb. 
 Architectural design made with Tetrahedrons and regular Pyramids based square.(2003)
 Klitzing, Richard. "3D Euclidean Honeycombs x3o3o *b4o  octet  O21".
 Uniform Honeycombs in 3Space: 11Octet
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 