Tetrahedral-octahedral honeycomb

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)

Image:TetraOctaHoneycomb-VertexConfig.svg of the cells surrounding each vertex.]]

There are two reflective constructions and many alternated cubic honeycomb ones; examples:

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.

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:

Its vertex arrangement represents an A₃ lattice or D₃ lattice.{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html|title = The Lattice D3}}{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A3.html|title = The Lattice A3}} This lattice is known as the face-centered cubic lattice in crystallography and is also referred to as the cubic close packed lattice as its vertices are the centers of a close-packing with equal spheres that achieves the highest possible average density. The tetrahedral-octahedral honeycomb is the 3-dimensional case of a simplectic honeycomb. Its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tet-oct honeycomb.

The D{{sup sub|+|3}} packing can be constructed by the union of two D₃ (or A₃) lattices. The D{{sup sub|+|n}} packing is only a lattice for even dimensions. The kissing number is 2²=4, (2ⁿ⁻¹ for n<8, 240 for n=8, and 2n(n−1) for n>8).Conway (1998), p. 119

:{{CDD|node_1|split1|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|split2|node_1}}

The A{{sup sub|*|3}} or D{{sup sub|*|3}} lattice (also called A{{sup sub|4|3}} or D{{sup sub|4|3}}) can be constructed by the union of all four A₃ lattices, and is identical to the vertex arrangement of the disphenoid tetrahedral honeycomb, dual honeycomb of the uniform bitruncated cubic honeycomb:{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds3.html|title=The Lattice D3}} It is also the body centered cubic, the union of two cubic honeycombs in dual positions.

:{{CDD|node_1|split1|nodes|split2|node}} ∪ {{CDD|node|split1|nodes_10luru|split2|node}} ∪ {{CDD|node|split1|nodes_01lr|split2|node}} ∪ {{CDD|node|split1|nodes|split2|node_1}} = dual of {{CDD|node_1|split1|nodes_11|split2|node_1}} = {{CDD|node_1|4|node|3|node|4|node}} ∪ {{CDD|node|4|node|3|node|4|node_1}}.

The kissing number of the D{{sup sub|*|3}} lattice is 8Conway (1998), p. 120 and its Voronoi tessellation is a bitruncated cubic honeycomb, {{CDD|branch_11|4a4b|nodes}}, containing all truncated octahedral Voronoi cells, {{CDD|node|4|node_1|3|node_1}}.Conway (1998), p. 466

The [4,3,4], {{CDD|node|4|node|3|node|4|node}}, Coxeter group generates 15 permutations of uniform honeycombs, 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.

{{C3 honeycombs}}

The [4,3^1,1], {{CDD|node|4|node|split1|nodes}}, Coxeter group generates 9 permutations of uniform honeycombs, 4 with distinct geometry including the alternated cubic honeycomb.

{{B3 honeycombs}}

This honeycomb is one of five distinct uniform honeycombs[http://mathworld.wolfram.com/Necklace.html], {{OEIS el|A000029}} 6-1 cases, skipping one with zero marks constructed by the $\{\backslash tilde\{A\}\}\_3$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

{{A3 honeycombs}}

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{{Quasiregular polychora and honeycombs}}