Tetrahedral-octahedral honeycomb

Tetrahedral-octahedral honeycomb
Type	Uniform honeycomb
Family	Alternated hypercube honeycomb
Schläfli symbol	h0{4,3,4}
Coxeter-Dynkin diagrams
Cell types	{3,3}, {3,4}
Face types	triangle {3}
Edge figure	[{3,3}.{3,4}]2 (rectangle)
Vertex figure	8 {3,3} 6 {3,4} (cuboctahedron)
Cells/edge	[{3,3}.{3,4}]2
Faces/edge	4 {3}
Cells/vertex	{3,3}8+{3,4}6
Faces/vertex	24 {3}
Edges/vertex	12
Coxeter groups	C~3 [4,31,1] A~3
Dual	rhombic dodecahedral honeycomb
Properties	vertex-transitive, edge-transitive, face-transitive

A A^~₃ symmetry construction has alternately colored tetrahedra around each edge

The tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is comprised of alternating octahedra and tetrahedra in a ratio of 1:2.

It is vertex-transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge.

It is part of an infinite family of uniform tessellations called demihypercubic tessellations, formed as an alternation of a hypercubic honeycomb and being composed of demihypercube and cross-polytope facets.

In this case of 3-space, 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's a similar honeycomb called gyrated tetrahedral-octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra.

[edit] Images

Wireframe (perspective)

This diagram shows an exploded view of the cells surrounding each vertex.

[edit] See also

• cubic honeycomb

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