Tetrahedral-octahedral honeycomb


Tetrahedral-octahedral honeycomb

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.

Images

ee also

*cubic honeycomb


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