TETRAHEDRAL-OCTAHEDRAL HONEYCOMB

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|bgcolor=#e7dcc3|Type||Uniform honeycomb
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|bgcolor=#e7dcc3|Family||Alternated hypercube honeycomb
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|bgcolor=#e7dcc3|Schläfli symbol||h₀{4,3,4}
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|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||


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|bgcolor=#e7dcc3|Cell types||{3,3}, {3,4}
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|bgcolor=#e7dcc3|Face types||triangle {3}
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|bgcolor=#e7dcc3|Edge figure||[{3,3}.{3,4}]²
(rectangle)
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|bgcolor=#e7dcc3|Vertex figure||8 {3,3}
6 {3,4}
(cuboctahedron)

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|bgcolor=#e7dcc3|Cells/edge||[{3,3}.{3,4}]²
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|bgcolor=#e7dcc3|Faces/edge||4 {3}
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|bgcolor=#e7dcc3|Cells/vertex||{3,3}⁸+{3,4}⁶
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|bgcolor=#e7dcc3|Faces/vertex||24 {3}
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|bgcolor=#e7dcc3|Edges/vertex||12
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|bgcolor=#e7dcc3|Symmetry group||Fm3m
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|bgcolor=#e7dcc3|Dual||rhombic dodecahedral honeycomb
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|bgcolor=#e7dcc3|Properties||vertex-transitive, edge-transitive, face-transitive
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The 'tetrahedral-octahedral honeycomb' is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of alternating tetrahedra and octahedra.
It is vertex-transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alterating 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 90 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra.

Contents

Images

See also

Images

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Wireframe (perspective)
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This diagram shows an exploded view of the cells surrounding each vertex.
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See also

★ cubic honeycomb