| Gyrated tetrahedral-octahedral honeycomb | |
|---|---|
| Type | convex uniform honeycomb |
| Schläfli symbol | h{4,3,4}:g |
| Cell types | {3,3}, {3,4} |
| Face types | {3} |
| Edge figures | rectangle trapezoid |
| Vertex figure | Triangular orthobicupola G3.4.3.4 |
| Cells/edge | [{3,3}.{3,4}]2 {3,3}2.{3,4}2 |
| Cells/vertex | {3,3}8+{3,4}6 |
| Symmetry group | P63/mmc |
| Dual | trapezo-rhombic dodecahedral honeycomb |
| Properties | vertex-transitive, face-transitive |
The gyrated tetrahedral-octahedral honeycomb or gyrated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of octahedra and tetrahedra in a ratio of 1:2.
It is vertex-uniform with 8 tetrahedra and 6 octahedra around each vertex.
It is not edge-uniform. All edges have 2 tetrahedra and 2 octahedra, but some are alternating, and some are paired.
This is a less symmetric version of another honeycomb, tetrahedral-octahedral 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 geometry can also be constructed with an alternation operation applied to a hexagonal prismatic honeycomb. The hexagonal prism cells become octahedrons and the voids create a trigonal bipyramid which can be divided into pairs of tetrahedrons of this honeycomb.