Tetrahedral-octahedral honeycomb
John Horton Conway calls this honeycomb a tetroctahedrille, and its dual a dodecahedrille.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps.
It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
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.
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) 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.
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.
Its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tet-oct honeycomb.
John Horton Conway calls this honeycomb a truncated tetraoctahedrille, and its dual half oblate octahedrille.
John Horton Conway calls this honeycomb a 3-RCO-trille, and its dual quarter cubille.
The dual of a runcic cubic honeycomb is called a quarter cubille, with Coxeter diagram , with faces in 2 of 4 hyperplanes of the
It is related to the runcinated cubic honeycomb, with quarter of the cubes alternated into tetrahedra, and half expanded into rhombicuboctahedra.
John Horton Conway calls this honeycomb a f-tCO-trille, and its dual half pyramidille.
The dual to the runcitruncated cubic honeycomb is called a half pyramidille, with Coxeter diagram .
A related uniform skew apeirohedron exists with the same vertex arrangement, but triangles and square removed.
The vertex figure is called a triangular orthobicupola, compared to the tetrahedral-octahedral honeycomb whose vertex figure cuboctahedron in a lower symmetry is called a triangular gyrobicupola, so the gyro- prefix is reversed in usage.
The geometry can also be constructed with an alternation operation applied to a hexagonal prismatic honeycomb.
The hexagonal prism cells become octahedra and the voids create triangular bipyramids which can be divided into pairs of tetrahedra of this honeycomb.
It has a gyrated form called the gyroelongated alternated cubic honeycomb with the same arrangement of cells at each vertex.
