Cubic-octahedral honeycomb Cubic-octahedral honeycomb In the geometry of hyperbolic 3-space, the cube-octahedron honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells. Wide-angle perspective views: It contains a subgroup H2 tiling, the alternated order-4 hexagonal tiling, , with vertex figure (3.4). A lower symmetry form, index 6, of this honeycomb can be constructed with [(4,3,4,3)] symmetry, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram . This lower symmetry can be extended by restoring one mirror as .
Cubic honeycomb and its dual triangular pyramidille. Four cells exist around each vertex: The "cantitruncated cubic honeycomb" can be orthogonally projected into the euclidean plane with various symmetry arrangements. Cells can be shown in two different symmetries. The linear Coxeter diagram form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of truncated cuboctahedron cells alternating. The dual of the "cantitruncated cubic honeycomb" is called a triangular pyramidille, with Coxeter diagram, . This honeycomb cells represents the fundamental domains of formula_2 symmetry. A cell can be as 1/24 of a