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 .
Cyclotruncated 5-simplex honeycomb on each hyperplane. Regular and uniform honeycombs in 5-space: Cyclotruncated 5-simplex honeycomb In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1. Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells. It can be constructed as six sets of parallel hyperplanes that divide