Order-4 dodecahedral honeycomb and order-4 hexagonal tiling honeycomb all which have octahedral vertex figures: This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells: The rectified order-4 dodecahedral honeycomb, , has alternating octahedron and icosidodecahedron cells, with a cube vertex figure. There are four rectified compact regular honeycombs: The truncated order-4 dodecahedral honeycomb, , has octahedron and truncated dodecahedron cells, with a cube vertex figure. It can be seen as analogous to the 2D hyperbolic truncated order-4 pentagonal tiling, t{5,4} with truncated pentagon and square faces: The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb, , has
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