Bicubic Honeycomb Symmetry is a Euclidean space group in the cubic family, usually numbered 229. It is the symmetry group of the bitruncated cubic honeycomb (composed of truncated octahedra) and of a body-centered cubic lattice. It is a doubling of the symmetry of an ordinary cubic lattice, but unlike it, it is not generated by its reflections alone. Instead, the reflections generate the corresponding cubic honeycomb sub-symmmetry group (chonnic), with the remaining symmetries of batchic represented by the 2-fold rotational symmetry of chonnic’s tetrahedral domain.
Batchic symmetry is also an example of a symmetry group with multiple distinct “omnitrucates” up to structure; that is, convex isogonal polytopes whose vertex stabilizers are trivial (for honeycombs, convexity is determined by vertex figures, circumspheres, and Delauney triangulatoins). Below, the 4 classes omnitruncates of batchic symmetry are represented by a tetrahedral domain of the corresponding chonnic subgroup, which then contains two vertices and the segments of edges in the corresponding honeycomb. The differences in structure depends on where the vertices fall on or between certain planes thru the 2-fold axis of the fundamental domain.
