Medial birhombatocubic honeycomb

The medial birhombatocubic honeycomb or mabirch is an isogonal honeycomb that consists of cuboctahedra, square prisms, triangular antiprisms, rectangular pyramids, and tetragonal disphenoids. 1 cuboctahedron, 2 square prisms, 2 triangular antiprisms, 5 rectangular pyramids, and 2 tetragonal disphenoids join at each vertex.

It is one of a total of five distinct honeycombs (including two transitional cases) that can be obtained as the convex hull of two opposite small rhombated cubic honeycombs. In this case, the ratio between the edges of the small rhombated cubic honeycomb a4o3b4o is less than b:a = $$\frac{\sqrt2}{2}$$ (producing a subsymmetrical form of the rectified cubic honeycomb where the octahedra have triangular antiprism symmetry in the limiting case). The lacing edges generally have length $$\frac{\sqrt{3a^2-2ab\sqrt2+2b^2}}{2}$$.

This honeycomb cannot be optimized using the ratio method, because the solution (with intended minimal ratio 1:$$\frac{\sqrt{340+136\sqrt2}}{17}$$ ≈ 1:1.35720) would yield a small birhombatocubic honeycomb instead.