Small birhombatocubic honeycomb

The small birhombatocubic honeycomb or sabirch is an isogonal honeycomb that consists of cuboctahedra, triangular antiprisms, square antipodiums, and tetragonal disphenoids. 1 cuboctahedron, 2 triangular antiprisms, 4 square antipodiums, 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 between b:a = $$\frac{\sqrt2}{2}$$ (producing a subsymmetrical form of the rectified cubic honeycomb where the octahedra have triangular antiprism symmetry) and b:a = 3 (producing the rectified bitruncated cubic hoeycomb). This includes the convex hull of two uniform small rhombated cubic honeycombs. The lacing edges generally have length $$\frac{\sqrt{3a^2-2ab\sqrt2+2b^2}}{2}$$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{\sqrt{340+136\sqrt2}}{17}$$ ≈ 1:1.35720.