Great birhombatocubic honeycomb

{{Infobox polytope }coxeter = ao4ob3bo4oa&#zc (b:a > 3) The great birhombatocubic honeycomb or gabirch is an isogonal honeycomb that consists of small rhombicuboctahedra, triangular antiprisms, rectangular pyramids, and tetragonal disphenoids.
 * image =
 * dim = 4
 * type = Isogonal
 * space=Euclidean
 * obsa = Gabirch
 * cells = 6N tetragonal disphenoids, 12N rectangular pyramids, 4N triangular antiprisms, N small rhombicuboctahedra
 * faces = 8N triangles, 24N+24N isosceles triangles, 3N squares, 12N rectangles
 * edges = 12N+24N+24N
 * vertices = 12N
 * verf = Laterowedged wedge
 * symmetry = R{{sub|4}}×2
 * army=Gabirch
 * reg=Gabirch
 * dual=Great biorthocubic honeycomb
 * conv = Yes
 * orientable = Yes
 * nat = Tame}}

It is one of a total of five distinct polychora (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 greater than b:a = 3 (which produces the rectified bitruncated cubic honeycomb 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.