Square-hexagonal duoantiprism

The square-hexagonal duoantiprism or shiddap, also known as the 4-6 duoantiprism, is a convex isogonal polychoron that consists of 8 hexagonal antiprisms, 12 square antiprisms, and 48 digonal disphenoids. 2 hexagonal antiprisms, 2 square antiprisms, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the octagonal-dodecagonal duoprism. However, it cannot be made uniform, as it generally has 3 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{66+26\sqrt3+\sqrt{1922+1104\sqrt3}}{97}}$$ ≈ 1:1.33530.

Vertex coordinates
The vertices of a square-hexagonal duoantiprism based on squares and hexagons of edge length 1, centered at the origin, are given by:


 * $$\left(±\frac12,\,±\frac12,\,±1,\,0\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac{\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,±1\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,±\frac{\sqrt3}{2},\,±\frac12\right).$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,0,\,±1\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$