Square-pentagonal duoantiprism

The square-pentagonal duoantiprism or squipdap, also known as the 4-5 duoantiprism, is a convex isogonal polychoron that consists of 8 pentagonal antiprisms, 10 square antiprisms, and 40 digonal disphenoids. 2 pentagonal antiprisms, 2 square antiprisms, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the octagonal-decagonal 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:$$\sqrt{\frac{85+30\sqrt2+\sqrt{655+450\sqrt2}}{93}}$$ ≈ 1:1.32536.

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


 * $$\left(±\frac12,\,±\frac12,\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),$$