Pentagonal-hexagonal duoantiprism

The pentagonal-hexagonal duoantiprism or phiddap, also known as the 5-6 duoantiprism, is a convex isogonal polychoron that consists of 10 hexagonal antiprisms, 12 pentagonal antiprisms, and 60 digonal disphenoids. 2 hexagonal antiprisms, 2 pentagonal antiprisms, and 4 digonal disphenoids join at each vertex. It can be obtained through the process of alternating the decagonal-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:$$\sqrt{\frac{400+165\sqrt3+\sqrt{19955+11500\sqrt3}}{482}}$$ ≈ 1:1.35539.

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


 * $$\left(±1,\,0,\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±1,\,0,\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(±1,\,0,\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(0,\,±1,\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(0,\,±1,\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(0,\,±1,\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,±\frac12,\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),$$