Pentagonal duoantiprism

The pentagonal duoantiprism or pedap, also known as the pentagonal-pentagonal duoantiprism, the 5 duoantiprism or the 5-5 duoantiprism, is a convex isogonal polychoron that consists of 20 pentagonal antiprisms and 50 tetragonal disphenoids. 4 pentagonal antiprisms and 4 tetragonal disphenoids join at each vertex. It can be obtained through the process of alternating the decagonal duoprism. However, it cannot be made uniform, and has two edge lengths. It is the second in an infinite family of isogonal pentagonal dihedral swirlchora.

The ratio between the longest and shortest edges is 1:$$\frac{\sqrt{5+\sqrt5}}{2}$$ ≈ 1:1.34500.

Vertex coordinates
The vertices of a pentagonal duoantiprism based on pentagons of edge length 1, centered at the origin, are given by:
 * $$±\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$±\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$±\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$±\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$±\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$±\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$±\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$±\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$±\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right).$$