Pentagonal duoexpandoprism

The pentagonal duoexpandoprism or pedep is a convex isogonal polychoron and the third member of the duoexpandoprism family. It consists of 20 pentagonal prisms of two kinds, 25 rectangular trapezoprisms, 50 wedges, and 25 tetragonal disphenoids. Each vertex joins 2 pentagonal prisms, 1 tetragonal disphenoid, 3 wedges, and 2 rectangular trapezoprisms. It can be obtained as the convex hull of two orthogonal pentagonal-decagonal duoprisms, or more generally pentagonal-dipentagonal duoprisms, and a subset of its variations can be constructed by expanding the cells of the pentagonal duoprism outward. However, it cannot be made uniform.

This is one of a total of five polychora that can be obtained as the convex hull of two orthogonal pentagonal-dipentagonal duoprisms. To produce variants of this polychoron, if the polychoron is written as ao5bc oa5cb&#zy, c must be in the range $$c < b+\frac{a(1+\sqrt5)}{4}$$. It generally has circumradius $$\sqrt{\frac{5a^2+5b^2+5ab+5c^2+(a^2+b^2+3ab+c^2)\sqrt5}{10}}$$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{5+3\sqrt5-\sqrt{50+18\sqrt5}}{2}$$ ≈ 1:1.10412.

Vertex coordinates
The vertices of a pentagonal duoexpandoprism, constructed as the convex hull of two orthogonal pentagonal-decagonal duoprisms of edge length 1, centered at the origin, are given by:
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{1+\sqrt5}{2},\,0\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{1+\sqrt5}{2},\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{1+\sqrt5}{2},\,0\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,0,\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right).$$