Pentagonal-decagonal duoprism

The pentagonal-decagonal duoprism or padedip, also known as the 5-10 duoprism, is a uniform duoprism that consists of 5 decagonal prisms and 10 pentagonal prisms, with two of each joining at each vertex.

The convex hull of two orthogonal pentagonal-decagonal duoprisms is either the pentagonal duoexpandoprism or the pentagonal duotruncatoprism.

Vertex coordinates
The vertex coordinates of a pentagonal-decagonal duoprism, centered at the origin and with unit edge length, are given by:
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,0,\,±\frac{1+\sqrt5}{2}\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,0,\,±\frac{1+\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,0,\,±\frac{1+\sqrt5}{2}\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right).$$

Representations
A pentagonal-decagonal duoprism has the following Coxeter diagrams:


 * x5o x10o (full symmetry)
 * x5x x5o (decagons as dipentagons, pentagon duoprism symmetry)
 * ofx xxx10ooo&#xt (decagonal axial)
 * ofx xxx5xxx&#xt (dipentagonal axial symmetry)