Pentagonal orthobirotunda

The pentagonal orthobirotunda is one of the 92 Johnson solids (J34). It consists of 10+10 triangles and 2+10 pentagons. It can be constructed by attaching two pentagonal rotundas at their decagonal bases, such that the two pentagonal bases are in the same orientation.

If the rotundas are joined such that the bases are rotated 36°, the result is the pentagonal gyrobirotunda, better known as the uniform icosidodecahedron. Conversely, the pentagonal orthobirotunda can be seen as a gyrate icosidodecahedron, since it is an icosidodecahedron with one half rotated.

Vertex coordinates
A pentagonal orthobirotunda of edge length 1 has vertices given by the following coordinates:


 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\sqrt{\frac{5+2\sqrt5}{5}}\right),$$
 * $$\left(0,\,-\sqrt{\frac{5+2\sqrt5}{5}},\,±\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{25+11\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,0\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0\right).$$

Related polyhedra
A decagonal prism can be inserted between the two halves of the pentagonal orthobirotunda to produce the elongated pentagonal orthobirotunda.