Gyroelongated pentagonal pyramid

The gyroelongated pentagonal pyramid is one of the 92 Johnson solids (J11). It consists of 5+5+5 triangles and 1 pentagon. It can be constructed by attaching a pentagonal antiprism to the base of the pentagonal pyramid.

Alternatively, it can be constructed by diminishing one vertex from the regular icosahedron, which is why this polyhedron can also be called the diminished icosahedron. This means the icosahedron can also be thought of as a gyroelongated pentagonal bipyramid.

Vertex coordinates
A gyroelongated pentagonal pyramid of edge length 1 has the following vertices:
 * $$\left(0,\,0,\,\sqrt{\frac{5+\sqrt5}{8}}\right),$$
 * $$±\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\sqrt{\frac{5+\sqrt5}{40}}\right),$$
 * $$±\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\sqrt{\frac{5+\sqrt5}{40}}\right),$$
 * $$±\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\sqrt{\frac{5+\sqrt5}{40}}\right).$$

Alternative coordinates can be obtained by removing one vertex from the regular icosahedron:


 * $$\left(±\frac{1+\sqrt5}{4},\,+\frac12,\,0\right),$$
 * $$\left(0,\,±\frac{1+\sqrt5}{4},\,±\frac12\right),$$
 * $$\left(±\frac12,\,0,\,\frac{1+\sqrt5}{4}\right),$$
 * $$\left(\frac12,\,0,\,-\frac{1+\sqrt5}{4}\right).$$