Pentagonal duotegum

The pentagonal duotegum or pedit, also known as the pentagonal-pentagonal duotegum, the 5 duotegum, or the 5-5 duotegum, is a noble duotegum that consists of 25 tetragonal disphenoids and 10 vertices, with 10 cells joining at each vertex. It is also the 10-4 step prism and the square funk tegum. It is the first in an infinite family of isogonal pentagonal hosohedral swirlchora and also the first in an infinite family of isochoric pentagonal dihedral swirlchora.

The ratio between the longest and shortest edges is 1:$$\frac{\sqrt{25+5\sqrt5}}{5}$$ ≈ 1:1.20300.

Vertex coordinates
The vertices of a pentagonal duotegum based on two pentagons of edge length 1, centered at the origin, are given by:


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