Pentagonal-pyritohedral icosahedral duoantiprism

The pentagonal-pyritohedral icosahedral duoantiprism, or pepidap, is a convex isogonal polyteron that consists of 10 pyritohedral icosahedral antiprisms, 8 triangular-pentagonal duoantiprisms, 6 digonal-pentagonal duoantiprisms, and 120 digonal disphenoidal pyramids. 2 pyritohedral icosahedral antiprisms, 1 digonal-pentagonal duoantiprisms, 2 triangular-pentagonal duoantiprisms, and 5 digonal disphenoidal pyramids join at each vertex. It can be obtained through the process of alternating the decagonal-truncated octahedral duoprism. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{75+9\sqrt5}{58}}$$ ≈ 1:1.28066.

Vertex coordinates
The vertices of a pentagonal-pyritohedral icosahedral duoantiprism, assuming that the edge length differences are minimized, centered at the origin, are given by: with all even permutations of the first three coordinates, and with all odd permutations of the first three coordinates.
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,0,\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}}\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,0,\,-\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5-\sqrt5}{40}}\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}}\right),$$