Hexagonal-pyritohedral icosahedral duoantiprism

The hexagonal-pyritohedral icosahedral duoantiprism, or hapidap, is a convex isogonal polyteron that consists of 12 pyritohedral icosahedral antiprisms, 8 triangular-hexagonal duoantiprisms, 6 digonal-hexagonal duoantiprisms, and 144 digonal disphenoidal pyramids. 2 pyritohedral icosahedral antiprisms, 1 digonal-hexagonal duoantiprism, 2 triangular-hexagonal duoantiprisms, and 5 digonal disphenoidal pyramids join at each verte. It can be obtained through the process of alternating the dodecagonal-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:$$\sqrt{\frac{21+9\sqrt3}{22}}$$ ≈ 1:1.28962.

Vertex coordinates
The vertices of a hexagonal-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.
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,0,\,±1\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac{\sqrt3}{22},\,±\frac12\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±1,\,0\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac12,\,±\frac{\sqrt3}{2}\right),$$