Pentagonal-hexagonal prismantiprismoid

The pentagonal-hexagonal prismantiprismoid or phipap, also known as the edge-snub pentagonal-hexagonal duoprism or 5-6 prismantiprismoid, is a convex isogonal polychoron that consists of 10 ditrigonal trapezoprisms, 6 pentagonal antiprisms, 6 pentagonal prisms, and 30 wedges. 1 pentagonal antiprism, 1 pentagonal prism, 2 ditrigonal trapezoprisms, and 3 wedges join at each vertex. It can be obtained through the process of alternating one class of edges of the decagonal-dodecagonal duoprism so that the dodecagons become ditrigons. However, it cannot be made uniform, as it generally has 4 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{25+3\sqrt5+\sqrt{3570+498\sqrt5}}{58}$$ ≈ 1:1.72663.

Vertex coordinates
The vertices of a pentagonal-hexagonal prismantiprismoid based on a decagonal-dodecagonal duoprism of edge length 1, centered at the origin, are given by:


 * $$±\left(0,\,\frac{1+\sqrt5}{2},\,±\frac12,\,\frac{2+\sqrt3}{2}\right),$$
 * $$±\left(0,\,\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt3}{2},\,-\frac12\right),$$
 * $$±\left(0,\,\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right),$$
 * $$±\left(±\frac{\sqrt{5+2\sqrt5}}{2},\,\frac12,\,±\frac12,\,\frac{2+\sqrt3}{2}\right),$$
 * $$±\left(±\frac{\sqrt{5+2\sqrt5}}{2},\,\frac12,\,±\frac{2+\sqrt3}{2},\,-\frac12\right),$$
 * $$±\left(±\frac{\sqrt{5+2\sqrt5}}{2},\,\frac12,\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right),$$
 * $$±\left(±\sqrt{\frac{5+\sqrt5}{8}},\,\frac{3+\sqrt5}{4},\,±\frac12,\,-\frac{2+\sqrt3}{2}\right),$$
 * $$±\left(±\sqrt{\frac{5+\sqrt5}{8}},\,\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt3}{2},\,\frac12\right),$$
 * $$±\left(±\sqrt{\frac{5+\sqrt5}{8}},\,\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt3}{2},\,\frac{1+\sqrt3}{2}\right).$$