Pentagonal-square prismantiprismoid

The pentagonal-square prismantiprismoid or pispap, also known as the edge-snub pentagonal-square duoprism or 5-4 prismantiprismoid, is a convex isogonal polychoron that consists of 4 pentagonal antiprisms, 4 pentagonal prisms, 10 rectangular trapezoprisms, and 20 wedges. 1 pentagonal antiprism, 1 pentagonal prism, 2 rectangular trapezoprisms, and 3 wedges join at each vertex. It can be obtained through the process of alternating one class of edges of the octagonal-decagonal duoprism so that the octagons become rectangles. 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:$$\frac{10+\sqrt5+\sqrt{295+39\sqrt5}}{19}$$ ≈ 1:1.67296.

Vertex coordinates
The vertices of a pentagonal-square prismantiprismoid based on an octagonal-decagonal duoprism of edge length 1, centered at the origin, rae given by:


 * $$\left(0,\,\frac{1+\sqrt5}{2},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(±\frac{\sqrt{5+2\sqrt5}}{2},\,\frac12,\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(±\sqrt{\frac{5+\sqrt5}{8}},\,-\frac{3+\sqrt5}{4},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(0,\,-\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$$
 * $$\left(±\frac{\sqrt{5+2\sqrt5}}{2},\,-\frac12,\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$$
 * $$\left(±\sqrt{\frac{5+\sqrt5}{8}},\,\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$$