Hexagonal-square prismantiprismoid

The hexagonal-square prismantiprismoid or hispap, also known as the edge-snub hexagonal-square duoprism or 6-4 prismantiprismoid, is a convex isogonal polychoron that consists of 4 hexagonal antiprisms, 4 hexagonal prisms, 12 rectangular trapezoprisms, and 24 wedges. 1 hexagonal antiprism 1 hexagonal 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-dodecagonal 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:$$\frac{5+2\sqrt3+\sqrt{102+46\sqrt3}}{13}$$ ≈ 1:1.68790.

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


 * $$\left(±\frac{\sqrt2+\sqrt6}{2},\,0,\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{\sqrt2+\sqrt6}{4},\,±\frac12,\,±\frac{1+\sqrt2}{6}\right),$$
 * $$\left(0,\,±\frac{\sqrt2+\sqrt6}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$$
 * $$\left(±\frac{\sqrt2+\sqrt6}{4},\,±\frac{3\sqrt2+\sqrt6}{4},\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$$