Square-hexagonal prismantiprismoid

The square-hexagonal prismantiprismoid or shipap, also known as the edge-snub square-hexagonal duoprism or 4-6 prismantiprismoid, is a convex isogonal polychoron that consists of 8 ditrigonal trapezoprisms, 6 square antiprisms, 6 square prisms, and 24 wedges. 1 square antiprism, 1 square 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 octagonal-dodecagonal duoprism so 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:$$\frac{8+3\sqrt2+\sqrt{450+186\sqrt2}}{23}$$ ≈ 1:1.69328.

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


 * $$\left(±\frac{\sqrt{2+\sqrt2}}{2},\,±\frac{\sqrt{2+\sqrt2}}{2},\,±\frac12,\,\frac{2+\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt{2+\sqrt2}}{2},\,±\frac{\sqrt{2+\sqrt2}}{2},\,±\frac{2+\sqrt3}{2},\,-\frac12\right),$$
 * $$\left(±\frac{\sqrt{2+\sqrt2}}{2},\,±\frac{\sqrt{2+\sqrt2}}{2},\,±\frac{1+\sqrt3}{2},\,-\frac{1+\sqrt3}{2}\right),$$
 * $$\left(±\sqrt{\frac{2+\sqrt2}{2}},\,0,\,±\frac12,\,-\frac{2+\sqrt3}{2}\right),$$
 * $$\left(±\sqrt{\frac{2+\sqrt2}{2}},\,0,\,±\frac{2+\sqrt3}{2},\,\frac12\right),$$
 * $$\left(±\sqrt{\frac{2+\sqrt2}{2}},\,0,\,±\frac{1+\sqrt3}{2},\,\frac{1+\sqrt3}{2}\right),$$
 * $$\left(0,\,±\sqrt{\frac{2+\sqrt2}{2}},\,±\frac12,\,-\frac{2+\sqrt3}{2}\right),$$
 * $$\left(0,\,±\sqrt{\frac{2+\sqrt2}{2}},\,±\frac{2+\sqrt3}{2},\,\frac12\right),$$
 * $$\left(0,\,±\sqrt{\frac{2+\sqrt2}{2}},\,±\frac{1+\sqrt3}{2},\,\frac{1+\sqrt3}{2}\right).$$