Square duotegmatic alterprism

The square duotegmatic alterprism is a convex isogonal polyteron that consists of 2 square duotegums and 32 isosceles triangular-triangular duotegums. 1 square duotegum and 12 isosceles triangular-triangular duotegums join at each vertex. It can be formed as an alterprism of a square duotegum or as the hull of two dually oriented square dispehnoids.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{2+\sqrt2}{2}}$$ ≈ 1:1.30656.

Vertex coordinates
Vertex coordinates for a square duotegmatic alterprism, assuming that the edge length differences are minimized, centered at the origin, are given by:
 * $$\left(0,\,0,\,0,\,±\frac{\sqrt2}{2},\,\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(0,\,0,\,±\frac{\sqrt2}{2},\,0,\,\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,0,\,0,\,\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,0,\,\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(0,\,0,\,±\frac12,\,±\frac12,\,-\frac{\sqrt[4]{8}}{4}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,0,\,0,\,-\frac{\sqrt[4]{8}}{4}\right).$$