Triangular-square prismantiprismoid

The triangular-square prismantiprismoid or tispap, also known as the edge-snub triangular-square duoprism or 3-4 prismantiprismoid, is a convex isogonal polychoron that consists of 4 triangular antiprisms, 4 triangular prisms, 6 rectangular trapezoprisms, and 12 wedges. 1 triangular antiprism, 1 triangular 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 hexagonal-octagonal 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:$$\frac{3+2\sqrt6}{5}$$ ≈ 1:1.57980.

Vertex coordinates
The vertices of a triangular-square prismantiprismoid, assuming that the triangular antiprisms are regular and are connected by uniform triangular prisms of edge length 1, centered at the origin, are given by:
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,±\frac{3+2\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,±\frac{3+2\sqrt3}{6}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac{3+2\sqrt3}{6},\,±\frac12\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac{3+2\sqrt3}{6},\,±\frac12\right).$$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac{2\sqrt6-3}{6},\,±\frac12\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{2\sqrt6-3}{6},\,±\frac12\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac12,\,±\frac{2\sqrt6-3}{6}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,±\frac{2\sqrt6-3}{6}\right).$$

Another variant obtained from the uniform hexagonal-octagonal duoprism has coordinates given by:


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