Triangular-hexagonal prismantiprismoid

The triangular-hexagonal prismantiprismoid or thipap, also known as the edge-snub triangular-hexagonal duoprism or 3-6 prismantiprismoid, is a convex isogonal polychoron that consists of 6 ditrigonal trapezoprisms, 6 triangular antiprisms, 6 triangular prisms, and 18 wedges. 1 triangular prism, 1 triangular antiprism, 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 hexagonal-dodecagonal duoprism so that 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:$$\frac{1+\sqrt5}{2}$$ ≈ 1:1.61803.

Vertex coordinates
The vertices of a triangular-hexagonal prismantiprismoid, assuming that the triangular antiprisms and triangular prisms are uniform of edge length 1, centered at the origin, are given by:
 * $$±\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,\frac{3\sqrt3+2\sqrt6}{6}\right),$$
 * $$±\left(0,\,\frac{\sqrt3}{3},\,±\frac{2+\sqrt2}{2},\,-\frac{\sqrt6}{6}\right),$$
 * $$±\left(0,\,\frac{\sqrt3}{3},\,±\frac{1+\sqrt2}{2},\,-\frac{3\sqrt3+\sqrt6}{6}\right),$$
 * $$±\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,\frac{3\sqrt3+2\sqrt6}{6}\right),$$
 * $$±\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{2+\sqrt2}{2},\,-\frac{\sqrt6}{6}\right),$$
 * $$±\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{1+\sqrt2}{2},\,-\frac{3\sqrt3+\sqrt6}{6}\right).$$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:
 * $$±\left(±\frac{\sqrt5-1}{4},\,\frac{3\sqrt3+\sqrt{15}}{12},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$±\left(±\frac{\sqrt5-1}{4},\,\frac{3\sqrt3+\sqrt{15}}{12},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$±\left(±\frac{1+\sqrt5}{4},\,-\frac{3\sqrt3-\sqrt{15}}{12},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$±\left(±\frac{1+\sqrt5}{4},\,-\frac{3\sqrt3-\sqrt{15}}{12},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$±\left(±\frac12,\,-\frac{\sqrt{15}}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$±\left(±\frac12,\,-\frac{\sqrt{15}}{6},\,±\frac12,\,-\frac{\sqrt3}{6}\right).$$

A variant based on a uniform hexagonal-dodecagonal duoprism has vertices given by:


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