Triangular trioantiprism

The triangular trioantiprism, or trittap, is a convex isogonal polypeton that consists of 18 triangular duoantiprismatic antiprisms and 108 digonal trisphenoids. 6 of each facet type join at each vertex. It can be obtained through the process of alternating the hexagonal trioprism. However, it cannot be made uniform.

The ratio between the longest and shortest edges is 1:$$\frac{\sqrt6}{2}$$ ≈ 1:1.22474.

Vertex coordinates
The vertices of a triangular trioantiprism, based on base triangles of unit edge length, centered at the origin, are given by:


 * $$\left(0,\,\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,0,\,-\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,0,\,-\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac12,\,\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,0,\,-\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,0,\,-\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,0,\,-\frac{\sqrt3}{3},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,0,\,-\frac{\sqrt3}{3},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,0,\,\frac{\sqrt3}{3},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,\frac{\sqrt3}{6},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,0,\,\frac{\sqrt3}{3},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,-\frac{\sqrt3}{3},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,-\frac{\sqrt3}{3},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,\frac{\sqrt3}{6}\right).$$