Triangular duoexpandoprism

The triangular duoexpandoprism or triddep is a convex isogonal polychoron and the first member of the duoexpandoprism family. It consists of 12 triangular prisms of two kinds, 9 rectangular trapezoprisms, 18 wedges, and 9 tetragonal disphenoids. Each vertex joins 2 triangular prisms, 1 tetragonal disphenoid, 3 wedges, and 2 rectangular trapezoprisms. It can be obtained as the convex hull of two orthogonal triangular-hexagonal duoprisms, or more generally triangular-ditrigonal duoprisms, and a subset of its variations can be constructed by expanding the cells of the triangular duoprism outward. However, it cannot be made uniform.

This is one of a total of five polychora that can be obtained as the convex hull of two orthogonal triangular-ditrigonal duoprisms. To produce variants of this polychoron, if the polychoron is written as ao3bc oa3cb&#zy, c must be in the range $$c < b+\frac{a}{2}$$. It generally has circumradius $$\sqrt{\frac{a^2+b^2+ab+c^2}{3}}$$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{3+\sqrt{15}}{6}$$ ≈ 1:1.14550.

Vertex coordinates
The vertices of a triangular duoexpandoprism, constructed as the convex hull of two orthogonal triangular-hexagonal duoprisms of edge length 1, centered at the origin, are given by:
 * $$\left(0,\,\frac{\sqrt3}{3},\,±1,\,0\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,±\frac{\sqrt3}{2}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±1,\,0\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,±\frac{\sqrt3}{2}\right),$$
 * $$\left(±1,\,0,\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±1,\,0,\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12,\,-\frac{\sqrt3}{6}\right).$$