Triangular ditetragoltriate

The triangular ditetragoltriate or triddet is a convex isogonal polychoron and the first member of the ditetragoltriate family. It consists of 6 triangular prisms and 9 rectangular trapezoprisms. 2 triangular prisms and 4 rectangular trapezoprisms join at each vertex. However, it cannot be made uniform. It is the first in an infinite family of isogonal triangular prismatic swirlchora.

It can be obtained as the convex hull of 2 similarly oriented semi-uniform triangular duoprisms, one with a larger xy triangle and the other with a larger zw triangle. If the two triangles have edge lengths a and b, the lacing edges have length $$(1-b)\frac{\sqrt6}{3}$$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{2+\sqrt6}{2}$$ ≈ 1:2.22474. This value is also the ratio between the two triangles of the wto semi-uniform duoprisms.

Vertex coordinates
The vertices of a triangular ditetragoltriate, assuming that the trapezoids have three equal edges of length 1, centered at the origin, are given by:
 * $$\left(0,\,\frac{\sqrt3}{3},\,0,\,\frac{3\sqrt2+2\sqrt3}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac{2+\sqrt6}{4},\,-\frac{3\sqrt2+2\sqrt3}{12}\right),$$
 * $$\left(0,\,\frac{3\sqrt2+2\sqrt3}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(0,\,\frac{3\sqrt2+2\sqrt3}{6},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,0,\,\frac{3\sqrt2+2\sqrt3}{6}\right),$$
 * $$\left(±\frac12\,-\frac{\sqrt3}{6},\,±\frac{2+\sqrt6}{4},\,-\frac{3\sqrt2+2\sqrt3}{12}\right),$$
 * $$\left(±\frac{2+\sqrt6}{4},\,-\frac{3\sqrt2+2\sqrt3}{12},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac{2+\sqrt6}{4},\,-\frac{3\sqrt2+2\sqrt3}{12},\,±\frac12,\,-\frac{\sqrt3}{6}\right).$$