Triangular-truncated icosahedral duoprism

The triangular-truncated icosahedral duoprism or trati is a convex uniform duoprism that consists of 3 truncated icosahedral prisms, 20 triangular-hexagonal duoprisms, and 12 triangular-pentagonal duoprisms. Each vertex joins 2 truncated icosahedral prisms, 1 triangular-pentagonal duoprism, and 2 triangular-hexagonal duoprisms. It is a duoprism based on a triangle and a truncated icosahedron, which makes it a convex segmentoteron.

Vertex coordinates
The vertices of a triangular-truncated icosahedral duoprism of edge length 1 are given by all even permutations of the last three coordinates of:
 * $$\left(0,\,\frac{\sqrt3}3,\,0,\,±\frac12,\,±3\frac{1+\sqrt5}4\right),$$
 * $$\left(0,\,\frac{\sqrt3}3,\,±\frac12,\,±\frac{5+\sqrt5}4,\,±\frac{1+\sqrt5}2\right),$$
 * $$\left(0,\,\frac{\sqrt3}3,\,±\frac{1+\sqrt5}4,\,±1,\,±\frac{2+\sqrt5}2\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}6,\,0,\,±\frac12,\,±3\frac{1+\sqrt5}4\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}6,\,±\frac12,\,±\frac{5+\sqrt5}4,\,±\frac{1+\sqrt5}2\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}6,\,±\frac{1+\sqrt5}4,\,±1,\,±\frac{2+\sqrt5}2\right).$$