Bitruncatodecachoron

The bitruncatodecachoron or bited is a convex isogonal polychoron that consists of 10 tetrahedra, 20 triangular antiprisms, 30 tetragonal disphenoids, and 60 digonal disphenoids. 1 tetrahedron, 3 triangular antiprisms, 3 tetragonal dispehnoids, and 6 digonal disphenoids join at each vertex. It can be obtained as the convex hull of two oppositely oriented truncated pentachora.

This polychoron generally has one degree of variation. If the edge length of the truncated pentachora are a (those surrounded by truncated tetrahedra) and b (of tetrahedra), its lacing edges have length $$\sqrt{\frac{3a^2+2b^2+2ab}{5}}$$ and it has circumradius $$\sqrt{\frac{2a^2+3b^2+3ab}{5}}$$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{\sqrt{35}}{5}$$ ≈ 1:1.18322.

Vertex coordinates
Coordinates for the vertices of a bitruncatodecachoron, based on two truncated pentachora of edge length 1, centered at the origin, are given by:


 * $$±\left(\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,\frac{\sqrt3}{3},\,±1\right),$$
 * $$±\left(\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{12},\,-\frac{2\sqrt3}{3},\,0\right),$$
 * $$±\left(\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,±1\right),$$
 * $$±\left(\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{4},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$±\left(\frac{3\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$$
 * $$±\left(\frac{3\sqrt{10}}{20},\,-\frac{5\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$$
 * $$±\left(-\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,±1\right),$$
 * $$±\left(-\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{2\sqrt3}{3},\,0\right),$$
 * $$±\left(-\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{2},\,0,\,0\right),$$
 * $$±\left(-\frac{7\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),$$
 * $$±\left(-\frac{7\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$$
 * $$±\left(-\frac{7\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right).$$