Bitruncatotetracontoctachoron

The bitruncatotetracontoctachoron or bitec is a convex isogonal polychoron that consists of 48 cubes, 144 square antiprisms, 288 tetragonal disphenoids, and 576 digonal disphenoids. 1 cube, 3 square antiprisms, 3 tetragonal disphenoids, and 6 digonal disphenoids join at each vertex. It can be obtained as the convex hull of two oppositely oriented truncated icositetrachora.

The bitruncatotetracontoctachoron contains the vertices of a square-octagonal prismantiprismoid and the square double prismantiprismoid.

This polychoron generally has one degree of variation. If the edge length of the truncated icositetrachora are a (those surrounded by truncated octahedra) and b (of cubes), its lacing edges have length $$\sqrt{2a^2+6b^2+6ab-(a^2+4b^2+4ab)\sqrt2}$$ and it has circumradius $$\sqrt{a^2+3b^2+3ab}$$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{14-9\sqrt2}$$ ≈ 1:1.12786.

Vertex coordinates
The vertices of a bitruncatotetracontoctachoron, assuming that the square antiprisms are uniform of edge length 1, centered at the origin, are given by all permutations of:
 * $$\left(0,\,±\frac{\sqrt2}{2},\,±\frac{1+\sqrt{1+\sqrt2}}{2},\,±\frac{1+\sqrt2+\sqrt{1+\sqrt2}}{2}\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac{1+\sqrt2+\sqrt{2+\sqrt2}}{2}\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt{2+2\sqrt2}-2}{4},\,±\frac{2+\sqrt2+\sqrt{2+2\sqrt2}}{4},\,±\frac{2+\sqrt2+\sqrt{2+2\sqrt2}}{4},\,±\frac{2+\sqrt2+\sqrt{2+2\sqrt2}}{4}\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt{2+2\sqrt2}}{4},\,±\frac{\sqrt2+\sqrt{2+2\sqrt2}}{4},\,±\frac{\sqrt2+\sqrt{2+2\sqrt2}}{4},\,±\frac{4+\sqrt2+\sqrt{2+2\sqrt2}}{4}\right).$$

An alternate set of coordinates, based on two uniform truncated icositetrachora of edge length 1, centered at the origin, are given by all permutations of:
 * $$\left(±\frac{3\sqrt2}{2},\,±\sqrt2,\,±\frac{\sqrt2}{2},\,0\right),$$
 * $$\left(±\frac52,\,±\frac12,\,±\frac12,\,±\frac12\right),$$
 * $$\left(±\frac32,\,±\frac32,\,±\frac32,\,±\frac12\right),$$
 * $$\left(±2,\,±1,\,±1,\,±1\right).$$