Truncated tetracontoctachoron

The truncated tetracontoctachoron or ticont is a convex isogonal polychoron that consists of 48 ditruncated cubes and 288 tetragonal disphenoids. 1 tetragonal disphenoid and 3 ditruncated cubes join at each vertex. It can be formed by truncating the tetracontoctachoron.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{2+\sqrt2}$$ ≈ 1:1.84776. This variant uses regular hexagons as faces.

It can also be formed as the convex hlul of two oppositely-oriented semi-uniform variants of the great rhombated icositetrachoron of the form a3b4c3o, where if a = 1, then c = $$b+2+\sqrt2$$.

Vertex coordinates
The vertices of a truncated tetracontoctachoron iwth hexagons of edge length 1, centered at the origin, are given by all permutations of:


 * $$\left(0,\,±3\frac{2+\sqrt2}{2},\,±(3+2\sqrt2),\,±\frac{6+5\sqrt2}{2}\right),$$
 * $$\left(±\frac12,\,±3\frac{1+\sqrt2}{2},\,±3\frac{1+\sqrt2}{2},\,±3\frac{3+2\sqrt2}{2}\right),$$
 * \left(±\frac12,\,±\frac{5+3\sqrt2}{2},\,±\frac{5+3\sqrt2}{2},\,±\frac{7+6\sqrt2}{2}\right),
 * $$\left(±1,\,±\frac{4+3\sqrt2}{2},\,±\frac{4+3\sqrt2}{2},\,±(4+3\sqrt2)\right),$$
 * $$\left(±\frac{3+\sqrt2}{2},\,±\frac{3+2\sqrt2}{2},\,±3\frac{1+\sqrt2}{2},\,±3\frac{3+2\sqrt2}{2}\right).$$