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.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform variants of the great rhombated icositetrachoron, where if the great rhombated icositetrachora are of the form a3b4c3o, then $$c = b+(2+\sqrt2)a$$ It is one of five polychora (including two transitional cases) formed from two great rhombated icositetrachora, and is the transitional point between the medial bicantitruncatotetracontoctachoron and great bicantitruncatotetracontoctachoron.

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.

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


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