Rectified tetracontoctachoron

The rectified tetracontoctachoron or recont is a convex isogonal polychoron that consists of 48 rectified truncated cubes and 288 tetragonal disphenoids. 3 rectified truncated cubes and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the tetracontoctachoron.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform variants of the small rhombated icositetrachoron, where the edges of the cuboctahedra are $$2+\sqrt2 ≈ 3.41421$$ times the length of the other edges. It is one of five polychora (including two transitional cases) formed from two small rhombated icositetrachora, and is the transitional point between the small birhombatotetracontoctachoron and great birhombatotetracontoctachoron.

The ratio between the longest and shortest edges is 1:$$\sqrt{2+\sqrt2}$$ ≈ 1:1.84776.

Vertex coordinates
The vertices of a rectified tetracontoctachoron with triangles of edge length 1, centered at the origin, are given by:


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