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.

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:


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