Rectified small prismatotetracontoctachoron

The rectified small prismatotetracontoctachoron or respic is a convex isogonal polychoron that consists of 48 cuboctahedra, 192 rectified triangular prisms and 144 square antiprisms. 1 cuboctahedron, 3 rectified triangular prisms, and 2 square antiprisms join at each vertex. It can be formed by rectifying the small prismatotetracontoctachoron.

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 half the length of the other edges. It is one of five polychora (including two transitional cases) formed in such a way, and is the transitional point between the small birhombatotetracontoctachoron and medial birhombatotetracontoctachoron.

The ratio between the longest and shortest edges is 1:$$\sqrt2$$ ≈ 1:1.41421.

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


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