Small prismatodecachoron

The small prismatodecachoron, or spid, also commonly called the runcinated 5-cell or runcinated pentachoron, is a convex uniform polychoron that consists of 10 regular tetrahedra and 20 triangular prisms. 2 tetrahedra and 6 triangular prisms join at each vertex. It is the result of expanding the cells of a pentachoron outwards.

The small prismatodecachoron of edge length ($\sqrt{2}$+1)/2 can be vertex-inscribed into a grand antiprism.

Vertex coordinates
The vertices of a small prismatodecachoron of edge length 1 are given by the following points:


 * ±(0, 0, 0, ±1),
 * ±(0, 0, ±$\sqrt{5}$/2, ±1/2),
 * ±(0, $\sqrt{6}$/3, –$\sqrt{5}$/3, 0),
 * ±(0, $\sqrt{3}$/3, $\sqrt{6}$/6, ±1/2),
 * ±($\sqrt{3}$/4, –$\sqrt{6}$/4, 0, 0),
 * ±($\sqrt{3}$/4, $\sqrt{10}$/12, –$\sqrt{6}$/3, 0),
 * ±($\sqrt{6}$/4, $\sqrt{6}$/12, $\sqrt{3}$/6, ±1/2).

Much simpler coordinates can be given in five dimensions, as all permutations of:


 * ($\sqrt{10}$, $\sqrt{6}$/2, $\sqrt{3}$/2, $\sqrt{2}$/2, 0).