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, and indeed the regular hexacosichoron as well.

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).

Representations
A small prismatodecachoron has the following Coxeter diagrams:


 * x3o3o3x (full symmetry)
 * xxo3ooo3oxx&#xt (A3 axial, tetrahedron-first)
 * x(ou)x x(xo)o3o(xo)x&#xt (A2×A1 axial, triangular prism-first)
 * (xoxxox)(uo) (oxxoxx)(ou)&#xr (A1×A1 axial)

Related polychora
A small prismatodecachoron can be cut in half to produce two identical tetrahedron atop cuboctahedron segmentochora, with the tetrahedral bases in dual orientations. The triangular cupofastegium can also be obtained as a wedge of the small prismatodecachoron, in triangular prism-first orientation.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Tetrahedron (10): Bidecachoron
 * Triangular prism (20): Biambodecachoron
 * Square (30): Decachoron
 * Triangle (40): Bitruncatodecachoron
 * Edge (60): Rectified small prismatodecachoron