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.

It can also be obtained as one of several isogonal hulls of 2 10-3 step prisms, which could be called the triangular-prismatic 10-3 double gyrostep prism.

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


 * $$±\left(0,\,0,\,0,\,±1\right),$$
 * $$±\left(0,\,0,\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$±\left(0,\,\frac{\sqrt6}{3},\,-\frac{\sqrt3}{3},\,0\right),$$
 * $$±\left(0,\,\frac{\sqrt6}{3},\,\frac{\sqrt3}{6},\,±\frac12\right),$$
 * $$±\left(\frac{\sqrt{10}}{4},\,-\frac{\sqrt6}{4},\,0,\,0\right),$$
 * $$±\left(\frac{\sqrt{10}}{4},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),$$
 * $$±\left(\frac{\sqrt{10}}{4},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right).$$

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


 * $$\left(\frac{\sqrt2},\,\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,0\right).$$

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)

Semi-uniform variant
The small prismatodecachoron has a semi-uniform variant of the form x3o3o3y with half of the symmetry of the uniform variant, which means it is sometimes called a small disprismatopentapentachoron. this variant uses 2 sets of 5 tetrahedra of sizes x and y, and 2 stes of 10 semi-uniform triangular prisms of forms x y3o and y x3o as cells, with 2 edge lengths.

With edges of length a and b, its circumradius is given by $$\sqrt{\frac{2a^2+2b^2+ab}{5}}$$ and its volume is given by $$(a^4+16a^3b+36a^2b^2+16ab^3+a^4)\frac{\sqrt5}{96}$$.

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