Icosicosahedron

The icosicosahedron, e, or compound of ten tetrahedra is a uniform polyhedron compound. It consists of 40 triangles which form 20 coplanar pairs, combining into golden hexagrams. The vertices also coincide in pairs, leading to 20 vertices where 6 triangles join. It can be seen as a compound of the two chiral forms of the chiricosahedron. It can also be seen as a rhombihedron, the compound of five cubes, with each cube replaced by a stella octangula.

This compound is sometimes considered to be regular, but it is not flag-transitive, despite the fact it is vertex, edge, and face-transitive. It is however regular if you consider conjugacies along with its other symmetries.

Its quotient prismatic equivalent is the tetrahedral decayottoorthowedge, which is twelve-dimensional.

Vertex coordinates
Coordinates for the vertices of an icosicosahedron of edge length 1 are given by: plus all even permutations of:
 * $$\left(±\frac{\sqrt2}{4},\,±\frac{\sqrt2}{4},\,±\frac{\sqrt2}{4}\right),$$
 * $$\left(±\frac{\sqrt2+\sqrt{10}}{8},\,±\frac{\sqrt{10}-\sqrt2}{8},\,0\right).$$

Related polyhedra
It has connections to all weakly regular polyhedra and polyhedron compounds. It can be decomposed into 10 tetrahedra, 5 stella octangulas, or 2 chiricosahedra. It and each chiricosahedron has a dodecahedron convex hull and an icosahedron convex core while each stella octangula has a cube convex hull and an octahedron convex core, which form a rhombihedron and small icosicosahedron respectively.