Icosahedral pyramid

The icosahedral pyramid, or ikepy, is a CRF segmentochoron (designated K-4.84 on Richard Klitzing's list). It has 20 regular tetrahedra and 1 regular icosahedron as cells. As the name suggests, it is a pyramid based on the icosahedron.

It is the first segment of the hexacosichoron in vertex-first orientation.

Vertex coordinates
The vertices of an icosahedral pyramid of edge length 1 are given by:
 * $$\left(0,\,0,\,0,\,\frac{\sqrt5-1}{4}\right).$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac12,\,0,\,0\right)$$ and all permutations of the first three coordinates.

Related polychora
The icosahedral pyramid can be broken into a pentagonal antiprismatic pyramid and two pentagonal scalenes.