Great stellated hecatonicosachoron

The great stellated hecatonicosachoron, or gishi, also commonly called the great stellated 120-cell, is one of the 10 regular Schläfli–Hess polychora. It has 120 great stellated dodecahedra as cells, joining 5 to an edge and 20 to a vertex in an icosahedral arrangement.

Vertex coordinates
The vertices of a great stellated hecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of: and all even permutations of:
 * $$\left(±\frac{\sqrt5-1}{2},\,0,\,0,\,0\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4}\right),$$
 * $$\left(±\frac{3-\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac12,\,0\right).$$

The first two sets of vertices form an icositetrachoron that can be inscribed into the great stellated hecatonicosachoron.

Related polychora
The great stellated hecatonicosachoron is the colonel of a five-member regiment that includes three other regular polychora, namely the grand stellated hecatonicosachoron, the great faceted hexacosichoron, and the grand hexacosichoron. Of these, the grand stellated hecatonicosachoron also shares the same faces, so the great stellated hecatonicosachoron is the captain of a two-member company.