Small stellated hecatonicosachoron

The small stellated hecatonicosachoron, or sishi, also commonly called the small stellated 120-cell, is one of the 10 regular Schläfli–Hess polychora. It has 120 small stellated dodecahedra as cells, joining 3 to an edge and 12 to a vertex in a dodecahedral arrangement.

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

The first two sets of vertices form a unit icositetrachoron that can be inscribed into the small stellated hecatonicosachoron. This corresponds to the fact there is a compound of 25 icositetrachora with the same vertices and edges as the small stellated hecatonicosachoron.

Related polychora
The small stellated hecatonicosachoron is the colonel of a regiment containing 26 uniform polychora. Of these, one other is regular (the great grand hecatonicosachoron), three are semiregular (the small ditrigonary hexacosihecatonicosachoron, ditrigonary dishecatonicosachoron, and great ditrigonary hexacosihecatonicosachoron), 3 are noble (the invertidodecahedronary hecatonicosachoron, pentagrammic prismatic heptacosicosachoron, and pentagonal prismatic heptacosicosachoron), 8 others have full symmetry, and the other 10 (including 4 discovered in 2020) have lower symmetries. The regiment also includes at least 6 uniform compounds, 8 fissary uniforms, and an unknown number (at least several hundred) of scaliforms (all but 2 of the latter have been discovered in 2020 or later).

A significant number of unit-edge uniform polychora can be vertex-inscribed into the small stellated hecatonicosachoron. These include the icositetrachoron, the tesseract, the small prismatodecachoron, the small ditrigonary icosidodecahedral prism, the great duoantiprism, and the pentagonal-pentagrammic duoprism.