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:
 * (±1, 0, 0, 0),
 * (±1/2, ±1/2, ±1/2, ±1/2),
 * (±(1+$\sqrt{5}$)/4, ±($\sqrt{5}$–1)/4, ±1/2, 0).

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 contatining 22 uniform members plus a number of fissary and compound members. Besides the small stellated hecatonicosachoron, one other member is regular, namely the great grand hecatonicosachoron.

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.