# Small stellated hecatonicosachoron

Small stellated hecatonicosachoron
Rank4
TypeRegular
SpaceSpherical
Bowers style acronymSishi
Info
Coxeter diagramx5/2o5o3o
Schläfli symbol{5/2,5,3}
SymmetryH4, order 14400
ArmyEx
RegimentSishi
Elements
Vertex figureDodecahedron, edge length (5–1)/2
Cells120 small stellated dodecahedra
Faces720 pentagrams
Edges1200
Vertices120
Measures (edge length 1)
Edge radius${\displaystyle \frac{\sqrt3}{2} ≈ 0.86603}$
Face radius${\displaystyle \sqrt{\frac{5+\sqrt5}{10}} ≈ 0.85065}$
Inradius${\displaystyle \frac{1+\sqrt5}{4} ≈ 0.80902}$
Hypervolume${\displaystyle 15\frac{5-\sqrt5}{4} ≈ 10.36475}$
Dichoral angle144°
Central density4
Euler characteristic-480
Number of pieces1440
Level of complexity4
Related polytopes
DualFaceted hexacosichoron
ConjugateGreat grand hecatonicosachoron
Properties
ConvexNo
OrientableYes
NatureTame

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:

• ${\displaystyle \left(±1,\,0,\,0,\,0\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right),}$

and all even permutations of:

• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac12,\,0\right).}$

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.