# Small stellated hecatonicosachoron

(Redirected from Sishi)
Small stellated hecatonicosachoron Rank4
TypeRegular
SpaceSpherical
Notation
Bowers style acronymSishi
Coxeter diagramx5/2o5o3o (         )
Schläfli symbol{5/2,5,3}
Elements
Cells120 small stellated dodecahedra
Faces720 pentagrams
Edges1200
Vertices120
Vertex figureDodecahedron, edge length (5–1)/2
Edge figuresissid 5/2 sissid 5/2 sissid 5/2
Measures (edge length 1)
Circumradius$1$ Edge radius$\frac{\sqrt3}{2} \approx 0.86603$ Face radius$\sqrt{\frac{5+\sqrt5}{10}} \approx 0.85065$ Inradius$\frac{1+\sqrt5}{4} \approx 0.80902$ Hypervolume$15\frac{5-\sqrt5}{4} \approx 10.36475$ Dichoral angle144°
Central density4
Number of external pieces1440
Level of complexity4
Related polytopes
ArmyEx, edge length $\frac{\sqrt5-1}{2}$ RegimentSishi
DualFaceted hexacosichoron
ConjugateGreat grand hecatonicosachoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count14400
Euler characteristic–480
Schläfli type{5,5,3}
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
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:

• $\left(\pm1,\,0,\,0,\,0\right)$ ,
• $\left(\pm\frac12,\,\pm\frac12,\,\pm\frac12,\,\pm\frac12\right)$ ,

and all even permutations of:

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

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 heptacosiicosachoron, and pentagonal-prismatic heptacosiicosachoron), 8 others have full symmetry, and the other 10 (including 4 discovered in 2020) have lower symmetries. The regiment also includes at least 7 uniform compounds, 10 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 uniform polychora can be edge-inscribed into the small stellated hecatonicosachoron. These include the tesseract, the icositetrachoron, the small prismatodecachoron, the small ditrigonary icosidodecahedral prism, the pentagonal-pentagrammic duoprism, the great duoantiprism, the pentacubic extended duoantiprism, and the snub hexecontatetrasnub-snub disoctachoron.

Uniform polychoron compounds composed of small stellated hecatonicosachora include: