# Small stellated hecatonicosachoron

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${\displaystyle 1}$
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
Number of pieces1440
Level of complexity4
Related polytopes
ArmyEx
RegimentSishi
DualFaceted hexacosichoron
ConjugateGreat grand hecatonicosachoron
Convex coreHecatonicosachoron
Abstract properties
Euler characteristic–480
Topological properties
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:

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

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 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.