Small stellated hecatonicosachoron

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Small stellated hecatonicosachoron
Rank4
TypeRegular
Notation
Bowers style acronymSishi
Coxeter diagramx5/2o5o3o ()
Schläfli symbol
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
Deep holesTriangles
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Inradius
Hypervolume
Dichoral angle144°
Central density4
Number of external pieces1440
Level of complexity4
Related polytopes
ArmyEx, edge length
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.

Cross-sections[edit | edit source]

Vertex coordinates[edit | edit source]

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:

  • .

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[edit | edit source]

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:

External links[edit | edit source]