Small stellated hecatonicosachoron
|Small stellated hecatonicosachoron|
|Bowers style acronym||Sishi|
|Coxeter diagram||x5/2o5o3o ()|
|Cells||120 small stellated dodecahedra|
|Vertex figure||Dodecahedron, edge length (√–1)/2|
|Edge figure||sissid 5/2 sissid 5/2 sissid 5/2|
|Measures (edge length 1)|
|Number of external pieces||1440|
|Level of complexity||4|
|Army||Ex, edge length|
|Conjugate||Great grand hecatonicosachoron|
|Abstract & topological properties|
|Symmetry||H4, order 14400|
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:
- Small stellated diacositetracontachoron (2)
- Small stellated chirohexacosichoron (5)
- Small stellated chirododecahedral hyperprismatochoron (6)
- Small stellated dishexacosichoron (10)
- Small stellated disdodecahedral hyperprismatochoron (12)
[edit | edit source]
- Bowers, Jonathan. "Category 1: Regular Polychora" (#10).
- Bowers, Jonathan. "Category 17: Sishi Regiment".
- Klitzing, Richard. "sishi".
- Wikipedia Contributors. "Small stellated 120-cell".