Small stellated hecatonicosachoron

Small stellated hecatonicosachoron
(OFF file)
Rank	4
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Sishi
Coxeter diagram	x5/2o5o3o ()
Schläfli symbol	{5/2,5,3}
Elements
Cells	120 small stellated dodecahedra
Faces	720 pentagrams
Edges	1200
Vertices	120
Vertex figure	Dodecahedron, edge length (√5–1)/2
Edge figure	sissid 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 angle	144°
Central density	4
Number of external pieces	1440
Level of complexity	4
Related polytopes
Army	Ex, edge length $\frac{\sqrt5-1}{2}$
Regiment	Sishi
Dual	Faceted hexacosichoron
Conjugate	Great grand hecatonicosachoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	14400
Euler characteristic	–480
Schläfli type	{5,5,3}
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

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

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:

• Small stellated diacositetracontachoron (2)
• Small stellated chirohexacosichoron (5)
• Small stellated chirododecahedral hyperprismatochoron (6)
• Small stellated dishexacosichoron (10)
• Small stellated disdodecahedral hyperprismatochoron (12)

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#10).

• Bowers, Jonathan. "Category 17: Sishi Regiment".

• Klitzing, Richard. "sishi".

• Nan Ma. "Small stellated 120-cell {5/2, 5, 3}".

• Wikipedia Contributors. "Small stellated 120-cell".