# Rectified small stellated hecatonicosachoron

Rectified small stellated hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymRasishi
Coxeter diagramo5/2x5o3o ()
Elements
Cells120 dodecahedra, 120 dodecadodecahedra
Faces1440 pentagons, 720 pentagrams
Edges3600
Vertices1200
Vertex figureSemi-uniform triangular prism, edge lengths 1+5)/2 (base) and (5–1)/2 (side)
Edge figuredoe 5 did 5/2 did 5
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {3}}+{\sqrt {15}}}{2}}\approx 2.80252}$
Hypervolume${\displaystyle 15{\frac {135+53{\sqrt {5}}}{4}}\approx 950.66851}$
Dichoral anglesDid–5–doe: 144°
Did–5/2–did: 144°
Central density4
Number of external pieces2520
Level of complexity14
Related polytopes
ArmyRahi, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
RegimentRasishi
ConjugateRectified great grand hecatonicosachoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count43200
Euler characteristic–480
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The rectified small stellated hecatonicosachoron, or rasishi, is a nonconvex uniform polychoron that consists of 120 regular dodecahedra and 120 dodecadodecahedra. Two dodecahedra and three dodecadodecahedra join at each triangular prismatic vertex. As the name suggests, it can be obtained by rectifying the small stellated hecatonicosachoron.

## Vertex coordinates

The vertices of a rectified small stellated hecatonicosachoron of edge length 1 are given by all permutations of:

• ${\displaystyle \left(0,\,0,\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,

along with all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 1,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$.

## Related polychora

The rectified small stellated hecatonicosachoron is the colonel of a 3-member regiment that also includes the facetorectified small stellated hecatonicosachoron and hecatonicosintercepted hecatonicosachoron.