# Rectified great hecatonicosachoron

(Redirected from Righi)
Rectified great hecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymRighi
Coxeter diagramo5x5/2o5o ()
Elements
Cells
Faces
Edges3600
Vertices720
Vertex figureSemi-uniform pentagonal prism, edge lengths (5–1)/2 (base) and (1+5)/2 (side)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\approx 1.90211}$
Hypervolume${\displaystyle 75{\frac {4+3{\sqrt {5}}}{2}}\approx 401.55765}$
Dichoral anglesSissid–5/2–did: 144°
Did–5–did: 144°
Central density6
Number of external pieces3720
Level of complexity14
Related polytopes
ArmyRox, edge length ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$
RegimentRighi
ConjugateRectified grand stellated hecatonicosachoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count43200
Euler characteristic–960
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits3
ConvexNo
NatureTame

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

## Vertex coordinates

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

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

along with even permutations of:

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

## Related polychora

The rectified great hecatonicosachoron is the colonel of a regiment with 15 members. Of these, one other besideds the colonel itself is Wythoffian (the rectified grand hecatonicosachoron), two are hemi-Wythoffian (the pentagrammal antiprismatoverted hexacosihecatonicosachoron and great pentagonal retroprismatoverted dishecatonicosachoron), and one is noble (the medial retropental hecatonicosachoron).

The rectified great hecatonicosachoron also has the same circumradius as the hexagonal-decagonal duoprism.

Uniform polychoron compounds composed of rectified great hecatonicosachora include: