Rectified great hecatonicosachoron

Rectified great hecatonicosachoron
	(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Righi
Coxeter diagram	o5x5/2o5o ()
Elements
Cells	120 small stellated dodecahedra, 120 dodecadodecahedra
Faces	720 pentagons, 1440 pentagrams
Edges	3600
Vertices	720
Vertex figure	Semi-uniform pentagonal prism, edge lengths (√5–1)/2 (base) and (1+√5)/2 (side)
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral angles	Sissid–5/2–did: 144°
	Did–5–did: 144°
Central density	6
Number of pieces	3720
Level of complexity	14
Related polytopes
Army	Rox
Regiment	Righi
Conjugate	Rectified grand stellated hecatonicosachoron
Convex core	Hecatonicosachoron
Abstract properties
Euler characteristic	–960
Topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame
Discovered by	{{{discoverer}}}

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.

Cross-sections[edit | edit source]

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

$\left(0,\,0,\,±1,\,±\frac{1+\sqrt5}{2}\right),$
$\left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{4}\right),$

along with even permutations of:

$\left(0,\,±\frac{\sqrt5-1}{4},\,±\frac12,\,±\frac{5+\sqrt5}{4}\right),$
$\left(0,\,±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{\sqrt5}{2}\right),$
$\left(±\frac{\sqrt5-1}{4},\,±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{1+\sqrt5}{4}\right),$
$\left(±\frac12,\,±\frac{1+\sqrt5}{4},\,±1,\,±\frac{3+\sqrt5}{4}\right).$

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.