Small prismatohecatonicosachoron

Small prismatohecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Sophi
Coxeter diagram	(x5o5/2o5x)/2 (/2)
Elements
Cells	720 pentagonal prisms, 120 great dodecahedra
Faces	1800 squares, 1440 pentagons
Edges	3600
Vertices	720
Vertex figure	Pentagrammic antiprism, edge lengths (1+√5)/2 (base) and √2 (side)
Edge figure	gad.pip.pip.pip
Measures (edge length 1)
Circumradius	$\sqrt{5+2\sqrt5} ≈ 3.07768$
Hypervolume	$15\frac{155+69\sqrt5}{4} ≈ 1159.83259$
Dichoral angles	Gad–5–pip: 162°
	Pip–4–pip: $\arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495°$
Central density	3
Number of external pieces	10920
Level of complexity	21
Related polytopes
Army	Rox
Regiment	Rox
Conjugate	Quasiprismatohecatonicosachoron
Convex core	Chamfered hecatonicosachoron
Abstract & topological properties
Euler characteristic	–480
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The small prismatohecatonicosachoron, or sophi, is a nonconvex uniform polychoron that consists of 120 great dodecahedra and 720 pentagonal prisms. 2 great dodecahedra and 10 pentagonal prisms join at each vertex.

A double cover of this polychoron can be formed by expanding the cells of the great hecatonicosachoron outward.

Gallery

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the rectified hexacosichoron.

External links

• Bowers, Jonathan. "Category 5: Pentagonal Rectates" (#75).

• Klitzing, Richard. "sophi".