Small prismatohecatonicosachoron

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Small prismatohecatonicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymSophi
Coxeter diagram(x5o5/2o5x)/2 (/2)
Elements
Cells
Faces
Edges3600
Vertices720
Vertex figurePentagrammic antiprism, edge lengths (1+5)/2 (base) and 2 (side)
Edge figuregad.pip.pip.pip
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {5+2{\sqrt {5}}}}\approx 3.07768}$
Hypervolume${\displaystyle 15{\frac {155+69{\sqrt {5}}}{4}}\approx 1159.83259}$
Dichoral angleGad–5–pip: 162°
Pip–4–pip: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }}$
Central density3
Number of external pieces10920
Level of complexity21
Related polytopes
ArmyRox
RegimentRox
ConjugateQuasiprismatohecatonicosachoron
Convex coreChamfered hecatonicosachoron
Abstract & topological properties
Flag count57600
Euler characteristic–480
OrientableYes
Properties
SymmetryH4, order 14400
Flag orbits4
ConvexNo
NatureTame

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.

Vertex coordinates

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