# 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.