# Pentagonal-prismatic heptacosiicosachoron

Pentagonal-prismatic heptacosiicosachoron
Rank4
TypeUniform
Notation
Bowers style acronymPaphacki
Elements
Cells720 pentagonal prisms
Faces1800 squares, 720 pentagons
Edges1200
Vertices120
Vertex figureGreat noble triangular hexecontahedron
Edge figure(pip 4 pip 4 pip 5)×3
Measures (edge length 1)
Circumradius${\displaystyle 1}$
Inradius${\displaystyle {\frac {\sqrt {25-10{\sqrt {5}}}}{10}}\approx 0.16246}$
Volume${\displaystyle {\frac {45{\sqrt {5}}}{2}}\approx 50.31153}$
Central density259
Number of external pieces1260000
Level of complexity3490
Related polytopes
ArmyEx
RegimentSishi
ConjugatePentagrammic-prismatic heptacosiicosachoron
Convex coreJoined hecatonicosachoron
Abstract & topological properties
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureFeral

The pentagonal-prismatic heptacosiicosachoron, or paphacki, is a noble uniform polychoron in the small stellated hecatonicosachoron's regiment. It consists of 720 pentagonal prisms, 60 of which meet at each of its 120 vertices.

## Vertex coordinates

The vertices of a pentagonal-prismatic heptacosiicosachoron of unit edge length are given by:

• ${\displaystyle \left(\pm 1,\,0,\,0,\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$

and all even permutations of:

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

The first two sets of vertices form a unit icositetrachoron that can be inscribed into the small stellated hecatonicosachoron. This corresponds to the fact there is a compound of 25 icositetrachora with the same vertices and edges as the small stellated hecatonicosachoron.