# Joined hecatonicosachoron

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Joined hecatonicosachoron
Rank4
TypeUniform dual
Notation
Bowers style acronymPibhaki
Coxeter diagramo5o3m3o ()
Elements
Cells720 pentagonal tegums
Faces3600 isosceles triangles
Edges1200+2400
Vertices120+600
Vertex figure600 cubes, 120 dodecahedra
Measures (edge length 1)
Dichoral angle${\displaystyle \arccos \left(-{\frac {5+2{\sqrt {5}}}{10}}\right)\approx 161.30059^{\circ }}$
Central density1
Related polytopes
DualRectified hexacosichoron
Abstract & topological properties
Flag count43200
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
NatureTame

The joined hecatonicosachoron, also known as the pentagonal-tegmatic heptacosiicosachoron or pibhaki, is a convex isochoric polychoron with 720 pentagonal tegums as cells. It can be obtained as the dual of the rectified hexacosichoron.

It can also be obtained as the convex hull of a hecatonicosachoron and a hexacosichoron, where the edges of the hexacosichoron are ${\displaystyle {\frac {5+3{\sqrt {5}}}{5}}\approx 2.34164}$ times the length of those of the hecatonicosachoron.

The ratio between the longest and shortest edges is 1:${\displaystyle {\frac {5+{\sqrt {5}}}{5}}}$ ≈ 1:1.44721. Each face is an isosceles triangle that uses one short and two long edges.

## Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: