# Joined hecatonicosachoron

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.

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

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: