Joined hecatonicosachoron

Joined hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform dual
Notation
Bowers style acronym	Pibhaki
Coxeter diagram	o5o3m3o ()
Elements
Cells	720 pentagonal tegums
Faces	3600 isosceles triangles
Edges	1200+2400
Vertices	120+600
Vertex figure	600 cubes, 120 dodecahedra
Measures (edge length 1)
Dichoral angle	$\arccos \left(-{\frac {5+2{\sqrt {5}}}{10}}\right)\approx 161.30059^{\circ }$
Central density	1
Related polytopes
Dual	Rectified hexacosichoron
Abstract & topological properties
Flag count	43200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

Not to be confused with Joined hexacosichoron.

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 ${\frac {5+3{\sqrt {5}}}{5}}\approx 2.34164$ times the length of those of the hecatonicosachoron.

The ratio between the longest and shortest edges is 1:${\frac {5+{\sqrt {5}}}{5}}$ ≈ 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:

• Pentagonal tegum (720): Rectified hexacosichoron
• Isosceles triangle (3600): Semi-uniform small rhombated hexacosichoron
• Edge (1200): Rectified hecatonicosachoron
• Edge (2400): Semi-uniform small disprismatohexacosihecatonicosachoron
• Vertex (120): Hexacosichoron
• Vertex (600): Hecatonicosachoron

External links

• Klitzing, Richard. "pibhaki".