# Medial hexagonal hexecontahedron

Medial hexagonal hexecontahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramp5/3p3p5*a
Elements
Faces60 hexagons
Edges60+60+60
Vertices60+20+12+12
Vertex figure60+20 triangles, 12 pentagons, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 127.32013°
Central density4
Number of external pieces120
Related polytopes
Convex corePentagonal hexecontahedron
Abstract & topological properties
Flag count720
Euler characteristic–16
OrientableYes
Genus9
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The medial hexagonal hexecontahedron is a uniform dual polyhedron. It consists of 60 asymmetric nonconvex hexagons.

It is the dual of the snub icosidodecadodecahedron.

Each hexagon has two long edges, two of medium length and two short ones.

If the medium edges have unit length, the short edge length is ${\displaystyle \frac{1}{2}-\frac{\sqrt{(1-\xi)/(\phi^{3}-\xi)}}{2} ≈ 0.22679}$ and the long edge length is ${\displaystyle \frac{1}{2}+\frac{\sqrt{(1-\xi)/(-\phi^{-3}-\xi)}}{2}\approx 2.06072}$, where ${\displaystyle \xi ≈ -0.37744}$ is the the only real root of the polynomial ${\displaystyle 8x^3-4x^2+1}$, and ${\displaystyle \phi}$ is the golden ratio.

${\displaystyle \xi}$ can also be written as ${\displaystyle -\frac{1}{2\rho}}$, where ρ is the plastic number.

Each hexagon has four equal angles of ${\displaystyle \arccos\left(\xi\right) ≈ 112.17513°}$, one of ${\displaystyle \arccos\left(\phi^2\xi+\phi\right) ≈ 50.95827°}$, and one of ${\displaystyle 360°-\arccos\left(\phi^{-2}\xi-\phi^{-1}\right) ≈ 220.34122°}$.

A dihedral angle is equal to ${\displaystyle \arccos\left(\frac{\xi}{\xi+1}\right) ≈ 127.32013°}$.

The inradius R ≈ 0.90505 of a medial hexagonal hexecontahedron with unit edge length is equal to ${\displaystyle \frac{\sqrt{66\left(78+\sqrt[3]{12\left(27765+539\sqrt{69}\right)}+\sqrt[3]{12\left(27765-539\sqrt{69}\right)}\right)}}{132}}$.