Medial hexagonal hexecontahedron

Medial hexagonal hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	p5/3p3p5*a
Elements
Faces	60 hexagons
Edges	60+60+60
Vertices	60+20+12+12
Vertex figure	60+20 triangles, 12 pentagons, 12 pentagrams
Measures (edge length 1)
Inradius	≈ 0.90505
Dihedral angle	≈ 127.32013°
Central density	4
Number of external pieces	120
Related polytopes
Dual	Snub icosidodecadodecahedron
Convex core	Pentagonal hexecontahedron
Abstract & topological properties
Flag count	720
Euler characteristic	–16
Orientable	Yes
Genus	9
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

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

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

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

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

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

External links

• Wikipedia Contributors. "Medial hexagonal hexecontahedron".
• McCooey, David. "Medial Hexagonal Hexecontahedron"