# Medial inverted pentagonal hexecontahedron

Medial inverted pentagonal hexecontahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramp5/3p5p ()
Elements
Faces60 irregular pentagons
Edges60+60+30
Vertices60+12+12
Vertex figure60 triangles, 12 pentagons, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 108.09572°
Central density9
Number of pieces60
Related polytopes
Abstract properties
Flag count600
Euler characteristic–6
Topological properties
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The medial inverted pentagonal hexecontahedron is a uniform dual polyhedron. It consists of 60 irregular pentagons, each with two short, one medium, and two long edges. Its dual is the inverted snub dodecadodecahedron.

If the pentagon faces have medium edge length 2, then the short edge length is ${\displaystyle 1-\sqrt{\frac{1-\xi}{\phi^3-\xi}} ≈ 0.47413}$, and the long edge length is ${\displaystyle 1+\sqrt{\frac{1-\xi}{-\phi^{-3}-\xi}} ≈ 37.55188}$, where ${\displaystyle \xi ≈ -0.23699}$ is the largest (least negative) real root of the polynomial ${\displaystyle 8x^4-12x^3+5x+1}$. ​​The pentagons have three interior angles of ${\displaystyle \arccos\left(\xi\right) ≈ 103.70918°}$, one of ${\displaystyle \arccos\left(\phi^2\xi+\phi\right) ≈ 3.99013°}$, and one of ${\displaystyle 360°-\arccos\left(\phi^{-2}\xi-\phi^{-1}\right) ≈ 224.88232°}$, where ${\displaystyle \phi}$ is the golden ratio.

A dihedral angle can be given as ${\displaystyle \arccos\left(\frac{\xi}{\xi+1}\right) ≈ 108.09572}$.

The inradius R ≈ 0.55808 of the medial inverted pentagonal hexecontahedron with unit edge length is equal to the square root of a real root of ${\displaystyle 8192x^4-12352x^3+3376x^2-104x+1}$.