# Medial inverted pentagonal hexecontahedron

Medial inverted pentagonal hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramp5/3p5p ()
Elements
Faces60 irregular concave pentagons
Edges30+60+60
Vertices12+12+60
Vertex figure60 triangles, 12 pentagons, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 108.09572°
Central density9
Number of external pieces60
Related polytopes
ConjugateMedial pentagonal hexecontahedron
Convex coreNon-Catalan pentakis dodecahedron
Abstract & topological properties
Flag count600
Euler characteristic–6
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 ${\displaystyle 1-{\sqrt {\frac {1-\xi }{\phi ^{3}-\xi }}}\approx 0.47413}$, and the long edge length is ${\displaystyle 1+{\sqrt {\frac {1-\xi }{-\phi ^{-3}-\xi }}}\approx 37.55188}$, where ${\displaystyle \xi \approx -0.23699}$ is the largest (least negative) real root of the polynomial ${\displaystyle 8x^{4}-12x^{3}+5x+1}$. ​​The pentagons have three interior angles of ${\displaystyle \arccos \left(\xi \right)\approx 103.70918^{\circ }}$, one of ${\displaystyle \arccos \left(\phi ^{2}\xi +\phi \right)\approx 3.99013^{\circ }}$, and one of ${\displaystyle 360^{\circ }-\arccos \left(\phi ^{-2}\xi -\phi ^{-1}\right)\approx 224.88232^{\circ }}$, where ${\displaystyle \phi }$ is the golden ratio.

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

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 ${\displaystyle 8192x^{4}-12352x^{3}+3376x^{2}-104x+1}$.