# Medial pentagonal hexecontahedron

Medial pentagonal hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramp5/2p5p ()
Elements
Faces60 irregular pentagons
Edges30+60+60
Vertices12+12+60
Vertex figure60 triangles, 12 pentagons, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 133.80098°
Central density3
Related polytopes
ConjugateMedial inverted pentagonal hexecontahedron
Convex coreNon-Catalan pentakis dodecahedron
Abstract & topological properties
Flag count600
Euler characteristic–6
OrientableYes
Genus4
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The medial 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 snub dodecadodecahedron.

If the pentagon faces have medium edge length 2, then the short edge length is ${\displaystyle 1+{\sqrt {\frac {1-\xi }{\phi ^{3}-\xi }}}\approx 1.55076}$, and the long edge length is ${\displaystyle 1+{\sqrt {\frac {1-\xi }{-\phi ^{3}-\xi }}}\approx 3.85415}$, where ${\displaystyle \xi \approx -0.40903}$ is the smallest (most negative) real root of the polynomial ${\displaystyle 8x^{4}-12x^{3}+5x+1}$. ​The pentagons have three interior angles of ${\displaystyle \arccos \left(\xi \right)\approx 114.14440^{\circ }}$, one of ${\displaystyle \arccos \left(\phi ^{2}\xi +\phi \right)\approx 56.82766^{\circ }}$, and one of ${\displaystyle \arccos \left(\phi ^{-2}\xi -\phi ^{-1}\right)\approx 140.73912^{\circ }}$, where ${\displaystyle \phi }$ is the golden ratio.

The inradius R ≈ 1.07828 of the medial pentagonal hexecontahedron with unit edge length is equal to the square root of a real root of ${\displaystyle 8192x^{4}-12352x^{3}+3376x^{2}-104x+1}$.

A dihedral angle can be given as acos(α), where α ≈ -0.69216 is the negative of a real root of ${\displaystyle 16x^{4}+x^{3}-9x^{2}-x+1}$.