# Great pentagonal hexecontahedron

Great pentagonal hexecontahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramp5/2p3p
Elements
Faces60 mirror-symmetric pentagons
Edges30+60+60
Vertices20+60+12
Vertex figure20+60 triangles, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 104.43227°
Central density7
Related polytopes
DualGreat snub icosidodecahedron
Abstract & topological properties
Flag count600
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The great pentagonal hexecontahedron is a uniform dual polyhedron. It consists of 60 mirror-symmetric pentagons, each with two short and three long edges.

If its dual, the great snub icosidodecahedron, has unit edge length, then the pentagon faces' short edges have approximate length 0.49069 (equal to a root of the polynomial $31x^6+53x^5-26x^4-34x^3+17x^2+x-1$ ), and the long edges have approximate length 0.64563 (equal to a root of the polynomial $x^6+2x^5-4x^4-x^3+4x^2-1$ ). ​The hexagons have four interior angles of $\arccos\left(\xi\right) ≈ 101.50833°$ , and one of $\arccos\left(-\phi^{-1}+\phi^{-2}\xi\right) ≈ 133.96670°$ , where $\xi ≈ -0.19951$ is the negative root of the polynomial $8x^3-8x^2+\phi^{-2}$ , and $\phi$ is the golden ratio.

A dihedral angle can be given as $\arccos\left(\frac{\xi}{\xi+1}\right) ≈ 104.43227°$ .

The inradius R ≈ 0.50974 of the great pentagonal hexecontahedron with unit edge length is equal to the square root of a real root of $856064x^6-3900416x^5+1443072x^4-149376x^3-6384x^2-128x+1$ .