# Great inverted pentagonal hexecontahedron

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

Great inverted pentagonal hexecontahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramp5/3p3p
Elements
Faces60 mirror-symmetric concave pentagons
Edges30+60+60
Vertices20+60+12
Vertex figure20+60 triangles, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 78.35920°
Central density13
Related polytopes
DualGreat inverted snub icosidodecahedron
Abstract & topological properties
Flag count600
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

If its dual, the great inverted snub icosidodecahedron, has unit edge length, then the pentagon faces' short edges have approximate length 0.23186 (equal to a root of the polynomial ${\displaystyle 31x^6-53x^5-26x^4+34x^3+17x^2-x-1}$), and the long edges have approximate length 0.81801 (equal to a root of the polynomial ${\displaystyle x^6-2x^5-4x^4+x^3+4x^2-1}$).

A dihedral angle can be given as acos(α), where α ≈ 0.20178 is a real root of the polynomial ${\displaystyle 209x^6-94x^5-137x^4+100x^3-9x^2-6x+1}$.

The inradius R ≈ 0.25744 of the great inverted pentagonal hexecontahedron with unit edge length is equal to the square root of a real root of ${\displaystyle 856064x^6-3900416x^5+1443072x^4-149376x^3+6384x^2-128x+1}$.