# Great inverted pentagonal hexecontahedron

Great inverted pentagonal hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramp5/3p3p
Elements
Faces60 mirror-symmetric concave pentagons
Edges30+60+60
Vertices12+20+60
Vertex figure20+60 triangles, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 78.35920°
Central density13
Related polytopes
DualGreat inverted snub icosidodecahedron
ConjugatesPentagonal hexecontahedron, great pentagonal hexecontahedron, great pentagrammic hexecontahedron
Abstract & topological properties
Flag count600
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

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.

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 ${\displaystyle 856064x^{6}-3900416x^{5}+1443072x^{4}-149376x^{3}+6384x^{2}-128x+1}$.