# Great pentagonal hexecontahedron

Great pentagonal hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramp5/2p3p
Elements
Faces60 mirror-symmetric pentagons
Edges30+60+60
Vertices12+20+60
Vertex figure20+60 triangles, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 104.43227°
Central density7
Related polytopes
DualGreat snub icosidodecahedron
ConjugatesPentagonal hexecontahedron, great inverted pentagonal hexecontahedron, great pentagrammic hexecontahedron
Convex coreNon-Catalan pentagonal hexecontahedron
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 ${\displaystyle 31x^{6}+53x^{5}-26x^{4}-34x^{3}+17x^{2}+x-1}$), and the long edges have approximate length 0.64563 (equal to a root of the polynomial ${\displaystyle x^{6}+2x^{5}-4x^{4}-x^{3}+4x^{2}-1}$). ​The hexagons have four interior angles of ${\displaystyle \arccos \left(\xi \right)\approx 101.50833^{\circ }}$, and one of ${\displaystyle \arccos \left(-\phi ^{-1}+\phi ^{-2}\xi \right)\approx 133.96670^{\circ }}$, where ${\displaystyle \xi \approx -0.19951}$ is the negative root of the polynomial ${\displaystyle 8x^{3}-8x^{2}+\phi ^{-2}}$, and ${\displaystyle \phi }$ is the golden ratio.

A dihedral angle can be given as ${\displaystyle \arccos \left({\frac {\xi }{\xi +1}}\right)\approx 104.43227^{\circ }}$.

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