# Great pentagrammic hexecontahedron

Great pentagrammic hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramp5/3p3/2p
Elements
Faces60 pentagrams
Edges60+60+60
Vertices60+20+12
Vertex figure60+20 triangles, 12 pentagrams
Measures (edge length 1)
Dihedral angle≈ 60.90113°
Central density37
Related polytopes
DualGreat inverted retrosnub icosidodecahedron
Convex corePentagonal hexecontahedron
Abstract & topological properties
Flag count600
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The great pentagrammic hexecontahedron is a uniform dual polyhedron. It consists of 60 mirror-symmetric pentagrams.

It is the dual of the great inverted retrosnub icosidodecahedron.

Each pentagram has three long and two short edges; the ratio between them is given by ${\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 1.77422}$, where ${\displaystyle \xi \approx 0.94673}$, the largest positive root of the polynomial ${\displaystyle 8x^{3}-8x^{2}+\phi ^{-2}}$, and ${\displaystyle \phi }$ is the golden ratio. (The long edges' length is approximately 3.06368 or a root of the polynomial ${\displaystyle x^{6}-2x^{5}-4x^{4}+x^{3}+4x^{2}-1}$, and the short edges' length is approximately 1.72678 or a root of the polynomial ${\displaystyle 31x^{6}-53x^{5}-26x^{4}+34x^{3}+17x^{2}-x-1}$.)

Each face has four equal angles of ${\displaystyle \arccos(\xi )\approx 18.78563^{\circ }}$, and one angle of ${\displaystyle \arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 104.85746^{\circ }}$.

A dihedral angle is equal to ${\displaystyle \arccos \left({\frac {\xi }{\xi +1}}\right)\approx 60.90113^{\circ }}$.

The inradius R ≈ 0.14897 of a great pentagrammic hexecontahedron with unit edge length is equal to the square root of a root of the polynomial ${\displaystyle 856064x^{6}-3900416x^{5}+1443072x^{4}-149376x^{3}+6384x^{2}-128x+1}$.