Great pentagrammic hexecontahedron

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Great pentagrammic hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramp5/3p3/2p
Elements
Faces60 mirror-symmetric pentagrams
Edges30+60+60
Vertices12+20+60
Vertex figure60+20 triangles, 12 pentagrams
Measures (edge length 1)
Inradius≈ 0.14897
Dihedral angle≈ 60.90113°
Central density37
Related polytopes
DualGreat inverted retrosnub icosidodecahedron
ConjugatesPentagonal hexecontahedron, Great pentagonal hexecontahedron, great inverted pentagonal hexecontahedron
Convex coreNon-Catalan pentagonal 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 , where , the largest positive root of the polynomial , and is the golden ratio. (The long edges' length is approximately 3.06368 or a root of the polynomial , and the short edges' length is approximately 1.72678 or a root of the polynomial .)

Each face has four equal angles of , and one angle of .

A dihedral angle is equal to .

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 .

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