Great pentagonal hexecontahedron

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

Great pentagonal hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	p5/2p3p
Elements
Faces	60 mirror-symmetric pentagons
Edges	30+60+60
Vertices	20+60+12
Vertex figure	20+60 triangles, 12 pentagrams
Measures (edge length 1)
Inradius	≈ 0.50974
Dihedral angle	≈ 104.43227°
Central density	7
Related polytopes
Dual	Great snub icosidodecahedron
Abstract & topological properties
Flag count	600
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

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 $31x^6+53x^5-26x^4-34x^3+17x^2+x-1$), and the long edges have approximate length 0.64563 (equal to a root of the polynomial $x^6+2x^5-4x^4-x^3+4x^2-1$). ​The hexagons have four interior angles of $\arccos\left(\xi\right) ≈ 101.50833°$, and one of $\arccos\left(-\phi^{-1}+\phi^{-2}\xi\right) ≈ 133.96670°$, where $\xi ≈ -0.19951$ is the negative root of the polynomial $8x^3-8x^2+\phi^{-2}$, and $\phi$ is the golden ratio.

A dihedral angle can be given as $\arccos\left(\frac{\xi}{\xi+1}\right) ≈ 104.43227°$.

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

External links

• Wikipedia Contributors. "Great pentagonal hexecontahedron".
• McCooey, David. "Great Pentagonal Hexecontahedron"