Great pentagonal hexecontahedron

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

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 $31x^6+53x^5-26x^4-34x^3+17x^2+x-1$), 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"