Medial inverted pentagonal hexecontahedron

Medial inverted pentagonal hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	p5/3p5p ()
Elements
Faces	60 irregular pentagons
Edges	60+60+30
Vertices	60+12+12
Vertex figure	60 triangles, 12 pentagons, 12 pentagrams
Measures (edge length 1)
Inradius	≈ 0.55808
Dihedral angle	≈ 108.09572°
Central density	9
Number of pieces	60
Related polytopes
Dual	Inverted snub dodecadodecahedron
Abstract properties
Flag count	600
Euler characteristic	–6
Topological properties
Orientable	Yes
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

The medial inverted pentagonal hexecontahedron is a uniform dual polyhedron. It consists of 60 irregular pentagons, each with two short, one medium, and two long edges. Its dual is the inverted snub dodecadodecahedron.

If the pentagon faces have medium edge length 2, then the short edge length is $1-\sqrt{\frac{1-\xi}{\phi^3-\xi}} ≈ 0.47413$, and the long edge length is $1+\sqrt{\frac{1-\xi}{-\phi^{-3}-\xi}} ≈ 37.55188$, where $\xi ≈ -0.23699$ is the largest (least negative) real root of the polynomial $8x^4-12x^3+5x+1$. ​​The pentagons have three interior angles of $\arccos\left(\xi\right) ≈ 103.70918°$, one of $\arccos\left(\phi^2\xi+\phi\right) ≈ 3.99013°$, and one of $360°-\arccos\left(\phi^{-2}\xi-\phi^{-1}\right) ≈ 224.88232°$, where $\phi$ is the golden ratio.

A dihedral angle can be given as $\arccos\left(\frac{\xi}{\xi+1}\right) ≈ 108.09572$.

The inradius R ≈ 0.55808 of the medial inverted pentagonal hexecontahedron with unit edge length is equal to the square root of a real root of $8192x^4-12352x^3+3376x^2-104x+1$.

External links

• Wikipedia Contributors. "Medial inverted pentagonal hexecontahedron".
• McCooey, David. "Medial Inverted Pentagonal Hexecontahedron"