Great inverted pentagonal hexecontahedron

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

Great inverted pentagonal hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	p5/3p3p
Elements
Faces	60 mirror-symmetric concave pentagons
Edges	30+60+60
Vertices	20+60+12
Vertex figure	20+60 triangles, 12 pentagrams
Measures (edge length 1)
Inradius	≈ 0.25744
Dihedral angle	≈ 78.35920°
Central density	13
Related polytopes
Dual	Great inverted 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 inverted snub icosidodecahedron, has unit edge length, then the pentagon faces' short edges have approximate length 0.23186 (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 0.81801 (equal to a root of the polynomial $x^6-2x^5-4x^4+x^3+4x^2-1$).

A dihedral angle can be given as acos(α), where α ≈ 0.20178 is a real root of the polynomial $209x^6-94x^5-137x^4+100x^3-9x^2-6x+1$.

The inradius R ≈ 0.25744 of the great inverted 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 inverted pentagonal hexecontahedron".
• McCooey, David. "Great Inverted Pentagonal Hexecontahedron"