Great hexagonal hexecontahedron

Great hexagonal hexecontahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	p5/3p5/2p3*a
Elements
Faces	60 irregular hexagons
Edges	60+60+60
Vertices	20+60+24
Vertex figure	20+60 triangles, 24 pentagrams
Measures (edge length 1)
Inradius
Dihedral angle	90°
Central density	10
Number of pieces	240
Related polytopes
Dual	Great snub dodecicosidodecahedron
Abstract properties
Flag count	720
Euler characteristic	–16
Topological properties
Orientable	Yes
Properties
Symmetry	H3+, order 60
Convex	No
Nature	Tame

The great hexagonal hexecontahedron is a uniform dual polyhedron. It consists of 60 irregular hexagons, each with two short, two medium, and two long edges.

If its dual, the great snub dodecicosidodecahedron, has an edge length of 1, then the hexagon faces have short edge length $\frac{\sqrt{2\left(\sqrt5-1-2\sqrt{\sqrt5-2}\right)}}{4} ≈ 0.18177$ , medium edge length $\frac{\sqrt{2\left(\sqrt5-1+2\sqrt{\sqrt5-2}\right)}}{4} ≈ 0.52533$ , and long edge length $\frac{\sqrt2}{2} ≈ 0.70711$ . The hexagons have one interior angle of $\arccos\left(-\phi^{-1}\right) ≈ 128.17271°$ , one of $360°-\arccos\left(\phi^{-1}\right) ≈ 231.82792°$ , and four of 90°, where $\phi$ is the golden ratio.

The great hexagonal hexecontahedron and the cube are the only finite non-degenerate isohedral polyhedra with right dihedral angles.

Vertex coordinates[edit | edit source]

A great hexagonal hexecontahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

$\left(±\frac{\sqrt{10}-\sqrt2}{8},\,±\frac{\sqrt{10}+\sqrt2}{8},\,0\right),$
$\left(±\frac{1+\sqrt5}{4},\,±\frac{\sqrt{\sqrt5-1}}{4},\,0\right),$
$\left(±\frac{\sqrt2}{4},\,±\frac{\sqrt2}{4},\,±\frac{\sqrt2}{4}\right),$

as well as all even permutations and even sign changes of:

$\left(\frac{\sqrt{2\left(\sqrt5-2\right)}}{4},\,\frac{\sqrt{2\left(1-2\sqrt{\sqrt5-2}\right)}}{4},\,\frac{\sqrt{2\left(4-\sqrt5+2\sqrt{\sqrt5-2}\right)}}{4}\right),$
$\left(\frac{\sqrt{10}-\sqrt2}{8},\,\frac{\sqrt{10}-3\sqrt2}{8},\,\frac{\sqrt{\sqrt5-1}}{2}\right),$
$\left(\frac{\sqrt{2\left(4-\sqrt5-2\sqrt{\sqrt5-2}\right)}}{4},\,\frac{\sqrt{2\left(\sqrt5-2\right)}}{4},\,\frac{\sqrt{2\left(1+2\sqrt{\sqrt5-2}\right)}}{4}\right),$
$\left(\frac{\sqrt{\sqrt5-1+2\sqrt{2\left(5\sqrt5-1\right)}}}{4},\,-\frac{\sqrt{2\left(3-\sqrt5-\sqrt{2\left(5\sqrt5-1\right)}\right)}}{4},\,\frac{1+\sqrt5}{4}\right),$
$\left(\frac{\sqrt{\sqrt5-1-2\sqrt{2\left(5\sqrt5-1\right)}}}{4},\,\frac{\sqrt{2\left(3-\sqrt5+\sqrt{2\left(5\sqrt5-1\right)}\right)}}{4},\,\frac{1+\sqrt5}{4}\right).$