Great deltoidal hexecontahedron

Great deltoidal hexecontahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	m5/3o3m
Elements
Faces	60 darts
Edges	60+60
Vertices	20+30+12
Vertex figure	20 triangles, 30 squares, 12 pentagrams
Measures (edge length 1)
Inradius
Dihedral angle
Central density	13
Number of external pieces	120
Related polytopes
Dual	Quasirhombicosidodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great deltoidal hexecontahedron is a uniform dual polyhedron. It consists of 60 darts.

It appears the same as the great rhombidodecacron.

If its dual, the quasirhombicosidodecahedron, has an edge length of 1, then the short edges of the darts will measure $\frac{\sqrt{5\left(5+\sqrt5\right)}}{3} ≈ 2.00500$ , and the long edges will be $\frac{\sqrt{5\left(85+31\sqrt5\right)}}{11} ≈ 2.52523$ . The dart faces will have length $\frac{\sqrt{10\left(157-31\sqrt5\right)}}{33} ≈ 0.89731$ , and width $\frac{5+\sqrt5}{2} ≈ 3.61803$ . The darts have two interior angles of $\arccos\left(\frac12+\frac{\sqrt5}{5}\right) ≈ 18.69941°$ , one of $\arccos\left(-\frac14+\frac{\sqrt5}{10}\right) ≈ 91.51239°$ , and one of $360°-\arccos\left(-\frac18-\frac{9\sqrt5}{40}\right) ≈ 231.08879°$ .

Vertex coordinates[edit | edit source]

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

$\left(±\sqrt5,\,0,\,0\right),$
$\left(±\frac{25-9\sqrt5}{22},\,±\frac{15-\sqrt5}{22},\,0\right),$
$\left(±\frac{5-\sqrt5}{6},\,±\frac{3\sqrt5-5}{6},\,0\right),$
$\left(±\frac{\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5-\sqrt5}{4}\right),$
$\left(±\frac{4\sqrt5-5}{11},\,±\frac{4\sqrt5-5}{11},\,±\frac{4\sqrt5-5}{11}\right).$