Great dodecacronic hexecontahedron

Great dodecacronic hexecontahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	m5/3m5/2o3*a
Elements
Faces	60 kites
Edges	60+60
Vertices	20+12+12
Vertex figure	20 triangles, 12 pentagrams, 12 decagrams
Measures (edge length 1)
Inradius
Dihedral angle
Central density	10
Number of external pieces	300
Related polytopes
Dual	Great dodecicosidodecahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–16
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great dodecacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 kites.

If its dual, the great dodecicosidodecahedron, has an edge length of 1, then the short edges of the kites will measure $2\frac{\sqrt{2\left(5-2\sqrt5\right)}}{3} ≈ 0.68499$ , and the long edges will be $2\frac{\sqrt{65-19\sqrt5}}{11} ≈ 0.86272$ . The kite faces will have length $\frac{\sqrt{10\left(157-31\sqrt5\right)}}{33} ≈ 0.89731$ , and width $\sqrt5-1 ≈ 1.23607$ . The kites have two interior angles of $\arccos\left(\frac58-\frac{\sqrt5}{8}\right) ≈ 69.78820°$ , one of $\arccos\left(-\frac14+\frac{\sqrt5}{10}\right) ≈ 91.51239°$ , and one of $\arccos\left(-\frac18-\frac{9\sqrt5}{40}\right) ≈ 128.91121°$ .

Vertex coordinates[edit | edit source]

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

$\left(±\frac{\sqrt5-1}{2},\,±\frac{3-\sqrt5}{2},\,0\right),$
$\left(±\frac{25-9\sqrt5}{22},\,±\frac{15-\sqrt5}{22},\,0\right),$
$\left(±\frac{5-\sqrt5}{6},\,±\frac{3\sqrt5-5}{6},\,0\right),$
$\left(±\frac{4\sqrt5-5}{11},\,±\frac{4\sqrt5-5}{11},\,±\frac{4\sqrt5-5}{11}\right).$