Great disdyakis triacontahedron

Great disdyakis triacontahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	m5/3m3m
Elements
Faces	120 scalene triangles
Edges	60+60+60
Vertices	30+20+12
Vertex figure	30 squares, 20 hexagons, 12 decagrams
Measures (edge length 1)
Inradius
Dihedral angle
Central density	13
Number of external pieces	120
Related polytopes
Dual	Great quasitruncated icosidodecahedron
Abstract & topological properties
Flag count	720
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The great disdyakis triacontahedron is a uniform dual polyhedron. It consists of 120 scalene triangles.

If its dual, the great quasitruncated icosidodecahedron, has an edge length of 1, then the triangle faces' short edges will measure $3\frac{\sqrt{15\left(65-19\sqrt5\right)}}{55} ≈ 1.00239$ , the medium edges will be $2\frac{\sqrt{15\left(5+\sqrt5\right)}}{5} ≈ 4.16732$ , and the long edges will be $\frac{\sqrt{15\left(85+31\sqrt5\right)}}{11} ≈ 4.37382$ . The kites have one interior angle of $\arccos\left(\frac16+\frac{\sqrt5}{15}\right) ≈ 71.59464°$ , one of $\arccos\left(\frac34+\frac{\sqrt5}{10}\right) ≈ 13.19300°$ , and one of $\arccos\left(\frac38-\frac{5\sqrt5}{24}\right) ≈ 95.21236°$ .

Vertex coordinates[edit | edit source]

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

$\left(±3\frac{4\sqrt5-5}{11},\,0,\,0\right),$
$\left(±\frac{5-\sqrt5}{2},\,±\frac{5+\sqrt5}{2},\,0\right),$
$\left(±3\frac{5-\sqrt5}{10},\,±3\frac{3\sqrt5-5}{10},\,0\right),$
$\left(±3\frac{25-9\sqrt5}{44},\,±3\frac{4\sqrt5-5}{22},\,±3\frac{15-\sqrt5}{44}\right),$
$\left(±\sqrt5,\,±\sqrt5,\,±\sqrt5\right).$