# Great disdyakis triacontahedron

Great disdyakis triacontahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm5/3m3m
Elements
Faces120 scalene triangles
Edges60+60+60
Vertices30+20+12
Vertex figure30 squares, 20 hexagons, 12 decagrams
Measures (edge length 1)
Inradius$3\frac{\sqrt{1205\left(39−16\sqrt5\right)}}{241} ≈ 0.77575$ Dihedral angle$\arccos\left(-\frac{179−24\sqrt5}{241}\right) ≈ 121.33625°$ Central density13
Number of external pieces120
Related polytopes
DualGreat quasitruncated icosidodecahedron
Abstract & topological properties
Flag count720
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

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

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