# Tridyakis icosahedron

Tridyakis icosahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm5/3m3m5*a
Elements
Faces120 scalene triangles
Edges60+60+60
Vertices20+12+12
Vertex figure20 hexagons, 12 decagons, 12 decagrams
Measures (edge length 1)
Inradius${\displaystyle \frac{15}{8} = 1.875}$
Dihedral angle${\displaystyle \arccos\left(-\frac78\right) ≈ 151.04498°}$
Central density4
Number of external pieces240
Related polytopes
DualIcosidodecatruncated icosidodecahedron
Abstract & topological properties
Flag count720
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The tridyakis icosahedron is a uniform dual polyhedron. It consists of 120 scalene triangles.

If its dual, the great cubicuboctahedron, has an edge length of 1, then the triangle faces' short edges will be ${\displaystyle 5\frac{\sqrt{15}-\sqrt3}{8} ≈ 1.33808}$, the medium edges will be ${\displaystyle \frac{3\sqrt{15}}{4} ≈ 2.90474}$, and the long edges will be ${\displaystyle 5\frac{\sqrt{15}+\sqrt3}{8} ≈ 3.50315}$. The triangles have one interior angle of ${\displaystyle \arccos\left(\frac35\right) ≈ 53.13010°}$, one of ${\displaystyle \arccos\left(\frac13+\frac{4\sqrt5}{15}\right) ≈ 21.62463°}$, and one of ${\displaystyle \arccos\left(\frac13-\frac{4\sqrt5}{15}\right) ≈ 105.24526°}$.

## Vertex coordinates

A tridyakis icosahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

• ${\displaystyle \left(±3\frac{5+\sqrt5}{8},\,±\frac{3\sqrt5}{4},\,0\right),}$
• ${\displaystyle \left(±5\frac{\sqrt5-1}{8},\,±5\frac{1+\sqrt5}{8},\,0\right),}$
• ${\displaystyle \left(±\frac{3\sqrt5}{4},\,±3\frac{5-\sqrt5}{8},\,0\right),}$
• ${\displaystyle \left(±\frac54,\,±\frac54,\,±\frac54\right).}$