# Triakis icosahedron

Triakis icosahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Bowers style acronymTiki
Coxeter diagramm5m3o (     )
Conway notationkI
Elements
Faces60 isosceles triangles
Edges30+60
Vertices12+20
Vertex figure12 decagons, 20 triangles
Measures (edge length 1)
Dihedral angle$\arccos\left(-\frac{24+15\sqrt5}{61}\right) ≈ 160.61255^\circ$ Central density1
Number of external pieces60
Level of complexity3
Related polytopes
ArmyTiki
RegimentTiki
DualTruncated dodecahedron
ConjugateGreat triakis icosahedron
Abstract & topological properties
Flag count360
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

The triakis icosahedron is one of the 13 Catalan solids. It has 60 isosceles triangles as faces, with 12 order-10 and 20 order-3 vertices. It is the dual of the uniform truncated dodecahedron.

It can also be obtained as the convex hull of a dodecahedron and an icosahedron, where the edges of the icosahedron are $\frac{15+\sqrt5}{10} ≈ 1.72361$ times the length of those of the dodecahedron. Using an icosahedron that is any number more than $\frac{1+\sqrt5}2 ≈ 1.61803$ times the edge length of the dodecahedron gives a fully symmetric variant of this polyhedron. The upper limit is $3\frac{\sqrt5-1}{2}$ , where the dodecahedron's vertices coincide with the icosahedron's face centers.

Each face of this polyhedron is an isosceles triangle with base side length $\frac{15+\sqrt5}{10} ≈ 1.72361$ times those of the side edges. These triangles have apex angle $\arccos\left(-3\frac{1+\sqrt5}{20}\right) ≈ 119.03935°$ and base angles $\arccos\left(\frac{15+\sqrt5}{20}\right) ≈ 30.48032°$ .