# Great triakis icosahedron

Great triakis icosahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm5/3m3o
Elements
Faces60 isosceles triangles
Edges30+60
Vertices20+12
Vertex figure20 triangles, 12 decagrams
Measures (edge length 1)
Inradius${\displaystyle 5\frac{\sqrt{61\left(41−18\sqrt5\right)}}{122} ≈ 0.27735}$
Dihedral angle${\displaystyle \arccos\left(3\frac{5\sqrt5-8}{61}\right) ≈ 81.00141°}$
Central density13
Number of external pieces420
Related polytopes
DualQuasitruncated great stellated dodecahedron
Abstract & topological properties
Flag count360
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great triakis icosahedron is a uniform dual polyhedron. It consists of 60 isosceles triangles.

If its dual, the quasitruncated great stellated dodecahedron, has an edge length of 1, then the short edges of the triangles will measure ${\displaystyle 5\frac{7-\sqrt5}{22} ≈ 1.08271}$, and the long edges will be ${\displaystyle \frac{5-\sqrt5}{2} ≈ 1.38197}$. The triangles have two interior angles of ${\displaystyle \arccos\left(\frac34-\frac{\sqrt5}{20}\right) ≈ 50.34252°}$, and one of ${\displaystyle \arccos\left(-\frac{3}{20}+\frac{3\sqrt5}{20}\right) ≈ 79.31495°}$.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac{5-\sqrt5}{4},\,±\frac{3\sqrt5-5}{4},\,0\right),}$
• ${\displaystyle \left(±5\frac{13-5\sqrt5}{44},\,±5\frac{7-\sqrt5}{44},\,0\right),}$
• ${\displaystyle \left(±5\frac{2\sqrt5-3}{22},\,±5\frac{2\sqrt5-3}{22},\,±5\frac{2\sqrt5-3}{22}\right).}$