Great triakis icosahedron

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

Great triakis icosahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	m5/3m3o
Elements
Faces	60 isosceles triangles
Edges	30+60
Vertices	20+12
Vertex figure	20 triangles, 12 decagrams
Measures (edge length 1)
Inradius	$5\frac{\sqrt{61\left(41−18\sqrt5\right)}}{122} ≈ 0.27735$
Dihedral angle	$\arccos\left(3\frac{5\sqrt5-8}{61}\right) ≈ 81.00141°$
Central density	13
Number of external pieces	420
Related polytopes
Dual	Quasitruncated great stellated dodecahedron
Abstract & topological properties
Flag count	360
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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

Vertex coordinates

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

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

External links

• Wikipedia Contributors. "Great triakis icosahedron".
• McCooey, David. "Great Triakis Icosahedron"