Triakis icosahedron

Triakis icosahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Bowers style acronym	Tiki
Coxeter diagram	m5m3o ()
Conway notation	kI
Elements
Faces	60 isosceles triangles
Edges	30+60
Vertices	12+20
Vertex figure	12 decagons, 20 triangles
Measures (edge length 1)
Dihedral angle	$\arccos\left(-\frac{24+15\sqrt5}{61}\right) ≈ 160.61255^\circ$
Central density	1
Number of external pieces	60
Level of complexity	3
Related polytopes
Army	Tiki
Regiment	Tiki
Dual	Truncated dodecahedron
Conjugate	Great triakis icosahedron
Abstract & topological properties
Flag count	360
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Nature	Tame

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 $\arccos\left(\frac{15+\sqrt5}{20}\right) ≈ 30.48032°$.

External links

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