Tridyakis icosahedron

Tridyakis icosahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	m5/3m3m5*a
Elements
Faces	120 scalene triangles
Edges	60+60+60
Vertices	20+12+12
Vertex figure	20 hexagons, 12 decagons, 12 decagrams
Measures (edge length 1)
Inradius	$\frac{15}{8} = 1.875$
Dihedral angle	$\arccos\left(-\frac78\right) ≈ 151.04498°$
Central density	4
Number of external pieces	240
Related polytopes
Dual	Icosidodecatruncated icosidodecahedron
Abstract & topological properties
Flag count	720
Euler characteristic	–16
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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 $5\frac{\sqrt{15}-\sqrt3}{8} ≈ 1.33808$, the medium edges will be $\frac{3\sqrt{15}}{4} ≈ 2.90474$, and the long edges will be $5\frac{\sqrt{15}+\sqrt3}{8} ≈ 3.50315$. The triangles have one interior angle of $\arccos\left(\frac35\right) ≈ 53.13010°$, one of $\arccos\left(\frac13+\frac{4\sqrt5}{15}\right) ≈ 21.62463°$, and one of $\arccos\left(\frac13-\frac{4\sqrt5}{15}\right) ≈ 105.24526°$.

Vertex coordinates

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

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

External links

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