# Tridyakis icosahedron

Tridyakis icosahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm5/3m3m5*a
Elements
Faces120 scalene triangles
Edges60+60+60
Vertices12+12+20
Vertex figure20 hexagons, 12 decagons, 12 decagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {15}{8}}=1.875}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {7}{8}}\right)\approx 151.04498^{\circ }}$
Central density4
Number of external pieces240
Related polytopes
DualIcosidodecatruncated icosidodecahedron
ConjugateTridyakis icosahedron
Convex coreNon-Catalan disdyakis triacontahedron
Abstract & topological properties
Flag count720
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

## Vertex coordinates

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

• ${\displaystyle \left(\pm 3{\frac {5+{\sqrt {5}}}{8}},\,\pm {\frac {3{\sqrt {5}}}{4}},\,0\right),}$
• ${\displaystyle \left(\pm 5{\frac {{\sqrt {5}}-1}{8}},\,\pm 5{\frac {1+{\sqrt {5}}}{8}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {3{\sqrt {5}}}{4}},\,\pm 3{\frac {5-{\sqrt {5}}}{8}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {5}{4}},\,\pm {\frac {5}{4}},\,\pm {\frac {5}{4}}\right).}$