Icosahedron

One of the five regular polytopes. An icosahedron has 20 (triangular) faces, 30 edges and 12 vertices (at each of which 5 edges meet). If $a$ is the length of an edge of the icosahedron, then its volume is given by

$$V=\frac{5}{12}a^3(3+\sqrt5)\cong2.1817a^3.$$

Figure: i050020a

Comments

The regular polytopes are also called the Platonic solids.

The symmetry group of the icosahedron plays a role in various branches of mathematics, and led F. Klein to his famous book [a2].

References

[a1]	H.S.M. Coxeter, "Regular polytopes" , Dover, reprint (1973)
[a2]	F. Klein, "Lectures on the icosahedron and the solution of equations of the fifth degree" , Dover, reprint (1956) (Translated from German)