# Icosahedron

From Encyclopedia of Mathematics

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) |

**How to Cite This Entry:**

Icosahedron.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Icosahedron&oldid=31504