View source for Icosahedron

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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

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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050020a.gif" />

Figure: i050020a

====Comments====
The regular polytopes are also called the [[Platonic solids|Platonic solids]].

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Regular polytopes" , Dover, reprint  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Klein,   "Lectures on the icosahedron and the solution of equations of the fifth degree" , Dover, reprint  (1956)  (Translated from German)</TD></TR></table>