Putnam 1984 B6

A sequence of convex polygons $(P_n),n\ge0,$ is defined inductively as follows. $P_0$ is an equilateral triangle with side length $1$ . Once $P_n$ has been determined, its sides are trisected; the vertices of $P_{n+1}$ are the interior trisection points of the sides of $P_n$ . Express $\lim_{n\to\infty}[P_n]$ in the form $\frac{\sqrt a}b$ , where $a,b$ are integers.