sum(c_j e^(i j t)) draws a closed curve in the complex plane as 0<=t<=2 pi. By Green's thm, the enclosed area (weighted by winding number) is pi sum(j |c_j|^2), which is interestingly independent of the phase of c_j. Applying this to the Fourier series for Snowflake-like fractals arranged around a regular m-gon yields the identity inf 1 /===\ %pi (- + k) | | m 2 3 2 s %pi inf ( | | (1 - s tan(-----------))) %pi (------ + cot(---)) ==== | | n 2 m \ n = 1 (-2) s + 3

--------------------------------- = ------------------------ .

/ 1 3 2 %pi ==== (- + k) sin (---) k = - inf m m The boundary dimension is 1/(1-lg(1+s^2)/2). The standard snowflake is m=3, s=1/sqrt(3), or m=6, s=-1/sqrt(3). This differs from my previous strange identities in that the (rapidly convergent) infinite product is squared. As s -> 1 (spacefilling), the sum converges extremely slowly. In the spacefilling (s=1) limit, this formula seems to ascribe half the spacefilled area to the interior. In other words, it gives an area midway between the s=0 m-gon and the s=1 2m-gram obtained by erecting isosceles right triangles on the sides of the m-gon. --rwg