Bugs #23

Bugs Fractal 97

Zoom out by another factor of two. The two shapes alternate over the entire horizontal axis at a spacing of a*\pi.

Recall the formula, bugR(x+yi) = a*sin(x/a) + yi, and whatever you may remember about the sin() function. bugR() is periodic with period 2*a*\pi. bugR(z) = 0 for all z = a*n*\pi, n an integer. For the even multiples, bugR(z+2n*a\pi) \approx z and for old multiples, bugR(z+(2n+1)*a\pi) \approx -\overline{z}. While these observations do not constitute a full proof, it does strongly suggest why the two shapes alternate along the real axis.