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

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