Bugs #23

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.

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