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

Recall the formula, , and whatever you may remember about the sin() function. bugR() is periodic with period . bugR(z) = 0 for all , n an integer. For the even multiples, and for old multiples, . While these observations do not constitute a full proof, it does strongly suggest why the two shapes alternate along the real axis.