Beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the nth polygon is the circumcircle of the (n + 1)st and successive polygons are rotated one half-turn at each iteration. The results are shown above for nested triangles through heptagons in alternating black and white and in a cyclic rainbow coloring. The animation above shows successive iterations of a nested octagon. The black region of a nested square has area A = sum_(n = 0)^∞ (-1)^n/2^n = 2/3 if the initial square has unit edge length.
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