This is a division of the unit circle based on the intersection of the functions y=x^2, y=-x^2, x=y^2, and x=-y^2; and the unit circle.
The series become an infinite series once the value of r is derived from the set of positive real numbers. At certain values of r, the intersect points between the functions and the unit circle invert. For instance at r=2^(1/2), the intersection of the squared functions and the circle create an equivalent square to resultant square of the intersection of the x and y axis and the circle at r=2^(1/2). This is due to the functions relationship with the points (1,1), (1,-1), (-1,1), and (-1,-1); which is where the squared functions themselves intersect. This reduces the number of intersect points of the circle at r=2^(1/2) and the squared functions from 8 total to 4 total.