square root and equal division of rectangles. If you want to have a rectangle where you can split it into n of the rectangles scaled down and turned pi/2 radians, the ratio will be 1/sqrt(n). This is beacause if the fourmula 1/x = x/n ; x= sqrt(n) because first you divide x by its self, then by itself again, then you will have its reciprocal. and beacause you divided by itself 2 times, you are dividing by its square, so n = x^2, and x = sqrt(n)

proof in the image (the ratio is 17:12 (roughly ((sqrt(2)):1))