Escher and Mandelbrot

If you take a complex number z with argument θ and square it, you double θ. The Mandelbrot/Julia iteration

z ↦ z² + c

does pretty much the same thing, but adds wiggles to the curve. Since the iterations stop when |z| > 2, the boundary at the zeroth iteration is a circle; after the first it’s a pear shape, and so on. We can map any point in the region between bands to a point in a rectangular tile that’s periodic once along the outside edge and twice along the inside edge. Here’s a site with a few different examples.

The transformation Escher used in “Print Gallery” takes concentric circles at r=1/rⁿ to a logarithmic spiral. The concentric boundaries between iterations for the Julia set at c = 0 are circles. There ought to be a transformation for Mandelbrot / Julia sets similar to the Droste effect but spiraling inward so that the frequency is smoothly increasing just as the distance can be made to smoothly increase.