Circle Inversion

Circle Inversion trades familiar quadratic growth for a reciprocal quadratic step: take 1/z², then shift it by c. It looks like a small swap. The behavior is not.

In ordinary polynomial iteration, once an orbit grows beyond a certain size it keeps heading away from the origin. Here that logic turns inside out: near zero becomes enormous, and an enormous value comes back near zero. Then c decides where that exchanged value lands.

The picture is radial, with nested rings and sharp axial traces. The origin is a pole of order 2—a point where the map is undefined—and its pull reaches across the parameter plane.

The map z → 1/z² + c sits near the broader rational family known as McMullen maps, z → zⁿ + λ/zⁿ. Curt McMullen introduced that family in the late 1980s, showing rational maps whose parameter spaces contain several kinds of hyperbolic components, including ones with Sierpiński-carpet Julia sets.

With n = 2, λ = 1, and a translation parameter c, this gives the Circle Inversion map. Its name comes from 1/z², which inverts the complex plane with respect to the unit circle and doubles the angle. Robert L. Devaney and others later studied the McMullen family, tracing how escape loci and Julia sets change with parameters. FractalPark renders exactly this map — the n = 2, λ = 1 case with c as the translation parameter.

Concentric bands and radial spines run outward from the pole at the origin. The bands mark regions whose orbits stay bounded or escape at different rates, making nested rings like a target or the ripples from a dropped pebble.

Sharp axial structures form distinct spokes. They come from the inversion’s angular part: 1/z² doubles z’s angle, creating preferred directions where the dynamics line up. The reciprocal quadratic form gives this parameter plane a more rigid radial symmetry than most.