A visual compilation of various neat properties that can be uncovered by iterating the inverse counterpart of the Mandelbrot function. The beginning of the video should make it apparent that the inverse function is an alternative way to produce Julia sets — the nested square roots approach the boundary of the c-dependent sets as the iteration depth tends to infinity. Stacking up the different stages of iteration is a nice way of highlighting the inner self-similar structure of the sets.

Because the square root is a multivalued function a binary choice between + and - has to be made during each iteration step. For the overall structure it is enough to choose signs randomly so that on average it appears as if all possible paths are taken, but as the total number of branch paths grows exponentially with iteration depth it quickly becomes untenable to actually evaluate all the paths. For this reason the deeper parts can be seen to fade into darkness as the sample density starts to drop off. However, it is possible to probe individual branch paths by generating a particular sign sequence along with the iteration.

When such square roots are nested all the way to infinity they either converge on a single value (not necessarily from a single direction) or become trapped in a periodically repeating cycle — just like the interior points of the normal Mandelbrot. The periodicity is primarily determined by the length of the repeating sign sequence, but each pattern also has an associated value of c where the dynamics of the nested radicals bifurcate. For the two trivial patterns – all plus or all minus – the bifurcation points correspond with the Mandelbrot main cardioid cusp (c = ¼) and the main cardioid/period-2 bulb tangent point (c = -¾) respectively. Below these points each sequence converges to a single value, but above them the dynamics split: the endpoint of the positive sequence is determined by the imaginary component of z₀, but the negative sequence settles into a cycle between the two conjugate values. The golden ratio also makes an appearance for certain values of c. The dynamics of the nextmost trivial (alternating) sign sequence splits at the period-2/period-4 tangent point located at c = -⁵⁄₄. Below this point the periodicity of the iterated sequence is two and above it, four.

As a cherry on top, each unique sign sequence can be associated with a modular arithmetic arising from certain angle-halving/doubling patterns. The angles are easy to see when observing the dynamics at c = 0 (with slightly non-zero z₀ to nudge the sequence off-center). For example, the all-negative sign sequence stabilizes onto thirds (⅓ and ⅔) and the alternating sign sequence onto fifths (⅕, ⅖, ⅗, ⅘). For the first few sequences the different angle fractions seem to be perfectly enumerated, but at P_1/9 the first gap emerges. As if to avoid repetition the fraction ³⁄₉ that simplifies to the previously covered ⅓ is skipped by the doubling pattern and a similar thing also happens for P_1/15 due to reasons of divisibility. For the next two four-sign patterns after that the modular arithmetic structures split into ¹⁄₁₇ and ³⁄₁₇. Neither can reach the other by angle-doubling alone, so each is associated with its own distinct sign pattern.

When combined, the set of all infinitely nested radicals form the Julia sets associated with the different values of c. For the finitely nested radicals the choice of z₀ determines where the sequence ends up — when started from an interior point the sequence stays within the set, and vice versa for the exterior. Whenever the set of infinitely nested radicals manage to form a continuous loop of numbers the associated value of c is within the Mandelbrot set. At the negative end of the Mandelbrot set (c = -2) the angle-doubling patterns simplify into Chebyshev recurrences (of the first kind) and all convergence points (all purely real) can be expressed as cosines. It just so happens that I've run into this same numerical structure before, only from a different direction. At c = 0 the dynamics are likewise quite straightforward. For these two special cases it is easy to demonstrate the continuity, but elsewhere it becomes a bit trickier.

When a gap does emerge between adjacent "radical fractions" (eg. P_k and P_k±ε) the boundary necessarily becomes disconnected. When there is no longer a clear separation between the interior and exterior the previously filled Julia set turns to dust thus making the associated c not part of the Mandelbrot set.

You can find more Mandelroot renders here.

The accompanying music is 'Touching Infinity' by Sergii Pavkin.