Fractal animations with Enzyme

This repository contains two complex-plane zoom animations:

• Mandelbrot adapted from a Julia community example
• Newton-fractal implementation that uses Enzyme to differentiate the selected polynomial.

Both animations evaluate an orbit for every pixel, color the result by the number of iterations required to escape or converge, and progressively shrink the complex-plane viewport.

Setup

Instantiate the Julia environment from the repository root:

julia --project=. -e 'using Pkg; Pkg.instantiate()'

Plots renders the heatmaps and GIFs. The Newton implementation additionally uses Enzyme for automatic differentiation.

Mandelbrot animation

mandelbrot.jl samples each pixel as the complex parameter c in

$$z_{n+1} = z_n^2 + c.$$

The escape-time function returns the first iteration for which abs2(z) >= 4. The animation zooms toward -0.5626805 + 0.6422555im, increasing the iteration limit as the viewport gets smaller so that the boundary remains detailed.

Run it with:

julia --project=. mandelbrot.jl

Newton-fractal animation

newton.jl visualizes Newton iteration for the polynomial

$$p(z) = z^8 + 15z^4 - 16.$$

For every pixel, the pixel coordinate supplies an independent starting value z_0. The orbit follows

$$z_{n+1} = z_n - \frac{p(z_n)}{p'(z_n)}$$

until abs(p(z)) falls below the convergence tolerance or the iteration limit is reached. Enzyme computes the complex derivative with forward-mode automatic differentiation. Because p is holomorphic, seeding the derivative with 1 + 0im produces the usual complex derivative p'(z).

The heatmap stores the convergence iteration for every pixel. log1p is applied only to spread those integer counts more evenly across the color palette.

Run the animation with Julia threads enabled:

julia --threads=auto --project=. newton.jl
open newton_fractal.gif

The default animation renders 120 frames at 600 by 600 pixels and zooms around a repelling period-2 point of the Newton map:

0.5309620407446378 + 0.19807575472155833im

Unlike an attracting root, a repelling periodic point lies on the Julia set, so arbitrarily small neighborhoods continue to intersect different attracting basins.

This is what the final image looks like:

Finding 'good' Newton zoom centers

Ironically, finding a good zoom center was harder than I thought. I initially spammed center values, then saw how the animation turned out. A little bit of digging had me solve the equation $N(z) = 0$, where $N(z)$ is the Newton-Raphson iteration expressed.

find_period2_centers.jl reproduces the search used to choose the default Newton zoom center. It numerically solves

$$N^2(z) - z = 0,$$

where N is the Newton map. It then removes the attracting fixed points, deduplicates numerical solutions, and retains cycles whose multiplier satisfies

$$\left|(N^2)'(z)\right| > 1.$$

Print the selected center and its cycle information with:

julia --project=. find_period2_centers.jl

Print every candidate found by the default grid search with:

julia --project=. find_period2_centers.jl --all

The final selection heuristic favors a strongly repelling point away from the real and diagonal symmetry axes. Other reported candidates can be passed as the center keyword to render_newton_animation.