1. The Mandelbrot set

   The set of complex numbers c for which the iteration z_{n+1} = z_n² + c (starting from z₀ = 0) does not diverge. First visualized by Benoit Mandelbrot in 1980, its boundary is a fractal of Hausdorff dimension 2.

   • Wikipedia — Mandelbrot set

2. Perturbation theory

   Deep zoom renders one high-precision reference orbit at c_ref, then integrates each pixel as a small double-precision delta from that orbit—far cheaper than arbitrary precision per pixel.

   • K. I. Martin — Superfractalthing Maths (PDF, archived)
   • Wikipedia — Plotting algorithms (perturbation & series approximation)
   • mathr — Deep zoom theory and practice

3. Series approximation

   Coefficients along the reference orbit approximate the perturbation as a polynomial in ε = c_pixel − c_ref, skipping early iterations when |ε| is tiny.

   • Same Wikipedia section (series approximation)

4. Interior orbits & cycle detection

   When the reference point stays in the set, Brent-style cycle detection spots repetition in the orbit (checked on double-precision samples) so the BigFloat loop can stop early; the orbit is then tiled to full length for workers.

   • Wikipedia — Cycle detection (Brent's algorithm)

5. Arbitrary precision & pixel scale

   Below about 10⁻¹³ viewport height in the complex plane, adjacent pixels differ by less than IEEE doubles can represent; the app switches pipelines and uses BigFloat for the reference orbit with controlled truncation.

   • Wikipedia — Plotting algorithms (overview)

6. Smooth coloring

   Escape counts are turned into a continuous value using the normalized iteration formula so palette bands do not line up with integer iteration steps.

   • Wikipedia — Mandelbrot set (continuous / smooth coloring)

7. Standard pipeline shortcuts

   At shallower zoom, the app uses direct double-precision iteration with analytical tests for the main cardioid and period-2 bulb to skip known interior regions.

   • Wikipedia — Plotting algorithms (optimization)