// sketch

Mandelbrot Explorer

Infinite complexity from a two-line equation.

Controls

› Double-click — zoom in
› Shift + double-click — zoom out
› + / − buttons — fine zoom
› R key — reset view
› Play — animate Julia sets

The Equation

Everything you see is generated by a single recurrence relation applied millions of times:

z_n+1 = z_n² + c

Here z and c are complex numbers — pairs of values (real, imaginary) that live on a 2D plane. Each pixel on screen maps to one value of c. Starting from z₀ = 0, we repeatedly square and add c. Two things can happen:

▸Stays bounded — the sequence never escapes to infinity. That point belongs to the Mandelbrot set (rendered dark).
▸Escapes — |z| grows past 2 and races to ∞. Colour = how many iterations it took to escape (the escape time).

Not an image — live maths

There is no pre-rendered texture here. Every pixel you see was calculated by your machine just now, using the iteration formula above — the CPU running millions of arithmetic operations to decide the colour of each point.

When you zoom in, the browser re-runs the entire calculation at the new coordinates, at full floating-point precision. The render progress bar at the bottom reflects real computation time — deeper zooms or higher iteration counts take measurably longer.

// this is what GPU shaders were invented to parallelise

Why It Matters

Beyond aesthetics, the same iterative mathematics appears in modelling coastlines, protein folding, financial volatility, and antenna design. The Mandelbrot set is the canonical example that simple deterministic rules can produce boundless complexity — a foundational idea in chaos theory and dynamical systems.

Why Complex Numbers?

A complex number c = a + bi has a real part a (horizontal axis) and an imaginary part bi (vertical axis). Squaring a complex number is just algebra: (a + bi)² = a² − b² + 2abi. The imaginary unit i satisfies i² = −1.

In plain terms: each pixel is a point on a 2D grid. The equation folds, stretches, and rotates that grid in a way that produces infinite self-similar structure — the same swirls and tendrils appear no matter how far you zoom.

Self-Similarity & Fractals

The boundary of the Mandelbrot set has infinite detail. Zoom anywhere on the edge and you find smaller copies of the whole set, decorated with spirals, filaments, and bulbs — all arising from z² + c.

Mandelbrot called structures like this fractals — objects whose complexity does not simplify as you zoom in. The set has a fractal dimension between 1 and 2 (approximately 2 at the boundary).

Julia Sets

Every point c in the Mandelbrot set has a corresponding Julia set. Whereas the Mandelbrot set fixes z₀ = 0 and asks which values of c stay bounded, a Julia set fixes c and asks which starting values of z₀ stay bounded:

Mandelbrot: z₀ = 0, vary c
Julia: c fixed, vary z₀

Hit Play to see this in action. The animation fixes c at a point tracing the Mandelbrot boundary (c = 0.7885·e^iθ) while θ slowly rotates. Each scan is a different Julia set — you watch the fractal breathe and morph as c travels.

Points inside the Mandelbrot set produce connected Julia sets (one solid piece). Points outside produce Cantor dust — infinitely many disconnected fragments. The Mandelbrot set is literally the atlas of all Julia sets.