A fractal explorer with a Rust tile server

Fractal Explorer lets you pan and zoom around the Mandelbrot set, Julia sets, the Burning Ship and the Tricorn the way you would use a web map. That is not a coincidence: it is built on the same idea as a map.

Fractals as map tiles

A web map never loads one huge picture. It loads small square tiles for the part you are looking at, at the zoom level you are looking at. The fractal explorer does exactly that, with Leaflet in the browser and a Rust (Axum) server that, instead of looking tiles up, calculates them.

A request for tile /tile/z/x/y works out which rectangle of the complex plane that tile covers (4 divided by 2^z wide), then runs the escape-time calculation for each of its 256×256 pixels: repeat z = z² + c, count how many steps it takes to blow up, and colour the pixel by that count. The result is encoded as a PNG and sent back.

What makes it look good

• Smooth colour: plain iteration counts produce visible bands. The server uses the normalised iteration count, which interpolates between steps and gives smooth gradients.
• Palettes: colours come from cosine palettes, a + b·cos(2π(c·t + d)), which is a tiny formula with a lot of variety.
• Detail control: the number of iterations grows with zoom level (150 + 45 × zoom, capped at 30,000) and a Detail slider multiplies it. Deep zooms look blobby when there are too few iterations, so you turn it up.
• Julia sets: right-click a point on the Mandelbrot set to open the Julia set for that point.
• Shareable links: “Copy link” stores the position, fractal and colours in the URL.

Why Rust, and why nginx caches the result

Every tile is 65,536 pixels, each iterated up to thousands of times in floating-point maths, which is the kind of work where a compiled language earns its keep. The release build uses full optimisation and link-time optimisation, so the first Docker build is slow, but the service itself is small: Axum, Tokio, and the png crate.

Tiles never change for a given fractal and settings, so nginx caches them on disk for 30 days. Popular areas become instant after the first visitor, and the Rust service only does work for tiles nobody has asked for yet.

The limit you hit eventually

The zoom stops at about 2⁴⁶, roughly 70 trillion times. That is not an arbitrary cap: 64-bit floating-point numbers run out of precision there, and adjacent pixels start mapping to the same number, so the image turns into blocks. Going deeper needs a different technique (arbitrary precision, or perturbation methods). I chose to stop at the limit of the simple approach.