Fractal Generator & Mandelbrot Explorer

The rule is short enough to write on a napkin: take a number, square it, add the number you started with, and repeat. Do that for every point on the plane. Some points stay small no matter how many times you go round — those form the black body of the set. The rest escape towards infinity, and how quickly they go is what paints the colours.

That is genuinely all of it. The endless complication at the edge is not put there by anyone; it falls out of two operations repeated.

Scroll or pinch to zoom, and the point under your cursor stays put — which is what makes it possible to chase a feature inward instead of losing it after two steps.

The detail is genuinely infinite, but the arithmetic is not. Graphics cards work with numbers carrying about seven significant digits, and once one pixel is narrower than that, neighbouring pixels round to the same value and the image goes blocky. Rather than pretend otherwise, the zoom indicator says when you are nearing that limit and when you have reached it.

A Julia set uses the same equation with the roles reversed: one fixed constant for the whole picture, and each pixel a different starting point. There is a Julia set for every single point of the Mandelbrot, which is a remarkable fact and easy to see for yourself — turn on the point picker and click anywhere.

Points deep inside the black body give solid blobs, points far outside give scattered dust, and points right on the boundary give the intricate ones. That is not a coincidence: it is the same boundary doing the same work.

Multibrot raises z to a higher power — the number of bulbs around the body is always one less than the power. Burning Ship takes the absolute value of each part before squaring, which breaks the symmetry and gives hard, angular structures.

Newton is different in kind. It runs Newton’s method for solving z³ = 1 from every starting point and colours by which of the three answers it reaches. The boundaries have a startling property: all three colours meet at every single boundary point, at every magnification.

Eight palettes, each chosen to hold up at both a shallow and a deep zoom. Smooth colouring removes the hard banding; turning it off puts the bands back, which is a good way to see the escape-time structure directly. Colour cycling animates the palette without recalculating anything, so it stays smooth however deep you are.

Save at 1080p, 1440p, 4K, square or phone proportions. The image is re-rendered at that size rather than scaled up, so a 4K wallpaper is genuinely 4K.

What is the Mandelbrot set?

Take a number, square it, add your starting number, and repeat. For some starting numbers the result stays small forever; for others it races off to infinity. The Mandelbrot set is the collection of numbers that stay small — the black interior. The colours around it show how quickly the others escaped.

Why does the detail never run out?

Because the boundary genuinely is infinitely intricate: at every magnification there is new structure, and small copies of the whole shape appear again and again. What does run out is the computer's precision. This explorer says so plainly when you approach that limit rather than letting the picture quietly go blocky.

What is a Julia set?

The same rule with the roles swapped. In the Mandelbrot, each pixel is a different value of c; in a Julia set, c is one fixed number and each pixel is a different starting point. Every point of the Mandelbrot has its own Julia set, and the ones from near the edge are the most beautiful — which is what the point picker is for.

What does the detail slider do?

It sets how many times the calculation repeats before giving up. Too low and deep zooms turn into flat blobs, because the detail is simply never computed. The Auto button picks a sensible number for how far in you are.

What is smooth colouring?

The escape count is a whole number, so colouring by it directly produces hard concentric bands. Smooth colouring works out the fractional part of the escape and blends between them. Turn it off to see the bands, which are genuinely useful for understanding the structure.

What is the Burning Ship?

The same equation with the absolute value of each part taken before squaring. That one change breaks the symmetry and produces sharp, angular structures instead of round ones. Zooming into the smaller body on the left is where it earns its name.

Can I make a wallpaper?

Yes. Choose a size — 1080p, 1440p, 4K, square or phone-shaped — and save a PNG. The image is rendered fresh at that resolution rather than being scaled up, so it is sharp at full size.

Can I share a view I found?

Copy the share link. It carries the exact coordinates, zoom, palette and settings, so whoever opens it lands in precisely the same place. Nothing is uploaded — the whole view is encoded in the link itself.

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