Fractals: detail at every scale

Start with the Mandelbrot view. Each point in the picture represents a complex number $c = x + iy$: its horizontal position gives x, and its vertical position gives y. For each point, begin with $z_0 = 0$, square it, add c, and repeat:

The sequence of values is called an orbit. Some orbits stay within a bounded region; others eventually grow without limit. The Mandelbrot set contains exactly the values of c whose orbits stay bounded forever. Its familiar shape is a map of which choices of c have that property.

The colours around the set show how quickly orbits escape. Nearby points often escape at similar times, producing the ribbons of colour that trace the boundary. Changing the palette changes those colours while keeping the underlying shape.

A broad view hides how complicated this boundary is. Move towards it and smooth-looking curves separate into filaments. Follow a filament and you may find a spiral, a branching junction or a miniature Mandelbrot set. Familiar shapes recur, but their surroundings change as you move through the picture.

This structure across scales is what makes a fractal rewarding to explore. Try Seahorse valley for tightly curled filaments, or Mini Mandelbrot to find a small version of the whole set. Aim near an edge: the middle of a solid region usually gives you much less to see.

In the Mandelbrot picture, each pixel supplies a different c and every orbit starts at zero. A Julia picture fixes c for the whole image and uses each pixel as the starting value instead:

Now the picture asks which starting points stay bounded under that particular rule. Changing c can turn a connected shape into a scattering of separate pieces. The Mandelbrot set acts as a guide to this family: values of c inside it give connected filled Julia sets; values outside it give disconnected ones.

Choose Julia in Controls and try the Rabbit and Dendrite presets. Then adjust the real or imaginary part of c a little at a time and watch how the shape changes.

Multibrot replaces the square with a higher power. Raising z to the power d produces a shape with d − 1 matching lobes around its centre. Try powers 3 and 4, then follow an edge to see how their smaller structures differ.

Burning Ship takes the absolute value of both components of z before squaring. That fold changes the rounded Mandelbrot landscape into a more angular one. The Small ship preset is a useful place to begin.

Tricorn changes the sign of the imaginary component before squaring, using the complex conjugate of z. The result has three prominent lobes and a different arrangement of folds and filaments.

Newton's method is a way to solve $p(z) = 0$. Start from a guess and use the polynomial and its derivative to make a new one:

The Newton view runs this calculation from every pixel. Each root gets a colour, and the starting points that converge to it form its basin of attraction. Shading within each basin reflects how many steps the calculation took.

Close to a root, nearby guesses usually reach the same answer. Near a basin boundary, a tiny change in the starting point can send the calculation towards a different root. Zoom into these boundaries to find the colours interwoven at smaller and smaller scales.

Drag the root handles to reshape the basins, or enter a polynomial of degree 2 to 8, such as z^5 − 1. For a different outcome, try the Cycle preset: some starting points approach a repeating orbit instead of a root and remain dark.

A computer can only follow an orbit for a finite number of steps. In the escape-time views, a dark pixel means the orbit has not escaped within that limit. Some of those points belong to the set; others simply need more time to escape. Increase Iterations near a boundary and watch whether dark patches resolve into coloured detail.

Three settings affect how much detail you can see:

• Iterations gives each orbit more time. Automatic iterations adjusts the limit as you explore; a manual increase can reveal points that escape unusually slowly.
• Display detail sets how many pixels the CPU renderer calculates for the preview. Native gives the finest display; lower settings finish sooner.
• Precision keeps nearby coordinates distinct as you zoom. Leave it on Automatic to let the explorer choose an appropriate numerical method.

Deep views can take time to finish. Let the picture settle after a move: an initial preview gives way to finer samples, and the view details panel shows the rendering progress.

At a deep enough zoom, two neighbouring pixels have coordinates that differ only after many decimal places. Ordinary arithmetic eventually rounds them to the same value. More pixels alone cannot solve that problem: the calculation needs greater numerical precision.

For Mandelbrot, Julia and Multibrot, the explorer uses perturbation rendering. It calculates a reference orbit with high precision, then follows the small differences between nearby orbits and that reference. For Mandelbrot, those differences obey:

Here Z is the reference orbit, δz is the difference from it, and δc is the difference between the two positions. Tracking these differences makes very deep views practical without repeating the full high-precision calculation at every pixel. The Deep tip and Deep spiral presets let you try this directly.

Scroll over a detail to zoom towards it, pinch on a touchscreen, or drag to move around. Double-click a point to centre it and zoom in. To follow a detail continuously, bring it to the centre and press Auto zoom.

With the picture focused, use the arrow keys to pan, plus and minus to zoom, Space to pause or resume Auto zoom, and R to reset. Close the coordinates and zoom panel with its × button; the Info button in the bottom bar brings it back.

Press Share view to copy a link that records the fractal, position, zoom and rendering settings, including the exact coordinates of deep views. The link opens paused so someone else can explore from the same place. Your address also updates as you move, and this browser remembers your settings for the next visit.

Choose Export image to render a picture without the controls. Screen, HD, 4K, 8K and 16K keep the current framing; the resolution sets the longest edge, with 16K giving 15,360 pixels. Export resolution is independent of Display detail.

PNG preserves the rendered pixel values. JPEG and WebP offer adjustable compression for smaller files. Large images take longer to render, with a progress indicator and a Cancel button. The download uses the view captured when you press Download.

A saved image contains a fixed amount of detail. To explore beyond it, return to the explorer, zoom into the area you want and render a new picture.