Fractal Explorer (Mandelbrot & Julia)
Explore the Mandelbrot and Julia sets right in your browser. Zoom into the infinite detail of z = z² + c with click-to-zoom, pan, smooth coloring, and adjustable iteration depth — then save your view as a PNG.
How to Use
- Pick a set: Mandelbrot for the classic map of escape behaviour, or Julia for a single fixed parameter c.
- For Julia, type a complex constant c as its real (cRe) and imaginary (cIm) parts — try (-0.8, 0.156) for a spiral.
- Click anywhere on the canvas to zoom in 2× on that point; shift-click or right-click to zoom back out. Use the mouse wheel to zoom toward the cursor, and drag to pan.
- Raise the iterations slider to reveal finer detail as you zoom deeper; lower it if rendering feels slow.
- Switch the colour scheme to taste, then use Save PNG to download the current view, or Reset view to return home.
The maths behind the picture
Both fractals here come from iterating a single, deceptively simple rule on complex numbers:
A complex number z = x + iy is squared with z² = (x² − y²) + (2xy)i. Starting from some z₀ we apply the rule over and over. If the magnitude |z| ever exceeds 2 (so |z|² > 4) the sequence is guaranteed to run off to infinity, and we say the point escapes. Points that never escape after the iteration limit are drawn black — they are (approximately) inside the set.
Mandelbrot set: for each pixel we treat its coordinate as the constant c, start z at 0, and ask whether z = z² + c stays bounded. The black region is the Mandelbrot set itself; the glowing colours outside record how many steps each escaping point survived.
Julia sets: we instead fix one constant c (the cRe and cIm inputs) and let each pixel be the starting value z₀. Different choices of c give wildly different Julia sets — connected blobs when c lies inside the Mandelbrot set, scattered dust when it lies outside. That is why the two views are deeply linked.
The smooth-colouring formula turns the integer escape count into a continuous value, so the colour bands blend instead of stair-stepping. Everything runs per-pixel in plain JavaScript on the 2D canvas, entirely on your device.
Reference card
About the Fractal Explorer (Mandelbrot & Julia)
Fractal Explorer (Mandelbrot & Julia) is a quick, free tool for everyday maths and number work. It works in your browser and keeps everything on your device. Explore the Mandelbrot and Julia sets right in your browser. Zoom into the infinite detail of z = z² + c with click-to-zoom, pan, smooth coloring, and adjustable iteration depth — then save your view as a PNG.
How it works
Enter what you have and read the result as it updates live. It all runs on your own device, so it is quick and private, with nothing to install.
Want the deeper story? The Knowledge Base explains the ideas behind the tools in more detail.
Frequently Asked Questions
What is the Mandelbrot set?
The Mandelbrot set is the collection of complex numbers c for which the sequence z → z² + c, started at z = 0, stays bounded forever (it never runs off to infinity). Points inside the set are coloured black here; points outside are coloured by how quickly they escape. That escape speed produces the famous swirling, infinitely detailed boundary.
How is a Julia set different from the Mandelbrot set?
Both use the same rule z → z² + c. The Mandelbrot set varies c across the plane while always starting z at 0. A Julia set fixes a single c and instead varies the starting point z across the plane. Every point c in the Mandelbrot set corresponds to a connected Julia set; points outside give dust-like, disconnected Julia sets. Switch the set selector and type a c value to see this for yourself.
What does zooming in reveal?
The boundary of these sets is a fractal: it has detail at every scale. Zooming in shows new spirals, seahorse tails, mini-copies of the whole Mandelbrot set, and filaments that never smooth out no matter how far you go. Increase the iteration count as you zoom so the fine structure stays sharp instead of washing into solid colour.
Why can it get slow or blocky when I zoom deep?
Every pixel runs the iteration loop up to the iteration limit, so cost grows with both the canvas size and that limit. To stay responsive the tool caps the internal render width and uses a single full render per change. Very deep zooms also hit the limits of ordinary double-precision arithmetic, which can make the image look blocky — that is a hardware floating-point limit, not a bug.
Does this run locally in my browser?
Yes. All of the maths runs in JavaScript on your device using the 2D canvas — no WebGL, no server, no uploads. Nothing about your view is sent anywhere, and Save PNG writes the file straight from the canvas in your browser.
How do I use the Fractal Explorer (Mandelbrot & Julia)?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Teaching complex dynamics
Show students how the simple rule z = z² + c produces infinite structure, and how Mandelbrot and Julia sets relate.
Exploring self-similarity
Zoom into the boundary to hunt for mini-Mandelbrots and repeating spirals that illustrate fractal self-similarity.
Generating art & wallpapers
Frame a striking region, pick a colour scheme, and save a high-contrast PNG for prints or desktop backgrounds.
Demonstrating chaos & sensitivity
Nudge the Julia constant c by a tiny amount and watch the whole picture transform — a vivid take on sensitive dependence.
Prototyping fractal coordinates
Read off the live centre coordinates and zoom level to record interesting regions for use in other renderers.
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