# Fractal Explorer - Mandelbrot, Julia Sets, L-Systems & Ferns

> Zoom into the Mandelbrot set, Julia sets, Burning Ship, Tricorn and Newton fractals with multithreaded rendering, plus L-systems and the Barnsley fern.

- URL: https://qilingtools.com/en/math/fractal
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Creative Effects](https://qilingtools.com/en/tags/effects), [Charts & Visualization](https://qilingtools.com/en/tags/chart), [Algorithms](https://qilingtools.com/en/tags/algorithm)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Explore fractals right in your browser. The escape-time explorer covers the Mandelbrot set, Julia sets, the Burning Ship, the Tricorn and the Newton fractal for zⁿ − 1: scroll to zoom, drag to pan, pinch on touch screens and double-click to zoom in. Images are computed in strips across several Web Workers (one per CPU core) with a quick low-resolution pass first and full detail after, and while you interact the previous image is instantly transformed to the new view, so panning and zooming feel smooth. The maximum iteration count grows automatically with the zoom level or can be set by hand; colouring is smooth (continuous) with six palettes and adjustable band density and offset, and changing colours never recomputes the image. Hovering over the Mandelbrot set previews the Julia set for that point in the corner, and a click opens it; in Julia mode, drag the parameter c around a mini map and watch the set morph live. Calculations use double precision, so around 10¹³× the image starts to turn blocky — the tool warns you when you get close. The current view is stored in the URL so you can share exactly what you see, and PNGs export at resolutions from 1280×720 up to 3840×2160. Two more tabs cover L-systems (Koch snowflake, Sierpiński triangle, dragon curve, fractal plant, Hilbert curve and more, with editable axiom, rules and angle) and iterated function systems (Barnsley fern, Sierpiński carpet, maple leaf and others, drawn by the chaos game with hundreds of thousands of points shaded by density).

## How to use

1. Choose a fractal and zoom or pan with the mouse wheel, dragging or pinching
2. Adjust the palette, band density and iterations; click the Mandelbrot set to open the matching Julia set
3. Copy the share link or export a PNG at your chosen resolution, or switch to the L-system and IFS tabs

## Features

- Mandelbrot, Julia, Burning Ship, Tricorn and Newton zⁿ − 1 fractals
- Multithreaded Web Worker rendering with progressive refinement
- Wheel, drag and pinch zoom with double-precision deep zoom and a precision warning
- Smooth colouring, six palettes and automatic iterations by zoom
- Live Julia preview on the Mandelbrot set and a mini map for choosing c
- Shareable views in the URL and high-resolution PNG export
- L-systems: Koch snowflake, dragon curve, fractal plant and more, with editable rules
- IFS: Barnsley fern, Sierpiński carpet and more, shaded by density

## FAQ

### How is the Mandelbrot set drawn?

For each point c in the plane, start from z = 0 and repeat z ← z² + c. If |z| never grows beyond 2, the point belongs to the Mandelbrot set and is drawn black; otherwise the number of iterations before it escaped decides the colour. The tool uses the smooth iteration count n + 1 − log₂(log|z|) instead of the whole number, which gives continuous colour gradients without visible banding steps.

### How are the Mandelbrot set and Julia sets related?

Both use the same iteration z ← z² + c. A Julia set fixes c and lets the starting point z range over the plane; the Mandelbrot set fixes the start at z = 0 and lets c range over the plane. When c lies inside the Mandelbrot set its Julia set is one connected piece, outside it shatters into dust, and near the boundary it is at its most intricate. That makes the Mandelbrot set a map of all Julia sets — click on it to open the one for that point.

### Why does the image turn blocky when I zoom in very far?

JavaScript numbers are double-precision floats with about 15–16 significant digits. Around 10¹³–10¹⁴× zoom, neighbouring pixels are closer together than the smallest step a float can represent, so many pixels get identical coordinates and the picture turns into blocks. Going deeper needs arbitrary precision or perturbation algorithms, which are far more expensive; the tool warns you as you approach the limit.

### What are L-systems and IFS?

An L-system grows a fractal by rewriting a string: the Koch snowflake starts from F--F--F and replaces every F with F+F--F+F each generation, then turtle graphics reads F as draw forward and + / − as turns. An iterated function system is a handful of affine maps; the chaos game applies a randomly chosen map to the current point over and over, and after a few hundred thousand points their distribution is the fractal itself — the Barnsley fern needs only four maps to draw a lifelike fern.

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