Mandelbrot Set Generator
Draw the Mandelbrot set, zoom into it with the mouse and download the image.
This is a free online Mandelbrot set generator. Move the mouse wheel over the image to zoom in and out exactly where the pointer is, drag the image to move around, and download what you see as a PNG image of any resolution, up to 4K.
The Mandelbrot set is the most famous fractal ever drawn: a shape with a finite area and an infinitely long border, where every magnification reveals new spirals, filaments and tiny copies of the whole set. This tool draws it with smooth coloring, eight color palettes, and a zoom that goes down to a magnification of about one trillion times.
Everything runs directly in your browser, on your own processor. No image is uploaded to a server, and nothing needs to be installed.
Fractal options
Use the mouse wheel over the image to zoom in and out where the pointer is, and drag the image to move it. On a phone, use one finger to move the image and two fingers to zoom.
How to generate a Mandelbrot set image?
To create and save your own view of the Mandelbrot set, follow these steps:
- Pick a place to explore - Use the buttons above the image to jump straight to the most famous regions of the set: the Seahorse Valley, the Elephant Valley, the Triple Spiral and a mini Mandelbrot. They are a good starting point, and you can zoom away from there.
- Zoom with the mouse wheel - Every notch of the wheel magnifies the image 1.3 times around the pointer, so the detail you are aiming at stays under the cursor. On a phone, put two fingers on the image and pinch.
- Drag the image to move - Hold the mouse button and drag, or slide one finger on a touch screen. The image follows immediately and is calculated again when you release it.
- Raise the iterations if the image turns flat - The deeper you zoom, the more iterations the border needs. "Increase the iterations when zooming in" does it automatically, and you can always type a higher value yourself.
- Choose the colors - Change the palette, the number of color cycles and the color offset until the bands look the way you want. The inside of the set is painted with a single color, which you can change too.
- Choose the resolution - Select one of the common resolutions, or type any width and height. The image is drawn again at the new size, with the same view.
- Download the image - Click "Download PNG image" to save exactly what you see on the canvas.
Write down the center and the zoom of a view you like: typing the same three numbers again brings you back to exactly the same place.
Settings
Below are the parameters you can configure to draw the Mandelbrot set.
| Field | Description |
|---|---|
| Center (real part) | The real part of the complex number at the center of the image, which is the horizontal position on the complex plane. The whole set lies between -2.1 and 0.6. |
| Center (imaginary part) | The imaginary part of the complex number at the center of the image, which is the vertical position. The set is symmetric around the horizontal axis, so a positive value and its negative show mirrored images. |
| Zoom | The magnification of the view. A zoom of 1 shows the whole set, and a zoom of 1000 shows a region a thousand times smaller. The mouse wheel changes this value. |
| Maximum iterations | How many times the formula is applied to each pixel before deciding that the point belongs to the set. Low values paint thick, smooth borders; high values reveal the thin filaments that connect the parts of the set, and take longer to draw. |
| Increase the iterations when zooming in | Raises the number of iterations automatically as the magnification grows, which keeps the border detailed without having to adjust it by hand. |
| Palette | The sequence of colors used to paint the points outside the set: Classic, Fire, Ice, Electric, Sunset, Rainbow, Forest or Grayscale. |
| Color cycles | How many times the palette repeats every 256 iterations. A low value creates wide, soft bands; a high value creates many thin bands that follow the shape of the border closely. |
| Color offset | Rotates the palette without changing the shape of the bands, which is an easy way to find a color scheme you like. |
| Smooth coloring | Uses the fractional iteration count instead of the whole number, which removes the visible steps between the color bands and produces a continuous gradient. |
| Invert the palette | Reads the palette backwards, so the colors near the border of the set become the colors far from it. |
| Color of the inside of the set | The color of the points that never escape, which form the black body of the set in the classic images. |
| Resolution / Width / Height | The size of the generated image in pixels. Choose one of the common display resolutions, or type any width and height. |
What is the Mandelbrot set?
The Mandelbrot set is the set of complex numbers c for which the sequence
z0 = 0 and zn+1 = zn² + c
never runs away to infinity. Each pixel of the image is one value of c: the horizontal position is the real part and the vertical position is the imaginary part. The formula is applied over and over to that pixel, and as soon as the value gets farther than a fixed distance from the origin it is certain that it will keep growing forever, so the point is not in the set. The number of steps it took to run away is what chooses the color of the pixel, and that is why the colored bands outside the set look like contour lines around it.
Points that survive all the iterations are considered part of the set and are painted with a single color. This is only an approximation: with more iterations, some of those points would eventually escape. That is exactly why zooming deeper requires more iterations, and why a region can look like a solid blob until you raise them.
The set is named after Benoit Mandelbrot, who worked at IBM and published the first computer pictures of it in 1980. The shape had been described earlier, in the work of Pierre Fatou and Gaston Julia around 1918, but nobody had been able to see it: drawing it takes millions of multiplications, which is why the Mandelbrot set became a symbol of what computers made visible.
Famous places of the Mandelbrot set
The border of the set is not the same everywhere: each valley has its own style of decoration, and each one has a name given by the people who explored it. The buttons above the image take you straight to them.
| Place | Where it is | What you see |
|---|---|---|
| Seahorse Valley | Around -0.7453 + 0.1127i, in the crack between the main cardioid and the big circle on its left | Tails curled like seahorses, made of thinner and thinner spirals. It is the classic first zoom of the set. |
| Elephant Valley | Around 0.2825 + 0.01i, on the right side of the main cardioid | A parade of shapes that look like elephants walking in line, each one smaller than the last. |
| Triple Spiral Valley | Around -0.0888 + 0.6547i | Groups of three spirals turning around each other, repeated at every scale. |
| Mini Mandelbrot | Around -1.7638 + 0i, on the antenna that points to the left | A complete, tiny copy of the whole set, surrounded by decorations that do not exist around the original. There are infinitely many of these copies. |
Frequently Asked Questions (FAQ)
Is this Mandelbrot generator free to use?Yes, this tool is 100% free and works directly in your browser, with no installation and no sign-up required.
Are the images uploaded to a server?No. The fractal is calculated by your own browser and the image never leaves your computer.
Can I use the generated images in my own projects?Yes. The images are generated by a mathematical formula, and you can use them in websites, wallpapers, posters, videos, games and printed material.
How deep can I zoom?Up to a magnification of about 1013, which is ten trillion times. Beyond that, the numbers used by the browser (64 bit floating point) do not have enough decimal digits to tell two neighbor pixels apart, and the image breaks into flat blocks. Programs that zoom deeper than that have to use arbitrary precision arithmetic, which is much slower.
Why does the image look flat and featureless when I zoom in?Because the number of iterations is too low for that magnification. Deep inside a valley, the points near the border need hundreds or thousands of steps before they escape, and with too few iterations they are all painted as if they were inside the set. Raise "Maximum iterations", or keep "Increase the iterations when zooming in" turned on.
Why is the image drawn in blocks that get sharper?The tool first paints big squares to show you a rough image immediately, and then refines it down to single pixels. This way the page never freezes, and you can keep zooming before the fine detail is finished.
What does "smooth coloring" do?Without it, every pixel is colored by a whole number of iterations, so the image has visible steps between the color bands. Smooth coloring adds the fractional part, calculated from how far the point was when it escaped, and turns those steps into a continuous gradient.
How do I go back to a view I liked?Note the three numbers of the options panel: the center (real part), the center (imaginary part) and the zoom. Typing them again brings the image back exactly, on any computer.
Is the Mandelbrot set really infinite?The set fits inside a circle of radius 2, so its area is finite, but its border has infinite length and infinite detail: no matter how much you magnify it, it never becomes a smooth curve. That is what makes it a fractal.
What is the relation between the Mandelbrot set and Julia sets?They come from the same formula. In the Mandelbrot set each pixel is a different value of c, always starting from z = 0. In a Julia set the value of c is fixed for the whole image and each pixel is a different starting value of z. Every point of the Mandelbrot set corresponds to a connected Julia set, and every point outside it to a Julia set broken into dust.
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