Counting in Binary is a Fractal

I can’t remember for sure but I think I read about this first in A New Kind Of Science by Stephen Wolfram (yes, the same guy who made Wolfram Alpha!). Counting in binary is actually a fractal, check out these 4 sequential images…

Real Time Demo

To see it in action, check out this WebGL shadertoy:

How to Render the Mandelbrot Set

The Mandelbrot set is a beautiful creation of mathematics discovered by a French-American mathematician named Benoit Mandelbrot. it is also a fractal, meaning that it’s infinitely detailed and that it’s self-similar and made up of smaller versions of itself.

Wikipedia does a better job of explaining the history and the background more than I do so please check out this link for more info!

Mandelbrot Set

I also wrote an HTML5 powered mandelbrot set viewer that you can use to explore the fractal and manipulate colors to create your own mathematical works of art.

HTML5 Mandelbrot Explorer

In this article I’m going to explain how to render a Mandelbrot set yourself.  It’s going to be from a programming slant more than a mathematical slant so if you want the raw unadulterated math, I recommend checking out Wikipedia or other sources.

Rendering the Mandelbrot Set

The first step is to have some way to draw individual pixels on a canvas of some sort. It doesn’t matter what method you go with, it just matters that you are able to draw pixels somehow.

Various ways include:

• direct pixel accesss in HTML5 (what my Mandelbrot Explorer uses)
• drawing pixels into an image file
• using DirectX or OpenGL to render it to a 2d screen buffer.
• using graph paper, a calculator and some colored pencils to create it by hand (Possible, but ouch! Send me a picture if you actually do this! hehe)

Viewport

Now that you have a rectangle that you are able to render pixels to, we need to define a viewport.

In the Mandelbrot Set image at the top of this article, my rectangle is about 500×500 pixels big, and the x axis ranges from -2.5 to 2.5 and the y axis also ranges from -2.5 to 2.5.

I like to define my viewport in terms of the center point, and the width and height, so our viewports parameters are a center point of (0,0) and width and height of 5.

We have now established the parameters of our viewport!

We now need to iterate through each pixel in our rectangle and do the following steps for each…

Pixel Space to Viewport Space

The first thing we need to do is convert from pixel coordinates to viewport coordinates.

How we do that is like this:
ViewportX = ViewportMinX + (PixelX / PixelWidth) * ViewportWidth
ViewportY = ViewportMinY + (PixelY / PixelHeight) * ViewportHeight

After we have converted our pixel’s location from pixel space to viewport space, we are ready to do some math.

The Magical Mandelbrot Function

Like I mentioned earlier, the Mandelbrot set is a work of mathematical art.  The function itself isn’t very complex but it involves imaginary numbers.  To calculate the Mandelbrot set itself, you plug the viewport location of the pixel into the function.  After that, you take the output of the function and plug it back into the input of the function.  You continue this until the output of the function goes above some value (the common value to use is 2.  You’ll see me compare against 4 because I’m comparing squared numbers).  If the output goes above the threshold, it has essentially “escaped”.

There are pixels which may take a very very very large amount of iterations to escape, and as far as I know, they haven’t proven that all values will escape (I could be wrong though), so besides waiting for the function to “escape”, you should also set a maximum iteration count to keep it from iterating forever (or for a very long time).

The number of iterations it took to escape is what you use to set the pixel color for that pixel.

Here is some code (pseudo javascript) to show you the details of that process:

```var g_maxIterations = 255; //TODO: set this to however many iterations you want to allow maximum
var currentX;  //TODO: need to set this to the viewport X location of the current pixel
var currentY;  //TODO: need to set this to the viewport Y location of the current pixel

var z = 0;
var zi = 0;
var inset = true;
var numInterations = 0;

var newz;
var newzi;

for(indexIter=0; indexIter 4)
{
inset = false;
numInterations = indexIter;
indexIter = g_maxIterations;
}
}

if (inset)
{
//we never escaped, this pixel is the default color
//TODO: render a default color pixel
}
else
{
//we escaped!  numInterations is how many iterations it took
//TODO: convert iterations to a color and render the pixel
}
```

Colorizing The Pixel

There are several ways you could turn an iteration count into a color for a pixel. There are some ways listed below, but this is definitely not an exhaustive list! Play around with your own techniques and see what sort of interesting things you can create!

• Make a maximum iteration time of 255.  Make your output image be an 8bit greyscale (or color palleted) image, taking the iteration count and writing that out raw as the output color.
• Make several ranges of iteration values (for instance… 0-255, 256-511, 512-767, etc) where you define a full RGB color at each edge of the value ranges.  From there, figure out where your iteration count falls within the value ranges, and do a lerp between the color to the left and the color to the right based on your distance in the specific value range.  This way, you have smoothly blending color gradients and can go well beyond 255 maximum iterations.  I use a variation of this in my HTML5 Mandelbrot Explorer.
• Use arbitrary math functions to figure out the RGB of each pixel.  Such as R = Iterations % 256, G = (Iterations * 3) % 256, B = (Iterations * 7 + 39) % 256.

After you have the color for your pixel, you are done. Render that pixel, then move to the next until you have rendered them all.

Zooming and Panning (Scrolling)

By virtue of setting up a viewport as a centerpoint and a width and height, and making the code use that information to convert from pixel space to viewport space, we have made it really simple to implement zooming and panning.

To zoom in, just set your viewport width and height to be smaller (like divide them by 2 for instance).   To zoom out, set your viewport width and height to be larger.  You want to make sure and keep the same aspect ratio (width / height) in your viewport as your rendering rectangle to avoid distortion though, so be careful.   OR, you may want the distortion… it’s up to you (:

To pan the screen, or scroll it left, right, up or down, you just change the centerpoint to be more to the left, the right, higher, or lower.

Very simple, but it is really fun to scroll around and zoom in and out on the fractal to discover new and interesting features to share with your friends.

If you zoom in far enough, you might notice that at some point you have vertical or horizontal lines instead of the fractal shape, and that if you zoom in a little bit more, you’ll get a solid color.

You might ask yourself “Hey, I thought he said fractals were infinitely detailed?”

Well they are infinitely detailed, and in theory you could zoom in FOREVER and always see more and more things, but computers themselves don’t have infinite precision (it would take a computer infinitely large to let you zoom in infinitely), and you are just seeing the edge of the precision of your computer.

If you are using a language like C++, you can change your code to use doubles instead of floats to get a little more breathing room, or another option is to use a scientific mathematics library that is capable of a lot more precision than floats or doubles.

That’s All!

That’s really all there is to it, not that complex is it?

As I keep mentioning, I made something that allows you to explore the Mandelbrot Set in your browser.  You can find that here: Mandelbrot Explorer