The Mandelbrot set

(Text starts below the image)

A so-called complex number is a number that is composed of two parts:

• A real part; i.e., an 'ordinary'(real) number
• An imaginary part which is a (real) multiple of the imaginary number i, which is defined as the square root of -1 (i = √-1)

Complex numbers where invented —discovered?— in order to be able to solve math expressions containing the square root of negative numbers.

• Example: √-9 = √(9 * -1) = √9 * √-1 = 3i (and -3i)

To visualize the Mandelbrot set on a flat surface such as a sheet of paper or a computer screen, you consider the x-coordinates of the surface to represent the real parts of complex numbers and the y-coordinates the imaginary parts. Every point on that surface can then be expressed with an (x, y) coordinate pair that represents a different complex number.

The next thing you do is subject each point; i.e., each coordinate pair on the surface; i.e., each complex number C, to a repeating operation as follows:

• C_{step 0} = C
• C_{step 1} = C_{step 0}² + C
• C_{step 2} = C_{step 1}² + C
• C_{step 3} = C_{step 2}² + C
• Etc.
• or, in more general: C_{step n} = C_{step (n-1)}² + C

Now, if you do this with complex numbers that are within certain limits (real values between -2.0 and +.6; imaginary values between -.125 and +1.25), then as you process a number through the steps listed above you will see one of two things happen:

• Regardless of how many steps you take, the result remains close to where you started, or
• After some steps the result 'veers' off of the page or screen on its way to infinity.

We can also color the points not belonging to the Mandelbrot set based on how many steps it takes before they veer off to infinity. The fewer steps the darker, the more steps, the lighter. This is why in the image above you see some grey-tones as well as black and white. The lighter greys all appear close to the 'border' of the Mandelbrot set.

Note: The image of the Mandelbrot set above is computed real time; i.e., we do not use a pre-computed image sitting on a server somewhere. The above image comprises 1,500 x 1,500 = 2,250,000 pixels. Since the maximum number of steps for each point (pixel) is set to 100 (you can change this in the iterations text box) a maximum of 100 x 2,250,000 / 2 = 112,500,000 calculations with complex numbers must be completed in order to compute the entire image ('/2' because the (default) image is symmetrical on the x-axis. If we have the value of a point on one side of the x-axis, we can use that same value on the other side of the x-axis).
Depending on your hardware and your browser, this may take more or less time. We have noticed, for instance (end of 2025), that on the Firefox browser, the computations are quite a bit faster than on the Chrome browser.

Riding the Mandelbus

To follow the trajectory of a point (complex number) through its steps, first click the Mandelbus button, then select a point (complex number) by clicking a location on the surface (step 0). The browser will compute the result for step 1 and draw a (red) line from the result of step 0 to the result of step 1. Another click computes step 2 and connects the result of step 1 with that of step 2, another one does step 3, etc. Thus, continued clicking displays the trajectory of the complex number you selected as you subject it to ever more steps.

At any time you can select the Reset button to reset the screen. If you try several points: some inside the Mandelbrot set, some outside the set (please Reset between trying points), you will notice that if you pick a point within the Mandelbrot set, the 'Mandelbus' remains in the area. Whereas if you pick a point outside the set (a point in the black space), the bus quickly escapes to infinity.

You will also notice that as you pick points closer to the 'edge' of the Mandelbrot set, it takes more steps to decide if the bus remains in the neighborhood or will escape to infinity.

This brings up an important question: Where exactly is the border of the Mandelbrot set?

Does the Mandelbrot set have a border?

By default the maximum number of steps (iterations) is set to 100. Please play with this number; e.g., try 100 and 500 and study the differences in results (note: be patient; setting iterations to 500 significantly increases the number of computations needed to compute the set). What do we see?

Well, the set seems to get a little larger when the max iterations is set to 100, and smaller when the max iterations is set to 500. There are also far larger grey areas in the 100-iterations set than in the 500-iterations set. Clearly then, it seems that if we increase the maximum number of steps (iterations), the border becomes less fuzzy; more precise.

We can explore this further by zooming in on the set's border; i.e., we take a small area near the set's border and 'blow it up' to look at the details inside.
To do this, first Reset the page. Then, with your mouse, click and drag a small rectangle area somewhere that includes a section of the set's border; then select Plot. You may also want to combine this with larger numbers of iterations. We will observe a few things:

• As we zoom in on the border of the set, the border becomes ever more complex, sometimes showing fantastic baroque-style tentacles, arms and spirals.
• Depending on where and how deep we zoom in, copies of the original Mandelbrot set appear on some of these tentacles. Those copies themselves have similar complex sets of arms and tentacles as the original set.

From this we conclude —strange at this may sound— that the Mandelbrot set does not have a clearly defined border. That no matter how deep we zoom in, we will always find more points that belong to the set.

Beauty bonus: Julia sets

Wheras there is only a single Mandelbrot set, each complex number has its own Julia set. Julia sets are computed using the same steps as the ones used in the Mandelbrot set, but instead of checking all the numbers on the surface for escape from the surface, we now check all the numbers on the surface for escape from a specific point on the surface. Hence, each point (complex number) on the surface having its own Julia set.

To generate Julia sets, first Reset the page; then select the Julia button; then click on a point on the Mandelbrot image surface.

The Julia set associated with the point you selected will appear in a separate browser tab.

Note that depending on how you pick the point for your Julia set, a Julia set can be fantastically complex and beautiful, or rather simple. Take a look at Julia examples to gain some idea of where interesting Julia sets can be found.

Also, note that once you have a Julia set displayed, it is worth playing with the 'iterations number. Lowering or increasing the number of iterations used for computing a Julia set can dramatically increase the complexity and visual appeal of the set.