Exploring the Mandelbrot Set

Below you can see and explore the graphical representation of the Mandelbrot set drawn on a complex plane. Read below about the Mandelbrot set, the complex plane, imaginary and complex numbers and fractals.

You may have already noticed the circle like figures that repeat at different sizes and positions. Click over any section and the image will zoom to that section. Keep clicking and you will discover and infinite world of nature like patterns. You may prefer to start by selecting one of our favorite places from the dropdown list. You can then zoom out or in and continue exploring. All points need to be recalculated at each zoom or movement, so please be patient.





What is the Mandelbrot Set?

The Mandelbrot set is a set of complex numbers that we can draw over a complex plane. We find the numbers that belong to the set using this formula:

zn+1 = zn2 + c

We input each coordinate into the formula as c and we start with z = 0 + 0i. the result is fed back to the function as a new z, while c does not change and the formula is iterated, which means that the result is fed back many times (400 times in our case). If the resulting number does not goes to infinity and stays between -2 and 2, the number does not diverges and belongs to the Mandlebrot set. Some examples:

We input the top left corner coordinate (-2 + 2i) and in the second iteration we reach -6i, so this number is not part of the set:

Iteration z c z2 + c
1 0 + 0i -2 + 2i -2 + 2i
2 -2 + 2i -2 + 2i -2 - 6i
3 -2 - 6i -2 + 2i -34 + 26i
4 -34 + 26i -2 + 2i 478 - 1766i
5 478 - 1766i -2 + 2i -2890274 -1688294i

Let's try 0 - 1i. As you can see, this point is part of the set as the resulting z is either 0 + 1i or -1 -1i:

Iteration z c z2 + c
1 0 - 0i 0 - 1i 0 - 1i
2 0 - 1i 0 - 1i -1 - 1i
3 -1 - 1i 0 - 1i 0 + 1i
4 0 + 1i 0 - 1i -1 - 1i
5 -1 - 1i 0 - 1i 0 + 1i
See Complex numbers at Math is fun to learn how to add and multiply complex numbers.

Points in the set are colored red. Those that go to infinity (diverge) are colored black. If you click the Use colors button, a different color is assigned to each point depending of the numbers of iterations needed to diverge. Points that do not diverge are black.

Follow these links to learn about the Mandelbroth set:

You may also want to read about imaginary and complex numbers, the complex plane and fractals: