Explore Fractals

We believe the best way to learn about fractals is to EXPLORE them! So let’s Fractalize!

 This is the famous Mandelbrot Set fractal generated by a very simple equation. Play with it, touch it, zoom into it. What you see above is the behavior of a starting point when you plug it into the equation, get an answer, and then feed that answer back into the equation. Wherever you point in the image is the starting value, and the lines that appear show the answer of the equation after each cycle. When you choose a starting point inside the black area, the answers spiral inward and stay finite. When you choose a starting point outside the black area, the answers expand and fly away to infinity. This is why we call this fractal the Mandelbrot Set – it is the set of all starting values that cause the answers to stay finite when calculated by this equation.

Next – let’s look at the Julia Sets and the Mandelbrot Set below. Every point in the Mandelbrot Set corresponds to a unique Julia Set. Trace along the boundary of the Mandelbrot Set in the left or top panel below, and watch the Julia Set in the other window. What do you notice?

You might expect that a complicated image like the Mandelbrot Set would require a complicated formula. But it’s not true! The wonder of fractals is that just by repeating a *simple* process over and over, we can create incredibly complex patterns. The equation that creates the Mandelbrot Set and Julia Sets is:

Z_n+1 = Z_n ² + C

That is, the next value of Z equals the old value of Z, squared, plus the starting value C.

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Now let’s play with a geometric fractal that repeats a simple branching process. Play with the points A, B and C to change the lengths and angles of the branches.

Besides trees, can you find other fractal patterns that remind you of natural fractals? Look for spiral fractals, fern fractals, brain fractals, cracking fractals, wavey fractals…

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If you want more power and control, you can download the amazing, FREE fractal explorer, XaoS, which lets you zoom into mathematical fractals. When you create a really beautiful fractal, please enter it in the Fractal Challenge! (Elementary, middle & high school students only.) Visit our Fractal Software page to learn about other powerful tools like IceFractal and UltraFractal.

Next, explore the Fractivities page which features many different projects to do at home, at school or outside in nature. We would love for you to participate in the Fractal Trianglethon! We need volunteers, teachers and students to help build the world’s largest Sierpinski Triangle.

Next, you may wish to explore our Online Fractal Course, which is a detailed exploration of fractals in nature and mathematics. While it is aimed at high school students, much of the material is appropriate for younger students, and more advanced students too.

See what people are saying about us on our feedback page and let us know what you think about fractals too!

Finally, if you want to help teach other people about fractals, please see our volunteers page. Thanks!