Sunday, July 23, 2006


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Question: What’s a fractal?


A) A picture is worth a thousand words, so here is the pictorial equivalent of a 5,000-word answer.

B) If you’re interested in the words, click on comments. If not, skip it and just enjoy the pictures.


Heidi said...

Fractals are strikingly beautiful mathematical objects that provide a method for looking at science, math and the natural world in entirely different ways. They have applications to so many fields I wouldn’t know where to begin in starting a list. They have mind boggling mathematical properties, yet they are startlingly simple to create.

The images you see on this post are all magnifications of only one fractal, the Mandelbrot Set. As with all fractals, it can be magnified infinitely many times, always showing more detail. The M-Set has been nicknamed “the thumbprint of God.” There are many reasons for this. It is patterned with whorls like a fingerprint. As the most well-known fractal, it represents a branch of mathematics that has been called the geometry of nature (creation). It is infinitely detailed. Because of this it has been said that it is more complex than the universe itself (depending on how you feel about the Planck length, but that’s a whole other post!).

This may sound really complicated, but what’s really cool is that it’s not! I teach it to all levels of my students. All you need is an understanding of basic algebra. Although the M-Set has been called more complex than the universe itself, it’s just a graph, a graph of a mathematical statement that you can write out using only SIX symbols.

For many reasons it is important to me to bring this into every class I teach. First, I want to show my students that math can be beautiful, because, sadly, all most of them have seen of math are things like long division and solving for x. Second, I want them to know that the field of math is alive; this graph was first seen in 1980, and more mathematics was created in the last century than in all previous human history. Third, I want to show them the mathematical power they have, that even knowing only basic algebra they can understand how this beautiful complexity is generated. Fourth, I want to show them some of the wildness of mathematics (a part of which is that fractals can be dimensions that are not whole numbers like 0, 1, 2, 3).

The defining characteristics of fractals are that they are the result of an infinite process, that they are self-similar, that they have detail on all scales of magnification, and that they can have dimension that is not a whole number.

I like fractals because they deal with infinity, because they are beautiful, because they have mind-blowing properties, and because they can bring math to life for my students.

If you have questions about fractals, leave them as a comment, and I’d be happy to explain further! 

Roman said...

Very nice pics. So, what actually the equation is?

Anonymous said...

For this one particular fractal, the Mandelbrot set, the equation is:


(where the ^2 just means z-squared - I just can't write it out better here)