Infinite Complexity in Finite Space

Occupy Math notices that last week’s post had no pictures at all – suggesting it’s time for another post serving our mission of showing math is pretty neat. Today we’re going to take a closer look at the Mandelbrot set. There are literally millions of renderings of this fractal on the web – and most of them look different from one another. Today’s post will suggest a proof of a property of the set that enables this incredible diversity of pictures.

The Mandelbrot set is more complex than the entire physical universe.

The Mandelbrot set is named after the gentleman that first explored its properties Benoit Mandelbrot. There is an excellent popular song about this fractal by Jonathan Coulton. The link that Occupy Math chose is one of several YouTube videos that use the song as a soundtrack. It zooms into the set showing that the diverse images are different subsets of the Mandelbrot set. Let’s talk briefly about what is going on in the Mandelbrot set. Using complex numbers, you pick a point in the plane (points in the plane are also complex numbers). You then generate a sequence of numbers, starting with the point you picked. To generate the sequence, you square the number you have now and then adding the original point to get the next number. If numbers generated this way run away from the center of the plane, the original point is not in the Mandelbrot set, otherwise it is.

The white parts of the picture above are points in the Mandelbrot set; the colored parts are colored with a pallet where the number of steps it took the sequence to get to a distance of at least two from the center of the plane are used to index the pallet and so pick a color. The shape is clearly complicated, but how can we see that it is more complicated than the physical universe?

At this point, Occupy Math wants to remind the readers of the Fibonacci sequence. This is a sequence of whole numbers that starts with 1,1 and then adds its last two members to get the next. The start of the sequence is 1,1,2,3,5,8,13,21,34,89,…

Since there is no largest whole number, the Fibonacci sequence generates new numbers forever.

Here is the punchline – the Fibonacci sequence indexes a feature of the Mandelbrot set. Examine the following picture. Each of the “lakes” of the Mandelbrot set has a spike coming out of it with an intrinsic branching factor. Several of the lakes are labeled with their branching factor. Since this is hard to see, zooms into the Mandelbrot set for 3, 5, and 8 are also provided.

The rule is this: the largest lake between two larger lakes has a branching factor that is the sum of the branching factors of the two larger lakes. Since we have found lakes that start the Fibonacci sequence, we get lakes with new branching factors, one for each number in the Fibonacci sequence, at all possible levels of zooming.



The fact that branching factors follow the (infinite) Fibonacci sequence means that the Mandelbrot set is infinitely complex.

If we add that bolded statement to the current thinking by physicists that the physical universe is finite, we see that the Mandelbrot set is more complex than the universe. This is a typical mathematical proof – we show our claim is correct, but miss a lot of the juice of the situation. In fact all branching factors appear for some lakes and so we don’t just get the Fibonacci sequence that starts 1,1; we get an infinite number of such sequences starting with any two whole numbers you care to choose.

Another interesting fact is that the Mandelbrot set contains an infinite number of (slightly distorted) copies of itself. These are called minibrots. Below is a minibrot from the fractal taxonomy. Scroll down to find another minibrot in the linked post. Several others appear in the taxonomy and at least tens of thousands exist on the web in various forms. Can you find minibrots in the images above?

With modern computers it would not be hard to have a math class take a field-trip into the Mandelbrot set. In past editions of the Science Olympics at the University of Guelph, Occupy Math has run a mathematical art contest with students exploring fractal space and saving the images they like. Occupy Math does not understand why we are not using these mathematical opportunities more broadly to spark interest in secondary math students. Is Occupy Math missing efforts in this direction? If so comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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