This post is a follow up to my earlier commentary on the joint history of math and art. I’m going to look at a source of interesting images, some of which might qualify as art. The quality that makes these images different is that they are not one or two dimensional. Instead, their dimension falls between whole numbers. You probably already heard of them, they are called fractionally dimensioned objects or fractals. Look at the sequence of images below.
The first image is the starting point, a simple equilateral triangle. The next image is generated by taking every line segment and replacing its middle third by the other two sides of an outward-pointing equilateral triangle that it is a side of. The image gets more and more complex and the final object, after you do the replacements forever (or at least until the changes are no longer visible), is called the Koch snowflake.
I picked the Koch snowflake for my example because its easy to demonstrate it possesses a certain weirdness. First, if you draw a circle continuing the original triangle, all the other pictures stay inside it. This means that the whole thing has area smaller than that of the circle. On the other hand, each step increases the length (perimeter) of the figure by 4/3 – we replaced a middle third with two lines the same length. If you keep multiplying by 4/3 over and over then you eventually get bigger than any finite number. That means the perimeter of the final Koch snowflake is infinite. Its not too hard to show that an object with finite area and infinite perimeter isn’t possible using the usual laws of geometry. So what gives?
The answer to this apparent contradiction is that the squiggly line along the perimeter of the snowflake cannot reasonably be said to be one dimensional. At this point we fall off the cliff into really high level mathematics. A reasonable figure for the dimension of the perimeter is slightly more than 1.26 or exactly the logarithm, base 3, of 4. It’s not a coincidence that the numbers “3” and “4” show up. If you’re up for it, this is the Hausdorff dimension, one of several different ways of measuring the dimension of an object.
The thing is that we’ve found a huge number of different types of fractals. You may have heard of Newton’s method for finding roots of an equation. It turns out that, if you use it
with complex numbers then it can be used to generate fractal images like the ones below from simple equations. The first image, for example, arises from “ex-cubed equals one”. To be fair, I spent some time finding a view window that looked good for each of these fractals.
The second image is a tiny little part of the entire image, for example. One of the intriguing things is that you often get effects you were not expecting when you search through fractal space. The upside-down versions of the main image scattered about in the second image and the faces in the third are examples of this phenomenon.
The next set of images are a type of fractal called a Julia set. Like the Koch snowflake, they arise from repeating a simple equation over and over until something happens. For these images, the thing that happens is that the value of the sequence of numbers you get by repeating the equation gets big. These pictures all arise from very simple mathematical equations – the hard part is finding the ones that look interesting. This sounds to me very much like a job for an artist.
Julia sets are named after Gaston Julia. He figured out that they were weird objects long before we had the computer power to make the sorts of visualizations I’ve shown you. This is an example where work in abstract math located something that was interesting for reasons that were really not apparent at the time of the original discovery. One of my methods of relieving stress is to make fractals and try and find new types of fractals. I’m also told that my choice of colors and forms is extreme and garish, suggesting that involving my colleagues in the fine arts is not the worst thing I could do.
I’ll return to one of my main themes to wind up. To start working with fractals you need only very basic programming skills and even more basic mathematical knowledge. One of the prices we pay for math anxiety and innumeracy, for contempt for math, is that artistically gifted people are far less likely to find fractals next to their brushes or pencils in their box of tools. If you know about other interesting types of fractals, or mathematical art projects, please share in the comments or send a tweet!
I hope I see you here again.
Department of Mathematics and Statistics
University of Guelph, Ontario, Canada