Beautiful Math

This article was published in the March/April 2003 issue of Imagine, a magazine published by Johns Hopkins University’s Center for Talented Youth. The version published differs slightly from the one published here, as they cleaned up some of my writing and condensed some of my long-windedness.

There is beauty in mathematics.

I became interested in math before I was ten. A teacher gave me an algebra textbook to keep me busy, which I quickly devoured. Basic set theory and Boolean algebra were a breeze, but I did get stuck with logarithms. Over the next few years I continued to study math at an accelerated pace, absorbing plane geometry, trigonometry, and calculus in due time. It was easy to learn, because I was interested. Mathematics is pure logic, a system where simple rules are assembled together to build more complex rules, a towering edifice of thought. Beauty lies in its elegance, in how everything fits together, in the absence of the chaos I found in the “real” world.

And yet there is another beauty that lurks in mathematics, quite apart from the cool beauty that excites the intellect but leaves the heart dulled. This beauty is a surprise, because it springs from chaos—from mathematics that mimics real-world processes that are unpredictable. From these netherworlds of mathematics where order seems seldom present springs fractal art, an art form that combines the precision of mathematical logic, the surprises of chaotic systems, and the passion of artistic expression.

Quite apart from studying mathematics, I also became interested in computers and programming. Like mathematics, computers are driven by logic, but it is a different approach in that it is not what you can prove, but what you can do. I was immediately attracted to computer graphics, as that is the fastest way to actually see what you can make the computer do. With parallel interests in computers and mathematics, I was perfectly set up to become consumed with fractals.

Fractals are shapes which have similar characteristics at different sizes. That is, the big details look like the small details, in the same way that the bumpiness on the surface of a rock is similar to the bumpiness of a hillside, or how small branches on a tree fork into even smaller branches just like the trunk forks into large limbs and then small branches. Fractals are plentiful in nature, and fractal structures, while incredibly complex, can often be expressed with simple equations. The complexity arises from feeding the results of each calculation back into the equation; this feedback produces chaos, but it also allows pockets of order to emerge.

Chaos is a rich area for mathematical study; rigorous bounds on regions of chaos can be computed, structures and rules extracted from seeming randomness. There is much to be learned from this; but this is not what a fractal artist like me does. Mathematical insight is a good thing, but I am primarily interested in visually exploring what the mathematics can do. The computer is a wonderful tool for this exploration, because the process of feeding numbers into equations, getting results, and feeding them back into equations, all to see what happens when there are hundreds or thousands of results, is time-consuming to do by hand! Computers do the work in a fraction of a second, leaving me free to worry about the equations themselves.

There are many methods of creating fractal art, as there are many mathematical systems which produce chaotic behavior. One of the most popular is the method which can be used to draw the famous Mandelbrot set (and many other fractals too). The approach is to consider the image to be composed of small squares, the pixels on the screen. Each square is assigned a two-dimensional (complex) number. This number is fed into an equation, producing a result; the result is fed back into the equation, producing another result. This produces a long string of results, one sequence for each square in the image. These sequences as a whole provide the basic structure of the fractal. To produce an actual colored image, these sequences are fed into another equation, which interprets the sequences in some way and produces a single value for each sequence—for each square. These values produce the colors for each square. The fractal image is a sort of graph of the fractal formula. There are thousands of different equations that can be used with this model, and thousands of different ways to interpret the sequences to produce colors.

One of the most fascinating aspects of images created this way is that they often have infinite detail. I can select a small portion of the image and magnify it, showing details that were too small to see before. And then I can take a small portion of the magnified image, and magnify it again. And again. And again. I can magnify until I’ve reached the limits of precision on my computer, and then beyond. Mathematically, the details have no end; as long as I have the patience to let the computer crunch numbers, I can continue to find new detail. This is the exploration side of fractal art—of looking at a detail of a fractal shape and knowing that I may be the only person to have ever seen it. It is possible to magnify fractal images immensely; for one animation I created, I started with an image just a few inches across, and started magnifying it. The animation always showed just a few inches at a time, but by the time the animation ends, the original image has been magnified so much that it would measure 1/50th of a light-year across. (The animation can be found at www.fractalus.com/50th)

Fractal art is based on mathematics, computed with mathematics, colored with mathematics. But it is not bound by mathematics. And it is at this point where images leave the realm of graphs by which you prove theorems and enter the realm of art. With other forms of art I have many tools at my disposal; I have particular types of brushes and knives, different types of canvas, different types of paint, which I use at my discretion to produce the image I want. My experience will cause me to have favorites, tools that are more comfortable for me, while other artists will prefer different tools. With fractal art, my tools are equations, my paints are computer-produced colors, and my canvas is the screen. Sometimes I don’t know exactly what one of my favorite tools will do when I plug its equation into the process; at other times I know exactly how to get the effect I want. These decisions aren’t based on what is mathematically sound, though; they’re based on what I can do, what equations I can use that will make the image what I want it to be. I do not further the understanding of mathematics in what I do… but I do produce art that others find beautiful.

Another wonderful aspect of fractal art is that while it is grounded in mathematics, and equations draw the images, it isn’t necessary to understand all of the math just to have fun with it. Modern fractal software will hide the math from me, letting me play with numbers and swapping one equation out for another in the blink of an eye. But when I feel the urge to modify the equation itself—as I often do—the equation is right there, under the hood, ready to be tweaked. I know many fractal artists who could not explain the mathematics at all; I know other fractal artists for whom exploring the equations is everything, and art is a lesser by-product. I am between the two extremes.

When people hear I create fractal art, they often ask me to describe my “paintings”. They are impossible to describe; they simply have to be seen. Sometimes they’re very geometric, orderly, structured. Others—especially those that borrow mathematical textures created for 3D computer graphics—have a much more organic feel, and the patterns are nearly impossible to see. In these, our brains impose order where none is visible, and we see shapes that aren’t there, making sense where there is none. People respond to all kinds of images, and they often surprise me when they tell me which ones they like.

There is beauty in mathematics. You just need to look in the right place.

Photo Credits: Speaking Order from Chaos, Very, Cherry Blossom Season, Banded Clouds, I Miss Winter, Fronds & ’Brot, Overwrought, Joynt, Woman on Bench #17, Detail Control, Edge of Darkness, Quetzalcoatl, In Defiance of Vision, Simple Pleasures, Old Wood, Apple Orchard, Searching for Meaning, Thunderhead, Popcorn Quilt Tweak, One for the Road: Copyright © 1997–2004 Damien Jones; software: Ultra Fractal 2, 3, 4, 5 and FractInt 20