I can’t very well argue that assessment. Fractals, in all their mesmerizingly infinite glory, are often symbols and tools of both mysticism and psychedelics. And it’s no wonder: what abstract image could be more philosophically poignant than a shape that mirrors itself at smaller and smaller scales… forever? The term “fractal” generally means a geometric shape where each of the shape’s parts are the same as the whole shape. They can also be number sequences, or real-world phenomena that aren’t as mathematically perfect. The Sierpinski Triangle, shown below, is a very basic fractal shape.
Fractals were popularized by a mathematician named Mandelbrot in the 1970’s, when computing was finally powerful enough to begin understanding and analyzing the phenomenon. This is pretty recent as far as science goes, and new theories and discoveries about fractals are coming to light even today.
But let’s begin at the beginning. What makes a fractal?
According to Mandelbrot’s (widely accepted) definition, there are technically two requirements for a shape to be considered a fractal. The first is self-similarity, or that infinite repetition mentioned earlier. In theoretical mathematics, that means perfect self-similarity like the Sierpinski Triangle above, but when used to analyze naturally occurring forms like mountains or trees, it refers to statistical self-similarity.
The second requirement is a non-integer dimension, creatively labelled a “fractal dimension.” Yes, you read that right: fractals are neither 2D, nor 3D, nor any other familiar whole-number dimension.
The reason for this relates to the underlying mechanics of dimensions. A one-dimensional line segment, for example, can be divided into 3 equal parts, and each of these parts are 1/3 the length of the whole line segment. A 2D square, on the other hand, must be broken into 9 equal parts to produce squares that are 1/3 the area of the original square. Note that 9 is simply 32. With a 3D cube, as you may be able to guess, producing a cube 1/3 the volume of the original requires 33 equal parts.
This logic applies to any 1D, 2D, 3D, or even 4D shape (not that this is nearly so easy or intuitive to confirm when higher dimensions are involved). With fractals, though, not so much. In the Sierpinski Triangle, which is flat but has infinite holes, finding that 1/3 area piece requires approximately 31.585 pieces. All fractals behave similarly, displaying odd scale factors that indicate dimensionalities between 1 and 2. Or for fractals that are less flat, like the Menger Sponge shown below, their fractal dimensions are between 2 and 3.
The discovery and construction of fractals is a subject unto itself. The Sierpinski Triangle is a perfect example of the many curious relationships between the construction of fractals and other math. At its most basic, and how it’s typically introduced, it can be created by removing triangles. That looks something like this:
It can also be created from Pascal’s Triangle (where each number is the sum of the two numbers above it). Simply color the odd and even numbers differently, and the emerging pattern is the Sierpinski Triangle.
Much to my surprise, it can even be constructed from the Tower of Hanoi, which I included in last week’s list of aesthetic math and logic puzzles. When drawing a graphical representation of the game’s “state space,” or a picture of all the possible actions to take in the puzzle, the resulting diagram highly resembles the Sierpinski Triangle.
The Tower of Hanoi, it turns out, is “fractal-ish,” a term I just made up to explain this phenomenon. What I mean is that this puzzle is self-similar (fractal requirement #1) without having a fractal dimension (fractal requirement #2). Isolate the moves performed before moving Disk 4, and the puzzle works exactly like solving the 3-disk variation. Isolate the moves performed before moving Disk 5, and the puzzle works exactly like solving the 4-disk variation.
In what I found to be the most fascinating twist, the Sierpinski triangle arises when one plays the “chaos game.” In the chaos game, you have a triangle with three outer corners distinguished as 1, 2, and 3. Start with any point inside the triangle, and begin “rolling a die” to randomly choose one of the three corners. Each time a corner is chosen, a new point is created halfway between the previous point and the corner. It’s random and unpredictable, because you never know which corner will be chosen next. The result, however, inevitably turns out like this:
If different shapes or different rules are selected, different fractals can be generated (although not all possible rule combinations create a fractal). This makes it an interesting method for the discovery and classification of new fractals.
The reason for this relates to the necessary path the points take when following rules like this. Fractals, due to their self-similarity, are composed of fractions in both distance and position. In essence, take any point on a Sierpinski triangle, and any of the three points that lie halfway to the three corners are also within the Sierpinski triangle. The result is that with the “formula” the chaos game applies (an Iterated Function System, to be precise), the set naturally “maps” to the Sierpinski Triangle.
The chaos game has been used on still more complicated shapes, like the Barnsley Fern shown below. The most widely-used practical application for IFS sets like these is actually image compression, where fractal patterns within images are stored with the much simpler IFS rules.
The relationship between fractals and “chaos” is a nuanced one. “Chaos theory” refers to the study of complex systems that cannot be analyzed by simple cause-and-effect methods—like a butterfly flapping its wings across the globe, small changes have huge effects in these unpredictable patterns. The subjects of study in chaos theory are nonlinear systems, which, it turns out, is lots and lots of things, particularly in nature. Fractals are a kind of nonlinear system, and that makes them very useful for modelling chaotic patterns. These patterns are so complex that a computer is required to calculate all the possibilities, making chaos theory another more recent area of scientific study.
Many natural phenomena have underlying fractal patterns, including trees, rivers, snowflakes, mountains, broccoli, coastlines, lightning, and clouds. That said, there’s a certain amount of debate in mathematics because these natural forms clearly do not repeat forever the way a mathematically-perfect fractal would. They are, perhaps, only “fractal-ish,” like the Tower of Hanoi.
Fractals become, therefore, important when trying to borrow from nature. In design fields this practice is known as biomimicry, or imitating nature’s solutions when solving human design problems. Fractals turn out to have a few key advantages over smoother lines and surfaces.
For example, fractals have much higher surface areas than other shapes. This is currently used in antennas both to decrease the amount of space the antenna takes up and to increase the number of frequencies a given antenna receives. It’s even been found that structures composed of fractals tend to be stronger than non-fractal structures, which has even been used to increase the durability of concrete.
Fractals are also utilized in some of today’s digital art, although their artistic merits may have a slightly longer history. One group of researchers found that analysis of Jackson Pollock’s splatter paintings revealed fractals—his work was as fractal as natural forms like trees, clouds, and coastlines. Fractal analysis was shown to help distinguish real Pollocks from fake with a success rate of 93%.
What’s more, these fractals may contribute to the aesthetics of the artwork. The same researchers found that viewing fractal patterns in any format, be they natural, artistic, or mathematically-generated, reduced stress dramatically in the viewer (by 60%, to be exact!). The researchers posit that this may be due to the extensive presence of fractals in nature. Because of their prevalence, humans may have adapted better visual fluency for fractals, making them easier and faster to process.
“Fractal art” itself is an abstract art movement which originally developed in the 1980s. Interestingly enough, it’s often created on mathematically dependent computer programs. The result is groups of artists who must know some math, intuitively or otherwise, to accomplish their visual goals. For these artists, math becomes an artistic tool.
The long and short of it is that fractals are fascinating, whether they’re being used to further new scientific advancements or just as mesmerizing visuals. And unlike many older areas of math and science, their relatively more recent timeline makes them a potential area for innovation and discovery. And what could be more inspiring than that?
To explore the beautiful visuals of fractals, check out the iOS app "Frax." You can find the free version here, the iPhone version here, and the iPad version here.
To design your own fractals, start with a more basic fractal generator.
For more robust fractal art, try the browser program Turtle Graphics Renderer. Then, explore downloadable fractal art software like ChaosPro or Mandelbulber.
To stay up-to-date on M•A•D Symmetry and learn some weird facts to impress your friends with, use the sign-up form below!