Think not of what you see, but what it took to produce what you see
Randomness is all around you…or so you think. Consider the various shapes of the morning clouds, the jagged points of Colorado’s Rocky Mountains, the twists and turns of England’s coastline and the forks of a lightning bolt streaking through a dark, stormy sky. Such irregularity is commonplace throughout our natural world. One can also find similar irregular structures in biology. The branch-like structures in your lungs called Bronchi, for instance, fork out in irregular patterns that eerily mirror the way rivers bifurcate into smaller streams. It turns out that these irregular structures are not as irregular and random as one might think. They’re self-similar, meaning the overall structure remains the same as you zoom in or out.
The mathematics that describes these irregular shapes and patterns would not be fully understood until the 1970s with the advent of the computer. In 1982, a renegade mathematician by the name of Benoit Mandelbrot published a book entitled “The Fractal Geometry of Nature”. It was a revision of his previous work, “Fractals: Form, Chance and Dimension” which was published a few years before. Today, they are regarded as one of the ten most influential scientific essays of the 20th century.
Mandelbrot coined the term “Fractal,” which is derived from the Latin word fractus, which means irregular or broken. He called himself a “fractalist,” and often referred to his work as “the study of roughness.” In this article, we’re going to describe what fractals are and explore areas where fractals are used in modern technology, while saving the more technical aspects for a later article.
Measuring Coastlines With the Koch Curve
Let’s jump right in with a good old fashioned gedankenexperiment. Your job is to measure how long the coastline of England is. Let’s say you start off with a yard stick and you complete your measurement, with the end result being X number of yard sticks. Now let’s try with a 12″ ruler. And then with a 6″ ruler. You will find that the smaller your measurement device gets, the longer your coastline measurement becomes. So how does one measure the length of a coastline accurately?
In order to answer that question, we must familiarize ourselves with a fractal technique called the Koch Curve, as seen on the left. Let us imagine an equilateral triangle. Take each side of the triangle and put another little triangle on it. And now take each side of the little triangles and do the same. This iterative process creates what is known as a mathematical “monster,” and is a paradox. The equation produces an infinite and therefore unmeasurable curve. But if you graph out the set of points the equation produces, you find a closed finite shape sometimes called a Koch Snowflake. The image on the left represents one side of our equilateral triangle.
This is in direct relation to our coastline measurement problem. The length of the coastline grows to infinite as we make the measurement device smaller, but there is clearly a closed shape to measure. Mandelbrot recognized this connection, and developed a measurement method that, instead of measuring the length of the coastline, measures the “roughness” of it. To understand this, we need to tweak our understanding of dimension.
A straight line is one dimension. A triangle is two dimensions. If we think of the Koch Curve lying somewhere in between the two, we can see that as we iterate, we increase the roughness. In the image on the left, E is “rougher” than C. And if we think of a coastline like the outer parts of our Koch Curve, we can measure this roughness in terms of the number of iterations of the fractal. This insight would catapult Mandelbrot’s notoriety within the scientific community.
Fractals in Hollywood
In 1978, Loren Carpenter was an engineer for Boeing, and was trying to develop software to make realistic looking mountainous terrain to be used as a backdrop for plane CAD models. He stumbled upon Mandelbrot’s first book in a book store, and read it cover to cover. Twice.
At that point in time, Mandelbrot was not well known, and the term ‘fractal’ was something few had heard of. In the book, Mandelbrot described how many irregular shapes in nature, including mountains, can be thought of as fractals. You can make a fractal by taking a smooth shape, breaking the surface into pieces, arrange those pieces in a self similar pattern, and repeating…similar to how we made the Koch Curve. Carpenter had found his solution.
Within 72 hours after discovering fractals, he had produced mountainous terrain on his computer. He made a rough terrain out of a handful of large triangles. He then wrote the code to break each one into 4 individual triangles. And then, through a process called iteration, repeated this dividing process many thousands of times. The end result of this relatively simple process was a realistic looking mountainous terrain.
Carpenter would go on to work with Lucasfilm and become a co-founder of Pixar Animation Studios.
Fractals in your Cell Phone
Many of us hackers are all too familiar with needing different antennas for different RF frequencies. Your cell phone has a handful of RF devices: WiFi, Bluetooth and a cellular transceiver to name a few. Most phones do not have individual antenna for each radio. The single antenna in your cell phone is what is known as a Sierpinski Carpet fractal. It turns out this style of antenna is the most efficient way of receiving signals for differing frequency bands, and works for all radios in the device.
I hope you enjoyed our little tour of fractals. There are many more examples of fractals in our everyday lives, of course. Let us know of any you’ve run across in the comments below.
Main Image – Fractal Science Kit
Thumbnail Image – Wikimedia Commons