Ever fired up the Xbox and taken a moment to appreciate the lifelike nature of a simulated landscape? No. Of course not. However, little known fact: oftentimes their creation is facilitated by fractals. A fractal is, most basically, a shape created by an endlessly repeating pattern. Though fractals are not commonly used, they can be remarkable, particularly in art. Allow me to show you just how these objects are created.
The infinite perimeter of a fractal
Say you start out with a line. Remove the middle third of the segment and create a point with two more segments of the same length, as shown. Then perform the same function on all the newly created segments. Keep repeating this operation on a smaller and smaller scale to generate this fractal. Now as you might notice, the length of the combined segments increases with each iteration. The equation for determining the length at a given iteration is , where s is the length of the original line (3 in this case) and n is the iteration. These patterns can be expressed mathematically by the series: or simplified: the sequences continuing on forever. The length increases without bound because here you have an infinite series that does not converge. That is, the lengths never approach a specific number with increasing iterations. In fact, the sum approaches infinity. This particular fractal is called a Koch curve.
Interestingly, the area within a fractal such as the Koch Snowflake can be calculated. This shape is created by connecting three of the aforementioned Koch curves. You need only take the sum of the area of its infinite equilateral triangles, pictured above. These theoretical fractals can have infinite perimeters encompassing finite area. And the neat factoids don’t stop there.
The fractal gets its name from its interesting place in geometry. Looking at it, you can see that the object is not quite one-dimensional, and not quite two-dimensional. Other fractals can be used to model rugged surfaces (think of a piece of crumpled aluminum foil). Is that considered two-dimensional or three-dimensional? The answer is, it’s somewhere in between. They are considered to have a fractional dimension such as 2.5 or 1.7, hence the name “fractal”.
crumpled aluminum foil
Furthermore, Fractals can approximate length or area to whatever degree of accuracy the calculator desires. This goes to show that the precision of measurement always depends on the unit you are using to measure. A man named Mandlebrot used this logic when proposing an estimate for the length of the coast of Great Britain, a paper where the notion of a fractal was first introduced. Measuring in kilometers yields a wildly different answer from measuring in centimeters, or measuring by the width of a grain of sand. The result increases in accuracy the smaller the unit. Herein lies the fundamental concept.
Still from Disney’s Frozen
Currently, fractals are most applicable in art and design. They are used to create remarkable bits of animation such as in Disney’s Frozen. For simulating a mountainous region, a programmer might iterate a fractal-like pattern to create its rough looking surface. But fractals can also be useful in technology. Fractenna is a company constructing more powerful and efficient antennae based on fractal geometry. Even architects often design campuses that follow fractal-like patterns. These layouts are often considered more practical. And, probably most importantly, one can actually calculate the dimension of a rough surface using fractal analysis. Consequently, we now know that the dimension of your average piece of broccoli is around 2.7.
Published in Deerfield Academy's science magazine, Door to STEM.