Fractals, Dimensionality, and More!
I’m sure we are all familiar with YouTube’s notorious suggestions list, filled with tantalizing click-bait that could steal your attention for hours. While most adults will tell you to stay away from these entertaining time sinks, on Friday night last week, I found myself succumbing to the temptation! Indeed, after perusing videos ranging from Thermodynamics, to Political Loyalties in the Middle East, to, of course, Funny Cat Videos (because who hasn’t delighted in those!), I finally stumbled upon a word I had heard few times, but never took the time to decipher: FRACTALS. This inevitably led to two more hours of watching mathematical proofs, documentaries, and applications all surrounding this deceptively simple concept. So, without further ado, allow me to introduce Fractals, Dimensionality, and More!
Fractal Definition
A fractal can be simply defined as an infinitely complex pattern that maintains self-similarity across different scales. Fractals are typically created through iterative processes. Sounds complicated? Don’t fret. let’s begin by defining some terminology:
Self-similarity: Essentially, to determine whether a figure satisfies this condition, conduct the “zoom” test. Focus in on a small portion of a geometric figure. If that zoomed in section exactly resembles the original geometry, then the figure is said to be self-similar. This is the defining characteristic of fractals that can be described mathematically and has versatile real-world applications!
Iteration: The process of iteration can be best described using an example. Take the function f(x) = x+1. If you iterate this function around zero, set x = 0 and solve for f(x). Then, take this output and plug it back into the function as the new x and once again, solve for f(x). Continue this process several times and eventually you get a series of iteration!
The Julia Set
Research into fractals began in the early 20th century, pioneered by French mathematician, Gaston Julia. The goal of many scientists and mathematicians at the time involved trying to explain complex and “random” occurrences in nature through mathematical expressions. In 1918, Julia published his 199 page paper, “ Mémoire sur l’iteration des fonctions rationnelles” in which he described what would become famously renowned as the Julia Set.
Now, if it took Julia nearly 200 pages to describe the derivation and proofs of this set, it would undoubtedly take me a lifetime! So, to simply explain the math behind these fractals, picture a complex plane, on which coordinates are expressed in the form a+bi. Julia sets are created using the recursive formula (a.k.a one that repeats itself several times):
where both z and c are complex numbers, and a and b are between -2 and 2. Define C as a fixed complex number and choose an initial Z value. For example, when we defined iteration above, we used 0 as our initial x value. When you iterate this function around the starting Z value, a series will result for which the magnitude of Z sub n either:
- Tends to infinity OR
- Remains bounded to a magnitude less than 2. Note that magnitude here is defined as the distance of a point on the complex plane from zero.
Thereafter, assign a color to each initial Z value on the complex plane depending on how fast it tends to infinity when iterated, or color it black if it remains bounded. The result is a fractal image that can be created for every complex number (C )with a and b between -2 and 2. Let’s look at an example:
Say we want to create a Julia Set fractal for C = 1+2i. Draw a complex plane:
Next, take one point, let’s say -1+1i, from this plane and define it as Z initial. Plug it into the recursive formula:
If the iteration leads to a number that “blows up” and goes to infinity, color it blue. Else, if it “bounces” between numbers and never exceeds 2, color it black. If you do this for every point on the plane (taking into consideration how fast iterations “blow up”), you’ll get an image somewhat like what is shown below:
NOTE: this is not the fractal for 1+2i, just an example to show how the coloring based on Z sub n creates a self similar figure!
Because there exist infinite C values with a and b between -2 and 2, there are infinite Julia sets that each have unique fractal images. However in the early 1900s, powerful computing tools for conducting millions of iterations were not yet developed, and thus it wouldn’t be until the 1970s for extensive fractal computation and visualization to “take shape” (haha).
Mandelbrot Set
Fast forwarding to the 1970s, Benoit Mandelbrot, a Polish born mathematician, famously quoted, “Think not of what you see, but of what it took to produce what you see.” Working at IBM, Mandelbrot harnessed this newest computing power to conduct Julia’s recursive formula iterations, but of course, not without a little twist.
Mandelbrot was motivated by “… a pragmatic desire to model nature in a way that captured roughness.” This goal directly contradicted with prevalent calculus theories that claim figures can be modeled by smooth curves. Take Riemann sums for example. Calculus idealizes Riemann sums by using them to approximate the area under smooth curves. However, upon closer inspection, these rectangles actually have a square roughness that cannot be captured using integration. Mandelbrot’s goal was to study the realistic roughness and irregularity that exists in figures.
So that’s the concept, now let’s look at the derivation. Mandelbrot began with a Julia set. However, instead of manipulating the initial Z value, he made all iterations begin at zero and changed the value of C using all numbers on the complex plane with a and b between -2 and 2. Thus, unlike Julia sets which are infinite and fix C, the Mandelbrot set converges to a singular fractal, albeit one much more complex than its predecessor.
As you can see, depending on the chosen value of C, iterations around zero may tend to infinity or remain bounded and are colored in the same manner as the Julia sets. Therefore, the Mandelbrot set (M) is officially defined as the range of complex numbers C for which the magnitude of iterations around zero remains bounded (a.k.a the black regions)!
And of course, to show the self-similarity of the Mandelbrot set, check out this one hour zoom of the incredible fractal. WARNING: it’s addicting!
Example of Infinite vs. Bounded Iterations
So in the past two sets we frequently brought up iterations that “tend to infinity” or “remain bounded.” But what exactly does that look like? Once again, what better way to demonstrate than through example:
Let’s start with the function f(x) = z² + 1. If we we set z initial = 0 and iterate, we get a series for which f(x) becomes very large very quickly:
f(0) = 1
f(1)= 2
f(2) = 5
f(5) = 26
f(26) = 678
Because f(x) shows no sign of getting smaller the longer we iterate, we can safely call this function diverging (tending to infinity).
Now let’s make a slight change. What if we iterate f(x) = z² -1?
f(0)=-1
f(-1) = 0
f(0) = -1
f(-1) = 0
As you can see, the resulting series bounces back and forth between 0 and -1. Because f(x) will never assume any other two values, we call this a bounded series.
Koch Snowflake
Now to take a break from the mathematical monsters we’ve tried to decipher, let’s take a quick look at some fractals that are more conceptual, but help enforce the idea of infinite complexity. In 1904, Swedish mathematician Helge von Koch concocted his paradoxical “Koch snowflake.” The rules are very simple. Begin with two triangles and superimpose them to create a six-sided star. For each of these sides, replace the middle third with another protruding equilateral triangle. Continue this process infinitely. The progression looks something like this:
Now what is intriguing about this snowflake is the fact that it has a finite area but an infinite perimeter. Imagine inscribing the snowflake inside a circle. It is obvious that the more iterations of side length fractioning will not cause the snowflake area to exceed that of the circle. However, as the sides continue to be fragmented, the snowflake’s perimeter grows exponentially forever!
Application: Koch’s Fractal Antenna
In 1988, Boston University professor Nathaniel Cohen decided to play around with electrical wire and made a ground-breaking discovery about antennas that drew inspiration from Koch’s 20th century snowflake. Cohen noticed that if he bent an electrical wire into a similar equilateral triangle geometry, he could receive and transmit many more electrical signals while minimizing the area occupied by the antenna. What became known as the Cohen fractal antenna, this technology revolutionized wireless phone innovation. Instead of requiring multiple antennas to communicate various types of signaling, small chips using self-similar fractal wiring allowed phones to become compact, and this technology is now present in almost all wireless devices!
The Cantor Set
Even before Gaston Julia hit the ground running, German mathematician Georg Cantor began simple fractal theory by playing around with a line in 1883. Fascinating right? Now don’t be too quick to judge, because Cantor’s “mathematical monster” puzzled the math community for years before more sophisticated iterative understanding could evolve. Essentially, Cantor took a line and divided it in three, and then erased the middle third. This he did for each consecutively smaller line. The result was something as shown to the left:
What is interesting about this set is that if we were to add all of the middle sections deleted through each iteration, we would get 1 a.k.a the length of the original segment! Incredible right? If we want to get into the math, of course, you can confirm this with the mathematical expression for Cantor’s set:
Dimensionality
As if fractals couldn’t get more complicated, there’s one more topic to touch on before you can walk away with a somewhat comprehensive grasp of the field. In practice, fractal dimensionality is officially defined as “a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in.”
Traditionally, we discuss figures and shapes in 3 dimensions. A line is one-dimensional, a triangle is two-dimensional, and a cube is three dimensional. But when we discuss fractal dimensionality, we often use non-integer numbers like 1.585 for the Sierpinski triangle and 1.262 for Koch’s snowflake. So, what does this tell us about the geometric figure? As usual, let’s turn to an example using the famous Sierpinski triangle fractal.
The Sierpinski triangle is created by starting with a solid triangle, and continuously cutting out the middle fourth triangle of the original triangle. Or in other words, inscribe an equilateral triangle within each solid triangle and then remove that area. The result looks something like this:
But what dimension does this figure assume? You might argue that you are beginning with a 2 dimensional triangle and therefore the Sierpinski triangle still possesses an area an must also be in the second dimension. Or, you might suggest that removing infinite triangles leaves only linear divisions that define the triangle and thus, it can be thought of as a line in the first dimension. Fractal dimensionality essentially says that the Triangle is somewhere in between the two, specifically in the 1.585th dimension. This number describes how the fractal fills space. But how do we arrive at this number?
Essentially, determining fractal dimension has to do with understanding how scaling down the figure affects the number of pieces that are needed to create it. For simplicity, imagine a line. If we divide the line in two, we have just scaled it down by a factor of two and now have two equal lines that compose the original. To save you the derivation, dimension can be calculated by dividing the log of the number of pieces created by the log of the scaling factor. In this case:
log(2)/log(2) = 1 ** A line therefore lies in the first dimension
Now let’s look at a square. If you scale a square down by a factor of two, you reduce each side length to one half its original size. If you compare this scaled down version to the original, you will notice that it takes four copies of the smaller square to recreate the original one! Thus, if you plug it into the formula for dimensionality you will get:
log(4)/log(2)= 2log(2)/log(2) = 2 ** And lo and behold, a square occupies the second dimension!
So now let’s look back at the Sierpinski triangle. If we take scaled down the original triangle by a factor of 2, how many of those smaller triangles would it take to recreate the original? If you said three, then you’re on the right track. If we plug these conditions into the formula we get:
log(3)/log(2) = 1.585 ** The non-integer dimension of this fractal!
Now, while this serves as a great introduction to the concept of dimensionality, this calculation is restricted to figures that display self-similarity. If we want to prove the dimensionality of a circle for example, we cannot connect smaller pieces together to reconstruct the original. Thus exploration into dimensionality becomes much more complex… so expect a follow up on this topic in the future :)
Applications
I know I’ve just bombarded you with information, but I think it’s important to take a step back to answer well, why does this matter? As you can probably tell, fractals have revolutionized not only human innovation in technology and arithmetic, but also have allowed us to explain seemingly “random” aspects of nature that once so elusive, can now be modeled and predicted using iterative math.
Lung and tree branch networks display self-similarity, as do human neural and circulatory systems. Each bifurcation very closely resembles its parent branch. This revelation in biological understanding has opened incredible doors for medical innovation. For example, instead of programming each branch of the lung blood vessel system for molecular modeling, scientists can now implement a simple fractal code to simulate this complex system.
“If all these biological networks are fractal it means they obey some simple mathematical rules which can lead to new insights into how they work.”
So where do you see fractals in nature, and what do you believe the future holds? You might be surprised as to how many “chaotic” aspects of the world around us are actually governed by the captivating and enigmatic realm of fractal geometry. With ever growing sophistication in mathematics and computing tools, the possibilities are “endless!” (get it?)
But of course, if complex math isn’t your style, hopefully incredibly witty pickup lines are… Enjoy!
Works Cited:
MITK12Videos. “What Is A Fractal (and What Are They Good for)?” YouTube, Massachusetts Institute of Technology, 11 June 2015, www.youtube.com/watch?v=WFtTdf3I6Ug
Sanderson, Grant. “Fractals Are Typically Not Self-Similar.” 3Blue1Brown, YouTube, 27 Jan. 2017, www.youtube.com/watch?v=gB9n2gHsHN4.
Jimi. “The Julia Sets: How It Works, and Why It’s Amazing!” YouTube, YouTube, 3 Apr. 2017, www.youtube.com/watch?v=mg4bp7G0D3s.
TheBITK. “The Mandelbrot Set — The Only Video You Need to See!” YouTube, YouTube, 12 June 2016, www.youtube.com/watch?v=56gzV0od6DU
Mills, Ashley. “Calculating Fractal Dimensions.” YouTube, YouTube, 29 Oct. 2015, www.youtube.com/watch?v=RFMZZ4pPKlk
Bourke, Paul. “Julia Set Fractal (2D).” Polygonising a Scalar Field (Marching Cubes), June 2001, paulbourke.net/fractals/juliaset/.
“Fractals and the Fractal Dimension.” Volume 49 | Journal of Transnational Law | Vanderbilt University, Vanderbilt University, 2007, www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html
