August 12, 2020
Beyond Calculus
Captivated by the motion of waves & mesmerized by the perfectly-imperfect symmetry of leaves, it’s crystal-clear we’re a pattern-seeking species. Externally, studies have proven that we use patterns to weigh our environment of danger — a disruption in our daily routine (particularly back in hunter-&-gather societies) signals to our conscious that something is off. Internally, patterns are inscribed in our DNA; in an energy-conservation effort, most biological processes are duplicative & are therefore likely to generate a visual form indicative of patterns.
In math, the branch of math centered around the study of continuous, patterned, yet irregular scalar symmetry is known as Fractal Geometry.
The story for modern fractals traces back to the 17th century when Rene Descartes first introduced the concept graphing a polynomial function. As elegant & as evolutionary as they were, we (ie — the math community) eventually noticed a glaring issue in this definition: they failed to easily, predictably trace out the patterns noticed IRL. Using absurdly-long & convoluted polynomials in an attempt to graph out the examples shown above highlighted obvious shortcomings.
They were missing a key, required, shift in frame of reference. The trick to mathematically mapping organic patterns is to not analyze them as they are but rather to think of what it took to produce them. They were missing the principle of iterations.
Evolutionary forces carve out the most effective processes for a species & then repeat that process — thus, nature is commonly, organically generative. A more familiar, universal example of this is the famed Fibonacci sequence seen in sunflower patterns & spiral shells. A core reason for its popularity is that its intricacy is generated by iterating a stunningly-simple sequence:
The Fibonacci sequence is the simplest example of an iterative function, but it does the trick in highlighting it’s generative nature; for each new iteration, the input used is simply the output from the preceding iteration. It’s a basic but simple example of a fractal.
Fractals, the crux of fractal geometry, are infinitely complex & detailed patterns that are self-similar across different scales; they’re mathematical objects created by recursions of functions in the complex space. As we’ll see shortly, fractal geometry brings us much closer to replicating the irregularities & intricacies that surround us.
We’ve covered the concepts of iterating a function (ie — using its previous output as the next input), however, before moving on to Julia’s contributions, it’s worth covering the basics of complex functions.
A complex number, as hopefully most remember, is a two-part number with a real & an imaginary component: a + bi. When seen in applied exercises, the short-hand notion for these two-part complex numbers is the single letter: z.
When graphed, complex numbers use a different coordinate system, the complex plane; logically, the plane runs along two expected axis: real & imaginary.
Seen above, the coordinate system is no longer Cartesian along an x-&-y axis, but rather structured along the real & imaginary axis. Graphing complex numbers, once algebraically separated into their two parts, is straightforward.
Complex numbers, much like their counterpart, natural numbers, share a property of magnitude, or “size.” However, while the magnitude of natural numbers is immediately identifiable as the absolute value, arriving at the size of a complex number is a bit more involved:
Seen above, we turn to the good-ole Pythagorean Theorem; the relative “size” of a complex number is the shortest distance between the point & the origin in the complex plane.
As we’ve foreshadowed & will shortly see, the entire behavior for fractal geometry derives from iterating a complex function (specifically z² + c). The property we care most about is magnitude; analogous to approximating the limit of a natural function as X approaches infinity, the fractal concept known as “escaping” tests if a complex function orbit escapes from the origin (aka, the magnitude approaches infinity under iteration).
The prequel to modern fractal geometry begins in the early 1900s with a young protagonist by the name Gaston Julia. A curious collegiate student fascinated with music & mathematics, he was particularly drawn to complex numbers & functions.
His contribution to modern fractal geometry started when his attention was piqued by the 1879 paper by Sir Arthur Cayley, The Newton-Fourier Imaginary Problem. In it, Cayley sought out the roots of the equation f(z) = z³ + c = 0 using the Newton-Raphson iterative method. Since there are three roots, he wondered if one could predict which of the three roots a given starting value of z would reach as a limit. He failed in his quest & left readers with the open challenge: “appears to present considerable difficulty.” While Cayley’s solution to finding “basins of attraction” for each root through iteration was indeed on the right track, the technology of his time limited his perspective; seen below, when iterated, the shape of these basins is infinitely complex — providing us with an accurate preview of the scalar-symmetry we see in fractals:
Unfortunately, life abruptly interrupted Gaston’s academic endeavors midway through his academic career at the University of Paris when he was drafted & scripted into joining the army for World War I. Compounding his misfortune, in 1915 he lost his nose & was nearly blinded; awarded the Legion of Honour for his valor, Julia, unfortunately, had to wear a black strap across his face for the rest of his life.
Released from service at the young age of twenty-five, Julia triumphantly returned to his favorite topic: iteration of complex polynomial functions. In an impressive & successful effort, he published a massive 199-page book memoir on the subject in 191. Titled Mémoire sur l’iteration des fonctions rationelles, it was published in the eminent Journal de Mathématiques Pures et Appliquées, won him the annual Grand Prix award from the French Academy of Sciences, & temporarily catapulted him to academic fame. The publication, along with contributions by Pierre Fatou, is the crux of fractal geometry; though they were quickly forgotten in their time, they incontrovertibly laid the foundations for future mathematicians.
The groundbreaking dissertation drew particular attention to a crucial distinction between iterations of points (or orbits) that converge to a repeated pattern versus those that escape (aka their magnitude goes to infinity). Those that repeat after multiple iterations, typically marked by their mesmerizing visual patterns, are known as Julia Sets. The mathematical equivalent of Julia set inverses, Fatou Sets are iterated complex functions that lead to escaped orbits or clusters of orbits. Before moving on, it’s worth clarifying that the formula here doesn’t deviate from a complex binomial with only a changing constant (though remember these are complex number so they both have two parts):
Now, the Julia Sets below are for our visual assuage but they were not constructed by Gaston Julia as they require modern computing; this, however, does make Gaston’s accomplishment appear more impressive as it suggests that he mentally “viewed” the mesmerizing intricacy of bounded, iterated sets through manual work or thought experiments.
Hopefully, the above provides a concrete example of the difference between these two types of sets; they’re all Julia sets except for the two Fatou sets on the bottom row. Again, each of these visualizations is simply a result of iterating the function Z² + C with different values of C (shown above). As exciting as these initial findings were, they fizzled out of the mainstream & further research remained dormant.
A quick thought experiment before we move on — the examples represent only eight different variations of C out of the entire complex plane. What if were to happen if we were to iterate every single point in search of Fatou & Julia sets?
Progress remained stagnant for decades. In fact, the now-accepted term fractals wasn’t coined until the 1970s, when one Benoit Mandlebrot brought the topic back to life.
Appropriately, Benoit’s entry to the field was inspired by nature; the first relevant publication for modern fractals is found in his 1975 investigative report, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.In it, he explores geographical curves & discovers that while they’re undefinable, they are:
statistically “self-similar,” each portion can be considered a reduced-scale image of the whole. In that case, the degree of complication can be described by a quantity D that has many properties of a “dimensions,” though it is fractional.
Mandelbrot, who was born [in 1924] in Poland, had read the work of both men & studied under Julia in the 1940’s, thus, he knew where to turn. Shortly after his observations studying the Coast of Britain, he began using a computer to map out every Julia sets both in search of possible matches & to confirm his suspicion that Julia sets were marked by this fractal property. Mandelbrot observed & tracked the result of iterating each constant (C ) in the complex plane by placing a white or black dot depending on whether the function converged to a Julia or Fatou set.
The resultant diagram, seen on the left (or above on mobile) is today known as the famous Mandelbrot Set. There are many variations with rich colors & captivating gradients, however, the original diagram was a simple black-&-white rendition. For a more mathematical definition, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected; or, in other words:
The Mandelbrot set is a dictionary of all Julia sets:
The visual above is the crux of fractal geometry. Tying it all together, it highlights the relationship between the Mandelbrot Set & all the corresponding Julia Sets within. Every dot in black (within the set) connects to a continuous Julia set, yet the middle lowest red dot, stemming from the white (outside of the set), connects to a cluster, a Fatou set.
Mandelbrot subsequently displayed images of the set & elaborated on its significance in speeches, papers & books. Much like Julia’s publication a near half-century ago, the Mandelbrot Set rocketed Benoit Mandelbrot to academic fame; consequently, the renewed interest established fractal geometry & eventually set the foundation for additional branches of math (such as chaos theory).
I won’t explore the philosophy here but it’s nothing short of poetic irony that it took imaginary numbers to mathematically map out reality. There is order behind the chaos; as Julia & Mandlebrot taught us, all we need to do is shift our perspective. This is hardly the beginning however, next, we’ll delve into some semi-original work as we attempt to bring significance to the Mandelbrot Set in three-dimensions…
