My middle-school mathematics classes were held in the basement, room 01. There was nothing pretty there, not the stained walls nor the humid smell. Most of the students had woken up less than an hour before, paying little attention to our teacher, laser focused on the clock’s minute hand. Customarily, a student asked, ‘what is the use of mathematics is in real life?’ Without hesitation, our teacher let out, ‘because there’s beauty to be found.’
He did not elaborate further. But, at that moment, the conviction in his voice was enough to make us understand, albeit loosely, that mathematics, like most subjects, need not be confined by practicality. The following article is an exploration of fractals. One of the innumerable beauties of mathematics.
The (Re)Birth of Fractals
In the early 20th century, mathematician-meteorologist Lewis F. Richardson asked himself a seemingly trivial question, namely, how long are the common borders between Spain and France, and Belgium and the Netherlands? His search for answers in these countries’ encyclopedias was staggering. Their disagreement on the distance was in the 20% range. A question that at first glance seemed trivial, proved to be far more complex.
Richardson turned his attention to Britain. Noting that using a smaller ruler or scale of measurement resulted in larger measurements of the coastline, he initially supposed the measurements would converge to a finite number representing the coastline’s true length. As Richardson later demonstrated in the infamous Coastline Paradox (a.k.a. the Richardson effect), this proved not to be the case. The measured length of coastlines and other natural features, increases without limit as the scale of measurement decreases towards zero (see Figure 1).
Richardson’s findings were largely ignored by the scientific community for being theoretical gimmicks. Yet, in the early 60s, a French-American mathematician named Benoît Mandelbrot stumbled upon the then-deceased Richardson’s paper and decided to pick up where the latter had left off. Mandelbrot, a gifted visual thinker with a real knack for pattern recognition greatly expanded on the coastline paradox in his 1967 paper titled How Long is the Coast of Britain?
Where others saw a dead end, Mandelbrot saw an entirely new realm of unanswered questions within mathematics, that of fractals and fractal dimensions.
Mandelbrot, being the heretic that he was, figured that discussing the length of a coastline was an entirely useless endeavor. Not only that, but he also concluded that the discipline of mathematics had been put in a straitjacket by Euclid’s one-, two-, and three-dimensional universes; x, y, and z. The fractal curves of the coastline paradox were unaccounted for in classical geometry, for they were not distortions of standard lines, planes, circles, or squares, but entirely new structures of their own.
Koch snowflakes (see Figure 2), based on Swedish mathematician Helge von Koch’s Koch Curve, are structures that help visualize and understand the features of fractals. They are constructed by taking an equilateral triangle, separating each side into three equal segments, using the middle segment as the base for a new self-similar equilateral triangle, and then iterating this process over and over. Each iteration increases the perimeter to 4/3 its previous size, in principle without limit to infinity. Still, if one were to draw a circle around the original triangle, it would never intersect with the iterated structure, paradoxically allowing an infinitely long line to be drawn inside a finite area.
Fractals are infinitely complex patterns that maintain self-similarity across scales, constructed by iterating simple processes and resulting in structures that exist in between our familiar dimensions (see Figure 3). Fractals are unique for having non-integer Hausdorff/fractal dimensions (a statistical index of complexity presented as a ratio of how the detail in patterns changes as the scale of measurement changes. Koch snowflakes have a dimension ≈ 1.26186). Their conception plagued physicists and mathematicians but was warmly received by artists, who had been toying with concepts of self-similarity for over two millennia.
Fractal patterns are instinctively appealing to the human eye and attempts to integrate them into the arts are by no means a novelty of the 20th century. The architecture of Hindu temples, designed to reflect their philosophy of the universe as “holonomic and self-similar,” and the geometric patterns found in Islamic design, particularly the arabesque which is sometimes interpreted as reflecting unity arising from chaos, are two examples among many (see Figure 4).
Early attempts to represent fractals in architecture and ornamental painting were limited by the technology available at the time, or lack thereof. It was not until the 1980s when computer-based modeling flourished and chaos theory (the study of underlying patterns in apparently random complex systems) gradually cemented itself in mathematics departments, that detailed images of fractals were generated. The results were so aesthetically appealing that they rapidly penetrated the walls of the scientific community and began appearing in popular culture and art.
The first breakthrough came in 1980 by Benoît Mandelbrot himself. After stirring up a storm with his coastline paper, Mandelbrot turned his focus to computer-based modeling, taking advantage of his position at IBM to feed equations into the world’s smartest computers for them to iterate and print out in image form. In 1980, he fed an IBM computer the simple equation fc(z) = z2+c, and had the program plot the set of all complex numbers c which did not grow to infinity when iterated from the starting point z = 0. The result is the now ubiquitous Mandelbrot set (see Figure 5).
By 1985 the Mandelbrot set had made its way to the cover of Scientific American, leaving behind ripples of excitement as it emanated across all fields of science and culture. James Gleick, the author of Chaos, put into words the structure’s complex beauty: “every new molecule would be surrounded by its own spirals and flame-like projections, and those, inevitably, would reveal molecules tinier still, always similar, never identical, fulfilling some mandate of infinite variety, a miracle of miniaturization in which every new detail was sure to be a universe of its own, diverse and entire.”
The exponentially increasing capabilities of personal computers in the 1980s allowed hobbyists and laymen to reproduce complex fractals, such as the Mandelbrot set, at home in their spare time. Macintoshes had enough power to iterate functions tens of thousands of times and produce incredibly detailed fractal galaxies populated by smaller galaxies of seahorse tails, antennae, islands, spirals, and other hypnotizing structures. By the end of the decade, the Mandelbrot set has become a symbol of mathematical aesthetic and beauty, as well as a rallying-banner for the belief that complex outputs need not have complex inputs, that simple processes could, and did, lead to chaotic outcomes. In a way, fractal geometry aligned mathematics with the late 20th century’s feeling for undomesticated, untamed, nature and art.
Do we like simplicity?
The emergence of fractal geometry in the second half of the 20th century, and its shift of mathematical aesthetic, was paralleled by an aesthetic change within the arts. The first half of the century had been dominated by Euclidian sensibilities, with a design aesthetic revolving around simple geometric shapes, minimalism, and functionality, most deeply ingrained in the Bauhaus school (see Figure 7).
The early popularity of Bauhaus spread to abstract art, where compositions remained strictly disciplined. Interwar period abstract art compositions are often built upon straight-line foundations, consisting of little more than simple geometric shapes, and depending on chromatic interactions to imbue the works with life and aesthetic appeal. This design philosophy – purely antithetical to fractal geometry – is most acutely represented by the work of German-born artist Josef Albers (see Figure 8).
The interplay between Euclidian and fractal geometry raises interesting questions about which types of structures resonate most with humans’ eye for beauty. In any natural object, say a tree, or a coral, is the most important feature the arrangement of straight lines, running from the trunk, through the larger branches, or the fractal aspects, found across scales, from the trunk all the way to the most minuscule twigs?
The stance of many Chaoticians was that simple shapes were inhuman, simply out of tune with the way nature organizes itself and with the human perception of it. They believed the key to inspiring a feeling of beauty lay in the harmonious arrangement of order and disorder. Strict geometric shapes, such as those found in the Bauhaus school, were characterized on a unique scale. Chaoticians argued that to be satisfactory, art must purge itself of scale, having unique elements that are perceived only to a far-away observer, and others that can only be discerned by putting one’s eye inches from the canvas.
Mitchell Feigenbaum, a pioneer of Universality – an observation that a diverse class of dynamical systems shares certain properties irrespective of their dynamical details – believed that “art is a theory about the way the world looks to human beings… What artists have accomplished is realizing that there is only a small amount of stuff that’s important, and then seeing what it was…”
Feigenbaum pointed to Turner and Ruysdael (see Figures 9,10), as artists representing schools of painting that lacked scale. If you look closely there is infinite detail, but you must take a step back to see the unity. Interestingly, centuries before the mathematics of fractals as iterative structures were defined, painters like Turner and Ruysdael displayed an innate understanding of nature’s processes. If one looks closely enough, they can see the iterative patterns of brush strokes layering paint over paint and bringing the water to life; capturing a principle of nature that eluded mathematicians and physicists for centuries to come.
The relationship between mathematics and arts has taken many forms throughout history, but very few instances have produced results rivaling the natural beauty and omnipresence of fractal geometry. As humans we feel a natural attraction to order within chaos, across all scales, an attraction best captured by William Blake’s line, to see the world in a grain of sand.
Gleick, James. Chaos: Making a New Science. Harmondsworth: Viking, Penguin Books Ltd., 1987. Physical copy.
Mandelbrot, Benoît. “How Long is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.” Science Vol.156, No.3775 (1967): 636-38. <http://www.jstor.org/stable/1721427>.
Trivedi, Kirti. “Hindu Temples: Models of a Fractal Universe.” The Visual Computer, Vol.5 (1989): 243-58.