Fractals for Dummies
Fractals are stunning. You may have seen one of these videos of a structure within a structure within a structure, on a seemingly infinite series. Beyond their amazing aesthetics and their hypnotic repetition effect, fractals are especially interesting because they seem to show that you can have an infinite number of levels, or scales, or iterations, within a limited structure. In other words: anything which is finite and fractal may contain the infinite within itself.
This video took 16 days of round-the-clock calculations to make. It shows a fractal pattern occurring over 350 million iterations, through a 10^198 times—too many zeros here to write with any other notation—zoom. It was made in 2014.
Since then, the same youtuber managed to compute another fractal video, this time showing 750 million iterations:
As you can guess, the limited number of iterations is only due to the limitation of computing power. A supercomputer could come up with more iterations, a super quantum computer with even more, and so on. The point is this: it is theoretically possible to calculate an infinite number of iterations by zooming continuously and indefinitely within a finite space. The pattern similarity across an indefinite number of occurrences and levels echoes the infinite.
No wonder why so many non-mathematicians “feel” that fractals make them dwell on the meaning of life. What does it mean to be finite—occupying a limited portion of space, having been born at such time and place, likely dying at some time and place—when even the smallest portion of space can hold infinity? It looks like the universe is made of an infinite number of parts which also have an infinite number of parts. An infinity of infinites within an infinite. Just like an “infiniteception.” But you need a fractal phenomena to witness it.
How the Fractals Were Found
The fractals are widely attributed to mathematician Benoît Mandelbrot (1924-2010). A student of the French école polytechnique, then a teacher at Harvard, Mandelbrot was a polymath who got a master degree in aeronautics but remained fascinated by the stock markets for all his life.
Over his long career, he was tasked with studying and teaching on complementary sets called the Julia sets and Fatou dusts, two complementary mathematical sets, the first ones worked on earlier by mathematician Gaston Julia who had been one of his teachers. Mandelbrot was lucky enough to have access to brand new IBM computers, still an academic luxury at the time. Using them to generate Julia sets, he isolated a sub-class of these which remained invariant no matter their scale. Such sets had a much higher degree of recursion and self-similarity than the other Julia sets.
Mandelbrot called these sets fractals. He coined this name out of the Latin fractus, meaning broken or shattered, as such sets were never smooth as an Euclidean right, but—almost—equally intended, curved, sharped and so on at any scale. In 1975, he published a book in French (Les objets fractals: forme, hasard, dimension), soon updated and followed by another work in English (The Fractal Geometry of Nature, 1982). The latter showed that fractals were not mathematical artifacts, as their “virtual” origin may hint, but a phenomenom which really occurs in nature.
The Romanesco cauliflower, used as the main illustrating picture for this article, the ammonite suture, the fern bush and some mountain ridges are but a small sample of all the natural objects that show a self-replicating aspect.
His sets made famous, Mandelbrot went back to what he deemed their most interesting application: the markets. In Fractals and Scaling in Finance (1997), he demonstrated that finance operators relied on unreliable tools but managed to “get it” by tweaking said tools from guesswork. In The Misbehavior of Markets: A Fractal View of Financial Turbulence (2006), he showed that one could not predict the future stock market values due to, inter alia, non-linearity and the absence of periodicity in market cycles, even if chaotic fluctuations did not happen completely at random. Nassim Taleb, whose books are easier to read for non-mathematicians, was hugely inspired by Mandelbrot.
Where Mandelbrot did not tread, or not too much—and he already did much—others would. In 1995, the mathematician prefaced the book Fractals in Petroleum Geology and Earth Processes, a multiple authorship effort exploring methods of crude oil reserves estimation, topographical effects of drilling and other related fields. Special effects for movies routinely use fractals to enhance their looks. In 2004, Google credited Julia, Mandelbrot’s former teacher, with having laid the cornerstone of the branch of mathematics from which they developed the PageRank technology.
As computed scientist Stephen Wolfram wrote:
Mandelbrot ended up doing a great piece of science and identifying… a fundamental idea. Put simply, that there are some geometric shapes, which he called “fractals”, that are equally “rough” at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space…
Nature is more complicated than classical geometry. The science of rights, squares, cones and so on is made of approximations. It works well for highly predictable phenomena with no scaling problem, such as the mechanics where Archimedes or Leonardo da Vinci excelled and as the stars, planets and comets whose Newton accurately predicted the position at particular dates. However, if we look to the Earth and the smaller instead of the so-called Platonician or geometrical sky, we do not see smooth shapes but rougher fractals.
Although Mandelbrot was born a Polish Jew, he picked up what he needed to in France before finding the fractals. This looks like an ironical revenge from René Descartes (1596-1650), the founder of the Cartesian philosophy, who advised on his Discourse on Methods to remain conscious of the smaller and mundane you can be sure of instead of inquiring on doubtful things that come with fashion and prestige. Newton always rejected Descartes’ theory of cosmos and, although his own theory of the universe was exemplary, the English astronomer cared a lot about fame and reputation, to the point of slandering his rival Leibniz over a personal dispute.
In several aspects, Mandelbrot was a Cartesian: he progressed through patient, cautious mathematical estimations, and he cared more about getting things right than about running to the fashionable. He took years to turn the drab book of 1975 into a mesmerizing view on nature, although his newer ideas were entirely derived from the older. Mandelbrot’s current pop status, added to his scientific discoveries, might be seen as a sort of ironic externality—or as fractal in itself, the world of science producing “pop star” scientists at a random scale.
The Newtonian synthesis, though, is no more dead than Euclidian geometry. Newton’s decidedly nonfractal model has been a breakthrough in itself and can still be used to track a variety of celestial bodies. Euclidian geometry is still studied dutifully by STEM students as well. Fractals just remind us that no matter the usefulness of finite mathematical models and Newtonian geometry, they are always “pictures in our heads”, as American journalist Walter Lippmann would say. The real world is rather fractalian and full of innumerable occurrences. It is daunting, thanks to its complexity seemingly going beyond our sagacity, and enthralling as well. Even the portion of space occupied by your body holds infinity! You just need some fractal lenses to see it. As Heidegger would say, here too the gods are present.
Perhaps this is what compelled Mandelbrot to say that he felt like a finder rather than an inventor. After all, he mostly explored an unsuspected field thanks to the progress of IT. He did not create fractals. Perhaps no one did. Nonetheless, fractals seem to play an important role in the rhythm of evolution and are more than likely to help us get order from the current chaos.
A Rainy Afternoon in Brittany
One rainy day in Brittany, while thinking of ways to occupy the three children who were visiting us, I thought of an interesting game:
Find out the length of the coast of Brittany. He who comes closest to the correct answer wins a piece of Kouign-Amann.
(For those not from Brittany, this is a local cake so delicious that, once you had a taste of it, you just can’t stop eating.)
After a short silence, the three children came back, all believing they had found the right answer. The older showed me the map of France which he had used to calculate, and announced proudly: about 260 kms. His younger brother announced that to his mind, the coast was almost its double, which is 500 kms. And in order to back up his conclusion, he showed us conscientiously his calculations made on the basis of a far more detailed map of Brittany used by trekkers. Finally, the youngest announced almost disdainfully that both his elder brothers had got it completely wrong:
The coast of Brittany, in my view, he said, couldn’t be measured as it is infinite in length: just ask a snail to go round all its rocks in order to realize that it is much longer than what my brothers had affirmed. My brothers think they are more intelligent just because they are bigger!
Mandelbrot too thought about the Brittany coast as a perfect example of fractal geography. His book on fractals in nature features a chapter on the most legendary West Coast of France where he shows that it all depends on the altitude from where one wants to scale. We could define the coast as a fraction, that is, as a non-integer number which cannot be reduced to one of the three dimensions classical geometry usually deals with.
I gave pieces of Kouign-Amann to all three children. And the story doesn’t stop there. The next day, it was raining too. (Spiteful tongues say a non-rainy day in Brittany is not a Breton day.) I took advantage of it to ask the children to describe what they saw in fractal images. Here is what they said—and if children can recognize and define fractal images, big children like us should be able to do it as well.
This is a Mandelbrot Set or Group:
Jeremy’s (age 6) comment:
You make a big circle, then smaller ones next to the big one, and still smaller ones on the smaller ones, and when they become too small you make points.
This one show a clear pattern and an easy way to build a fractal image:
Kevin’s (age 7) comment:
There is a cross within the cross within the cross within the cross within the cross!
This one is called a Koch snowflake:
Bertrand’s (age 8) comment:
You cannot measure it with a ruler; it never stops.
You will notice that the outer line of the snowflake (d), though of an infinite length if we keep zooming again an again, fits into a limited surface. In other words, the infinite length appears within a limited surface.
This cube is called the Menger sponge:
It’s like a Lego with a cube that helps us to make a bigger cube that helps us to make a bigger cube that helps us to make a bigger cube.
The closer you come or the further you move away, the more it is the same.
Each portion can be seen at any scale: each part is (visibly) a copy of the whole. This is self-similarity. Within a snowflake, you can see a structure with six sections, and if you zoom on a single section you can observe that it too is composed of six sections, ad infinitum. The same is true with clouds, mountains, thunder lightnings, trees, rivers, drying soils and even galaxies.
By now I hope you gained a better understanding of what the fractals are. To sum it up :
- Fractals were discovered during the second half of the twentieth century by mathematician Benoit Mandelbrot at Harvard, on the basis of French mathematician Gaston Julia’s work;
- Without going into mathematical details, a fractal object or phenomena can be acknowledged thanks to its self-similarity and its way to reproduce the same pattern or structure at any scale;
- Fractals cannot be understood through classical or Euclidean geometry, as their structures are always too jagged to be adequately summed up to rights, triangles and other classical forms, and as they never become smoother;
- Though found in labs from computer-powered calculus, fractals are first and foremost a natural phenomena;
- Fractals prove that a finite space, no matter its size, can contain an infinite number of parts or levels of observation.
And now, what about you?
By using relatively simple mathematical formulas, mathematicians are able to create complex geometric constructions, from an infinitely small pattern which can be repeated up to the infinitely large, that reproduce nature as faithfully as possible. Not perfectly, in fact, far from it, but far more faithfully than traditional geometry did.
Fractals also have applications much beyond narrowly scientific fields. They can be used in way more professional, or down-to-Earth fields, such as management, making sense of the world and optimizing our daily routines.
This is what my blog is all about.
—Bruno Marion, the Futurist Monk