Beauty and the Fractal

Beauty and the Fractal

This article draws from my book Chaos, a User’s Guide, chapter “Our Species Is Evolving: And You? (Beauty)”.

Remember assisting Beauty and the Beast when you were a child? Or, perhaps, as an adult? I do remember the 1991 Disney version. My smaller siblings enjoyed it, and they were right. Not only is this version of the anime more cultish and intemporal than the version Disney produced last year, but the tale itself is an all-time classic. A man-beast frightens people because of his weird, hairy outlook. Beyond said appearance, though, he is a fully human individual. This story, which teaches children about inner beauty and encourages them to seek beyond the superficial, is so typical it has become a reference number (425 C) in an international classification system for folktales from all over the world.

Beauty and the Beast‘s lesson can relate to fractal phenomena as well. At the beginning, fractals were either dry, colorless mathematics, or merely something from the lab. Mathematician Benoit Mandelbrot, the man who coined the word “fractal”, was criticized for focusing on laboratory artifacts, only produced by equations and computers. It is easy to laugh at this criticism today. But it was not completely groundless either: as Mandelbrot acknowledged, 19^th century mathematicians did their best to flee the “real” world and focus on disembodied numbers. Much of the 1850-1920 mathematicians’ research looks like a weird series of mind games clothed into the utmost academic officiality. Even to a full-fledged professional of quantity, this may look silly, and Mandelbrot’s fractal theory was not privileged to come up after a track record some would say is meaningless.

But fractals are much more than that. They are, first and foremost, a natural phenomena. It does not matter exactly how they were found. The very concept of fractal is mostly a set of goggles which allow whoever wears them to see profound similarities between galaxies and snowflakes, between very “real” cauliflowers and stock market cycles, between clouds and mountain ridges. Such similarities were noticed thanks to the concept, which has mathematical origins, before they were collected. And they are stunning.

Beautiful Chaos

Although fractals are the main inspiration behind my views, I must admit I gained more fame by talking about chaos. Fractals are often perceived as something between the fascinating and the mundane. They are beautiful enough to get displayed as a wallpaper, or hung on a wall, but that’s it—whereas chaos, being considered as a threat, compels a deeper interest. You may have googled “chaos theory” at least once in your lifetime. Conversely, Mandelbrot is almost only known by the scientists or those willing to use his discoveries in specific fields such as bankers or engineers.

Yet both themes are closely related. To turn chaos into a friend instead of a perpetual threat, getting the best out of whatever’s out there, making order out of the mess—which you may find more interesting than mere wallpapers—requires at least a basic understanding of the principles behind these stunning fractal pictures.

Chaos is often mistaken with disorder. Thus, one may confuse it with the lack of order and perceive it a something purely negative. In truth, it is entirely something else. From a philosophical point of view, chaos means something raw or undetermined enough to let ample room for potentialities. In a completely ordered world, nothing is possible beyond what is already there: early Enlightenment philosopher Gottfried Leibniz (1646-1716) ascribed to God the ability to determine everything which would exist, thus believing that chaos was merely an illusion and that everything we’d meet with was completely pre-ordained by God’s eternal calculations. But what Leibniz believed to be the product of divine perfection looks today like a dollhouse. How interesting is it to live in a world where you cannot truly take any decision?

Chaos is closer to what Aristotle viewed as potency. A chaotic system may be brimming with possibilities: the problem, from our point of view, is that all of these cannot be actuated simultaneously. Only some possibilites can be turned into action at the same time. If you were blessed with exceptional genetics, you might be able to turn into a triathlon medalist or into a powerlifting champion, but not both, each discipline requiring a very different physique to succeed. As a teenager, in your “raw”, undeveloped state, you might practice both sports, but if you truly want to achieve a professional level, you have to choose at some point. Which means more structured action and less chaos.

On the other hand, the very possibility to become this or that has something exciting. Children play the fireman, the doctor, the explorer, and many other “jobs” well before they have to choose a particular course of study. A friend of mine told me he felt the same in videogames where one has to mold, or build over, some virgin matter: molding a virtual virgin field in Simcity 4 before you lay the first stone, as if you were God himself, or riding the oceans over a couple caravels in Europa Universalis 4 and meeting with all these raw lands waiting to be explored, charted, built over, “is absolutely thrilling.” Even more than the act of building anything later.

In the Genesis, chaos is said to predate the created world. God is said to create the earth and sky but simply “moves” over the waters. We know today that life cannot appear without water. Indeed, on the Earth, the first living beings appeared in the oceans well before they evolved far enough to venture on the shores. Water has no shape of its own, but without it most of the shapes we know about would not exist, this including our own lives.

From a scientific point of view, chaos has a precise meaning as well. A system is said to be chaotic when it shows some fundamental unpredictability, i.e. when you can never compute with absolute certainty what its state will be in the future, the ability to change according to feedbacks and a high sensitivity to initial conditions.

The butterfly effect is a prime example of chaos. Can the flap of a butterfly’s wings in Brazil set off a tornado in Texas? Mathematically, yes, it can: the chain of causes and effects created by the butterfly will feed, here and there, and up to forming a hurricane. Just like an underground river which disappears from the landscape to resurface elsewhere with twice more water, a chain like that can benefit immensely in ways too complicated and haphazard to foresee, then take unexpected paths. Some of these chains become black swans. When entrepreneurs embrace agility, they mostly keep up with that process to the point of embodying it. This may decide whether a venture succeeds, sometimes turning into an entirely different thing of what it was first, or fails.

Shiva’s Dance

Chaos is beautiful because it isn’t a mess. The latter is just a lack of order—and it may bear much less potentialities than something chaotic. Mess rhymes with darkness as the pure absence of light, as being far from any ordering principles. Spiritual peoples from the Antiquity, confusing chaos with mess, saw matter and nature as “dumb”, “brute”, messy and hopelessly far from the divine. Yet even the Genesis does not say that, the primordial waters of chaos being used to shape the Earth and sky.

Chaos is not void in the sense of hopeless emptiness: it it partly deterministic—otherwise there would be no fluid mechanics—though very sensitive to initial conditions and unprevisible. Even chance is a positive principle, as it sometimes creates a bridge between the dark world of endless potentialities and the apparently clearer world of what actually exists. Even chaos has a modicum of order: without that, no order could be drawn from chaos, and there would be no fractals either. A complete lack of order is not chaotic but entropic.

An “entropic soup” would spell death to everything and everyone in the universe, this state being almost synonymous with a completely rigid order, the only difference being the complete lack of both ordered shapes and potentialities. A fertile world is a chaotic one. Wheat crops that receive no pesticide must struggle against parasites, which makes them develop antibodies. Thanks to that effort, no-pesticide wheat grains contain more antioxydants than Monsanto-grown grains. Likewise, children raised in an overly sanitized world are more prone to allergy than those who played hide and seek around the trees. At some point, even something as trivial as a wheat crop is a chaotic system. And it is not void of fractals either.

Chaotic systems are never completely balanced. In thermodynamics, complete equilibrium means no temperature flow, ergo maximum entropy. Of course, a lack of equilibrium or an overspilling variance is dangerous as well: if the average summer on Earth was 120ºC instead of, say, 30ºC, and if the average winter was -80ºC instead of something like 0ºC, the flourishing life we know of would simply not exist. Chaotic systems exist in a perpetual state of half-equilibrium, between completely balance and excessive imbalances, characterized by flows and cycles.

A chaotic world is full of cooperation and competition, of dance and meditation, of focused seriousness and humor. Chaos is inherently antinomian and the variance of a moving world makes apparently contradictory potentialities appear one after another. There are different levels of order and the boundaries separating this from that are not always clear. Fractal objects have no evident limitation between one scale and another: the Brittany coast has a length which seems to vary depending on the observer’s altitude. Where does the subjective part end, and where does the objective one begin?

Economist Joseph Schumpeter (1883-1950) famously theorized how economy needed “creative destruction” to function adequately. According to him, the economy may naturally get overloaded with unfit or useless ventures, and some have to fail so that others can grow on their ashes. (As a sidenote, before the subprime crisis, many chided this argument as heartless and un-sympathetic. After 2008, when billionaire banks benefitted from a “too big to fail” policy that impoverished middle classes were forced to pay for, a lot of people understood that bailing out the losers at the expense of society is not always the most sympathetic or humanitarian choice.) Companies have to keep up with the trends and needs to survive, and the innovation surge cannot be separated from the accelerated flow and risks of failure.

Some would say all this is not exactly aesthetical. Indeed, all of this reflects the truth of a chaotic, fractalized world, or, in Ken Wilber’s words, is rather a third-person than a first-person view. But everything’s in everything, and this causes a bit of self-recursion. What the world is can be perceived as beautiful. This remains difficult to explain in English, Shakespeare’s language being Western and enforcing a staunch distinction between so-called facts and value judgments. With an Eastern language, though, the beauty of chaos is encapsulated in a single embodiment: the dance of Shiva. This four- or six-armed godhead dances over people’s lives, destroying buildings and plans, which is sad but also makes room for new projects and expressions.

Shiva’s lesson is not only philosophical, and much less purely economic. It has aesthetic value. Hindu culture has been valuing dancing since time immemorial. Nietzsche, who famously asserted that one must have chaos within himself to give birth to a “dancing star”, was raptured by these dances, with Indian movies today oscillating—of course!—between sublime dance numbers and clumsy attempts.

Now we all dance to fractal rhythms. We have to replace our own ideas, to learn, unlearn, and update ourselves if we want to remain atop the current. Times are not that tough, though: we are much better off with Shiva treading on failed startups and employment than on dead bodies. What we witness through chaos is an endless series of micro-destructions and reconstructions. When you start witnessing how so many of our individual activities mirror much wider trends, thus embodying a fractal element that one would hardly find in the world I was born in, you can grasp the beauty of the whole cycle.

Belief in unequivocal, unambiguous progress, tends to breed manichaeism. It is tempting to divide the world between the good reformers and the fearful conservatives, or between the well-meaning revolutionaries and the ill-intentioned reactionaries. But even the Renaissance had to take its root in Antiquity, before the Middle Ages, to thwart the latter’s obscurantism. More recently, Schumpeter has long been deemed a conservative because he unequivocally rejected Marxism, but how could an apostle of creative destruction be labelled a defender of the status quo? You can’t blame chaos for being ambiguous or bringing unexpected results.

Fractal Biorhythms

As we’ve seen, chaotic systems are hypersensitive to initial conditions. But when is the initial, and when is the rest?

We are all made of DNA. Our genes are our initial basis. DNA seems to account for more than 50 per cent of our intellectual abilities  and we are of course sensitive to genetic expression. But the environment matters a lot, too, and any moment can kickstart a new chain of cause and effects. In a purely natural environment, the butterfly’s move is the most initial condition, in a human and multi-conscious one many choices can play that role. Which is a source of great fragility and of great power, potentially.

Some moments matter more than others. According to behavioral researchers Chip and Dan Heath, moments that matter tend to follow such patterns that we may engineer them consciously. These moments may affect us a lot, to the point of leading us towards entirely new directions, whereas seeing “Coca-Cola” ten thousand times over the years may not affect us beyond mere-exposure effect. In all likelihood, such moments turn our awareness and even our subconscious intentions to unforeseen directions; the “Coca-Cola” logo fosters nothing of the kind and remains background noise.

If you are into healthy eating, you must have heard about specific diets like the Mediterranean Diet, the Paleo Diet, or about specific eating schedules like Intermittent Fasting. All these limit the number of factors coming into our foodstuff or food consumption schedule. They allow their practicioner to modulate the entropy of careless, unhealthy eating through more restraining and controlled conditions.

These diets tend to impose order over chaos, but at the same time, they trigger wholly new reactions: many intermittent fasters report higher levels of energy in the morning, and a friend of mine who ate paleo for more than a year told me how he developed a nut addiction (!) from repeatedly replacing grains with salted almonds and cashews. These changes, in turn, can make overall health better: my friend lost the hay fever he had had since his teenage years, studied with more ease and considers his paleo diet time a key factor behind his academic success—his nut addiction notwithstanding.

During one of my trips to a so-called under-developed country, I was greatly astonished when a friend I met there fixed a meeting with me saying “tomorrow after the rain.” You can picture my perplexity at this unsettling assault on my slightly obsessive habit of demanding punctuality. Today, I can imagine perfectly how to handle my need for punctuality and respect for the natural cycles: my Outlook or Google calendar could simply incorporate an “after the rain” function that would be permanently synchronized online with the weather forecast of the place of meeting!

The more one gets an “integral” vision of it all, the more these uncountable occurrences of chaos and fractality become infused with an aesthetic vibe. There is something beautiful in being able to get a summer time in winter, a winter time in summer—thanks to having jobs or taking vacations under other latitudes—thus escaping linearity of the classic year-round weather cycle, while also making a conscious effort to synchronize our own biorhythms with our deeper needs. Aesthetic may be divorced from deeper truths just as it may express them. I suppose everyone has to make his or her own mind.

As a conclusion

Aesthetics means more than what meets the eye. There is beauty in meanings, in equations, and perhaps in still unknown dimensions. As said in my book, Chaos: a User’s Guide:

The universe as we perceive it is probably a projection. What is a projection? For instance, a geographical map is a two-dimensional projection of the real three-dimensional world. The film that you see on your screen is also another type of projection: in 2D, or 3D on the newer screens.

The world that we see is of a few dimensions only, in principle four: the three dimensions of space and time as the fourth. But this world as we perceive it could very well be, in fact, nothing but a projection of a universe that has a number of other dimensions, perhaps an infinite number…

Well, the projection of something of a larger dimension on to something of a smaller dimension creates what we may call ‘folds’, such as when we try to slip a very big shirt into a box that is too small for it. Try it and you’ll see!

You will notice that these folds naturally take a fractal form and perhaps this is how we can understand the fractal nature of our universe. In the same way, our life is perhaps a fold, a fold of consciousness. Indeed, the projection of a consciousness of an infinite dimension into a human dimension that is much smaller, turns our life into a fold of consciousness.

What I call learning to identify a fractal image is, in fact, being able to see a pattern in the folds, to see order in a world of chaos. In fact, on a very small section of a dress, one does not see the totality of the folds and all seems then to be flat or linear. At the worst, it rises at places or it goes down at others. We cannot see the whole landscape. But as soon as we go further away or rise in altitude, we increase the range of our vision; then the folds and the fractal nature emerge.

To know more about how to be a fractal, resilient manager, on how to implement optimized fractal processes, or to read my musings on fractal love, check my book: Chaos, a User’s Guide, available here.