UPDATED 15 APR 12020 (WIP)

Ascensio Naturae:

Order from Chaos

Today we take a look at a group of mathematical topics including dynamical systems, chaos, and fractals; and examine by analogy the rise of Life itself from the early Earth and throughout history; with particular focus on the dichotomy between order and chaos and the spontaneous emergence of both from one another.


We must begin by talking about change; how a system evolves over time. To this end, we will introduce the mathematical notion of a dynamical system: a dynamical system is simply a map from a space to itself: \[(\mathcal T,\, \mathcal M) \quad \mathcal T : \mathcal M \to \mathcal M\] All this really means is that dynamics describes how points in a space move over time, or rather, how the space itself evolves—a remarkably simple idea which can be applied to great utility across susprisingly many fields and real life applications. Let's take a look.

Dynamical systems can represent many things—for a few, consider the population of predators and prey in an ecosystem changing over time: if there are lots of lions and not many zebra, the lions will overhunt the zebra and begin to starve, followed by a period of less lions in which the zebra may flourish. This system can extend to include the plants, competing herbivores, or the entire world ecosystem.

What about the flow of water, or the atmosphere: in a storm, if we look at the position of one particle, where does it flow over time? What happens to the weather year by year in a German village, as the climate reacts to feedback loops in response to rising greenhouse gas levels or solar irradiation? How does life evolve—how can we characterize the changes to, for example, the genome of shark species, as they adapt to an ever changing ocean?

These are all examples of phenomena which may be modeled by dynamical systems, and in fact, the study of dynamics is a large and growing mathematical field. With the right models, you really can find out what happens to those water particles, to the phenotypes of living organisms, to the population of lions—even in the future. This is an incredibly powerful and valuable tool.

So, how do we find the right models? What is their nature? How can we begin?

Vector fields

Vectors are a mathematical object represented by lists of numbers. For example, maybe you're counting the populations of lions, zebra, and antelope with the vector \((195,\, 851,\, 4915)\). Since there are three numbers to look at, you might imagine this as a point in 3-dimensional space, and consider that space all possible combinations of these animals.

One very common way to model dynamical systems across all industries and fields of study is the vector field, which is effectively a space of vectors (or points in \(n\)-dimensional space) with another \(n\)-dimensional vector attached to each point, representing its velocity over time.

The reason this approach is so popular is twofold:

  1. There is a wealth of mathematical understanding and methodology around vector calculus, which allows you to analyze and compute such models.
  2. Many complex systems in the real world can be accurately modeled by a handful of parameters, allowing the construction of such vector-based models.

For example, the shape above, called a Lorenz attractor, is defined simply by \[\begin{aligned} v_x &= \sigma(y-x)\\ v_y &= x(\rho-z)-y\\ v_z &= xy-\beta z \end{aligned}\] Where \(\sigma\), \(\rho\), and \(\beta\) are special constants. Shocking as it may be, this system is a simplified 3-parameter model, as introduced above, for atmospheric convection, describing the change over time of directional temperature variation in the atmosphere, even though the atmosphere is a complex system with unthinkable quantities of various particles and mechanisms.

Much of applied mathematics is in search of such simple parameterized models, as this is what allows us to economically understand complex systems. Here are some other examples:

Couple this idea of dynamics with some physics and astronomical observations, and you could perhaps write the program necessary to keep a communications satellite on the correct orbital trajectory, providing internet to real world customers below.

Dynamics are truly universally applicable—no matter who you are, or what you do, building an understanding of dynamical systems can equip you with the predictive and analytical tools necessary to do great things.

They are used all the way from rocket science and meteorology to technical investing and the social sciences, with lots of interplay between the math. The more you learn about each case, the more connections you'll be able to make when trying to do something brand new.


Now we will examine the concept of emergence, which is the notion of complex phenomena arising from simple parts; together with recursion and self-similarity, the conceptual building blocks of many fractals.

There are some more obvious examples here: how complex waves arise from individual water droplets, or clouds and the weather arise from simple gas molecules, or how a deadly storm emerges from simple interactions of those same individually harmless molecules.

Maybe you find yourself wondering about the interesting shapes and patterns that arise from a leaf trying to maximize its exposure to sunlight; an electron trying to find the quickest path to the ground; or water droplets trying to stay whole. As is already clear, this concept of emergence is responsible for countless natural phenomena.

We don't often think of ourselves as natural things or in terms of our true underlying biology, so you might be surprised to consider that you, too, are an emergent process.

Your brain the complex interaction of countless individual neurons; your veins an intricately optimized delivery system for nutrients to cells; your very DNA the complex code that defines you in simple pairs of ribonucleic acids—much like the most intricate digital movies are patterns of 1s and 0s.

Even the Mandelbrot set pictured above, perhaps the most famous fractal with seemingly endless intricacy and beauty, is effectively defined by the simple transformation \[z=z^2+c\] recursively applied to complex numbers, where the points in the set are those \(c\) for which the transform does not diverge taken from \(z_0=0\).

Building fractals

In fact, many fractals emerge from simple rules—this is precisely why you may have noticed a certain fractal-ish look in some of the emergent patterns above. They often use recursion, which is the repeated application of a transform or rule, theoretically ad infinitum.

\[f(x,y) = \left(\begin{matrix}a&b\\c&d\end{matrix}\right) \left(\begin{matrix}x\\y\end{matrix}\right) + \left(\begin{matrix}e\\f\end{matrix}\right)\]

For example, the Barnsley fern above is constructed by recursive application of the rule in the middle, given four sets of constants \(a,b,c,d,e,f\) and a mix of these four transforms.

Some of the most popular fractals (Sierpiński's triangle, the Cantor set, the Koch snowflake, the Hilbert curve, etc...) can be constructed by applying a simple rule over and over to the segments of a simple 2D shape, allowing us to visualize their emergence; and share an interesting property of self similarity, that is, you can break up the fractal shape into a number of smaller perfect copies of itself, all of which are just as complex as the whole.

For example, you may have noticed that each leaf in the Barnsley fern looks identical to the larger fern, and is made up of many smaller versions of itself. Such fractals actually have some useful properties, which is why they appear to form in the real world, like in actual ferns.

Space-filling forms

One class of fractal materialized in such organisms is the space-filling curve, or space-filling tree. These objects are the solution to a difficult problem, which combats such effects as the square-cube law when scaling organisms to larger size.

Life has adapted to use a space-filling tree-like system to solve this. Your heart pumps blood through the massive aorta, which splits into smaller arteries, which themselves split again and again into smaller and smaller arteries. By continuing this process long enough, we eventually reach a point where every single cell has blood delivered in a timely fashion. The same concept works all the way from a hamster up to an elephant.

A small difference makes all the difference

Like yin and yang, order and chaos are opposites that intermingle. Chaos emerges from order, and surprisingly, order emerges out from chaos all the same. Most of the universe's matter is today organized in ordered entities – superclusters, galaxies, stars, planets, rocks, and yet these are the children of the chaos that followed the birth of the universe. Even humanity, with our sterilized hospitals, perfect geometric shapes, and rigorous mathematics: we too are the result of chaos.

If order is like attraction, chaos is like repulsion. You can only really go into a hole one way, but you can shoot out from a hole in a straight line in almost any direction, and after a long enough time you will be arbitrarily far away from where you would've been if you chose any other angle.

Consider life on Earth. After billions upon billions of years of dead rock, meteorite impacts, rampant volcanism and violent chemical reactions on the young planet's surface, after uncountably many chemical reactions which ended just as they had begun, each a little different in one way or another: one event did something strange. It did not fade away, instead, it became the predecessor of life itself, of self-replicating nucleic acid. From this one simple chemical sprung the last 4,000,000,000 years of evolutionary history. Quadrillions upon quadrillions of organisms, each playing the same game a little bit differently, each dealt a slightly different hand.

Life is chaotic.

Arguably the most important mathematical feature of chaos is sensitivity to the starting conditions. While chaos and randomness might seem similar to the untrained eye, chaos may very well be deterministic – the key determining factor is that a chaotic system will yield wildly different results if you ever so slightly nudge the parameters one way or another.

Today, there is an incredible diversity among lifeforms. We live in unthinkably complex ecosystems, and life has adapted to fill almost every niche one can imagine. In the oceans alone, there are whales the size of large buildings, and there are bacteria smaller than the smallest cells in your body, and everything in between. Where did it all come from? How did we get here?

Indeed, we share a common heritage. The human diver, the micro-organisms in the coral reef, the fish which feed upon its plants, the mitochondria which burn their glucose, the sharks which hunt the fish, and even the parasites within them are all cousins from a time not so long ago. Every animal, every plant, indeed every living being in the world share common ancestors. The tree of life bifurcates not once, but all the time, and at every level.

Some day in the past, they were siblings. Two organisms of the same species, each a little different in their unique genetics, each exposed to a slightly different experience of the environment, each put to different tests of natural selection, eventually became quite different. After many generations, they are unrecognizable, and distinguished.

If your great\(^{50}\) grandfather was instead his sister, perhaps today you would be a chimpanzee. If the same happened to your great\(^{100}\) grandfather, maybe you would be the cow whose steak you ate last night; your great\(^{1000}\) grandfather, and maybe you'd be the grass the cow grazed upon. Or maybe your bloodline would have died off millions of years ago.

That is sensitivity.


The second distinguishing factor of chaotic systems is topological transitivity, which basically means, if we are talking about a dynamical system (remember: a transformation from a space to itself), that you can get from any region to any other region eventually.

More rigorously, if \((\mathcal T,\mathcal M)\) is a chaotic system such that \(\mathcal T:\mathcal M\to \mathcal M\), then for any non-empty regions \(A,B\in \mathcal M\), there is a point \(x\in A\) such that \(\mathcal T^nx = y\) for some \(y\in B\) and some number \(n\).

Is this true about life? Indeed, it is. As a matter of fact, we have seem the same kind of wings evolve independently from at least three wildly different branches on the tree of life: the birds, which are dinosaurs; the bats, which are mammals; and the flying insects.

On a similar note, evolution is transitive: just as fish evolved into land-dwelling amphibians like frogs; some of the mammals eventually returned to the water, in sea lions, dolphins, hippos, and whales.

This is not to say that nothing stays the same: chaotic systems can very much have periodic curves which stay on a simple loop, these are unstable, and a small shift away from one can lead to extreme divergence. You might think of punctuated equilibrium, the theory of long periods of limited morphological change in a species followed by rapid bursts of evolution, as an analogy to this. Things might seem stable, but in chaos, the opposite end of the spectrum is always just a tiny nudge away.

Chaos is important because it is everywhere: chaos is responsible in part for everything you are, everything that is, and everything that ever will be.

In some sense, chaos is the reason why we live our lives as if we have free will, why we accept determinism on an academic level but live our lives as if the future is unforeseeable. Even our minds are chaotic – the very neural networks that make up our brains can explode with myriad imaginations and trains of thought with the slightest electrical signal in a single measly neuron; yet patterns still arise: rationality, beauty, love – all the things that make life special arise from the chaos.

One beautiful example of a partly chaotic dynamical system from whose simple rules complex phenomena emerge is Dr. John Horton Conway's Game of Life, in which a grid of cells evolves under these rules:

  1. Any live cell with 2 or 3 live neighbors remains alive,
  2. Any dead cell with 3 live neighbors becomes alive,
  3. All other cells die.

In this game, under these very rules, self-sustaining life itself emerges, contained systems which move and evolve depending on their surroundings. I highly recommend having a look at it online if you haven't seen it before.

John Conway died yesterday due to the novel coronavirus and COVID-19. This essay is dedicated in his memory. He was a great contributor to a wide variety of fields and his work will continue to inspire and intrigue future generations.

@jwmza · April 12, 12020


Morrison J. (2020) Fractals, Dynamics, Chaos.

Siefken J. (2019) Chaos, Fractals, and Dynamics.

Hirsch M.W., Smale S., Devaney R.L. (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos.

Lorenz E. (1995). Essence of Chaos.

Mandelbrot, B. (2010). Fractals and the art of roughness. At TED2010.

With various references to Wikipedia and imagery from Unsplash. https://unsplash.com