Is nature really chaotic and fractal?

In common usage, the word “chaos” means disorder, but is that so in physics? Not really. Chaos in physics stands for “unpredictable” and refers to physical systems that change their state over time.

A physical system is simply a slice of universe that we decide to consider as somehow separable from its surrounding environment. Sometimes we assume the collective effect of the system’s surroundings, but more often we prefer to assume that the system is isolated. Does true isolation exist? No: it’s artificial.

For example, a crystal is traditionally defined as a solid possessing an ordered structure that is infinitely periodic in the three spatial dimensions. Even assuming that such a “perfectly ordered” system could exist (and it cannot), such a system would still have a finite size in reality.

It is easy to realise that such perfect structures are therefore more imaginary than real: crystals, as they are found in nature, are not only of a finite size, but also possess many defects and are far from perfect.

Luckily for us all, perfection is subjective, and in physics a crystal is usually described by a set of rules, commonly annotated as equations, defining a set of symmetry operations that can be repeated recursively on a set of points representing atomic centres. These unfold into an infinite, 3D, periodic structure.

These rules are arbitrarily defined as “perfect” simply because they describe infinitely self-repeating patterns. Therefore, any real structure is, by comparison, very imperfect.

How can we predict the discontinuities and irregularities of a real structure if our model does not allow for defects? We cannot. The system we describe could therefore be labelled as “chaotic”, because it is unpredictable using those “perfect” rules.

We can thus conclude that the issue of chaos has nothing to do with reality, and a lot to with its human interpretation. 

Such “imperfections” also give rise to an interesting phenomenon: the fractal geometry of nature, which is also the title of a book written by a famous mathematician who studied these ubiquitous structures: Benoit Mandelbrot.

What are fractals? Fractals are technically geometric structures with a fractional dimension, for example 2.3. To understand what I just wrote, however, I’ll give you an example: suppose you are told to trace a straight line. We know from elementary school that straight lines are just an infinite set of points lying in one dimension, and a straight line itself is therefore infinite.

Can you actually draw a straight line? No, but you can possibly draw a segment! A segment is an infinite set of points delimited by two extreme points. If for a straight line you need a whole dimension to trace it all, for a segment you will certainly need less than that!

In other words, you will need a fraction between 0 and 1 to trace it: therefore a segment constitutes a very simple, yet fractal, geometry! Can you predict, using the equation of a straight line, y = a * x + b, all possible segments that lie on a single dimension? Yes, but such an equation would generate an infinite, uncountable, uncomputable and therefore inherently unorderable set of values for the coordinates of the segment extremes.

We can now confidently state that nature seems fractal, but is that truly so? One may argue that the answer to this question has more to do with philosophy than physics, and in a way that would be correct. But what if fractals are just an emergent property our innate inability to grasp infinity?