Fractals are both physical phenomenon and mathematical objects, exhibiting a repeating pattern that is displayed at various levels of magnitude, what is called scale invariance, creating self-similar patterns under magnification.1 Fractals are sometimes called the pictures of nature, and if chaos theory, as the name implies, is the chaotic and unpredictable dimension to nonlinear systems, then we might say fractals represent their orderly side. We see lots of order in our world. The Earth goes around the Sun today and it will do the same tomorrow and the next day for millions of years. If we take a butterfly, we see that one side of it is almost exactly mirrored in the other. We see the same in the regular geometric forms of snowflakes. One way of understanding this order is through the concept of symmetry. In mathematics, symmetry can be defined as an object or process that is invariant to a transformation. In more familiar terms, this means something that stays the same despite a change. With the example of the butterfly, its external physiology had a reflection symmetry, meaning we can simply flip one side over and we will get the other. The same is true for the snowflake, and we could imagine some symmetry in the pattern of earth’s orbit.
Symmetry & Order
Without these symmetries to the world, the scientific endeavor would be very difficult because science is about the creation of compact representations of the world. It is like we are creating maps of the world and it is only through finding these symmetries and encoding them in models that we can describe a wide variety of phenomena with simple equations. Without these symmetries, our scientific map of the world would have to be the same size as the world itself. Symmetry and asymmetry are then two of the most powerful concepts in mathematics and science for talking about order and chaos. They help us to understand these abstract concepts in a distinctly geometric and visual form. With the phenomena of chaos, we see a breaking of symmetry. Two things that started out similar became increasingly dissimilar. The symmetry between them became broken, and the result was after a short period of time complete asymmetry. Fractals have what is called scale invariance, that is, they have a symmetry with respect to scale, meaning the scale can change but the structure will repeat itself over various levels of magnitude. This scale invariance is also called self-similarity, and it is this type of symmetry under magnification that gives fractals an amazing type of structure and order. Fractals are both mathematical constructs that derive out of iterative functions and real world phenomena.
As with most of nonlinear systems, the core ideas behind fractals is of feedback and iteration. The creation of most fractals involves applying some simple rule to a set of geometric shapes or numbers and then repeating the process on the result. For example, the most famous fractal called the Mandelbrot set, so named after the discoverer of the concept of fractals, is a product of a simple iterative map on complex numbers. We will not go into the details of complex numbers, but the iterative map itself is quite simple. There are many extraordinary things about fractals but the first thing we will note is the infinite variety that these simple iterative functions can produce.
Whereas most Euclidian shapes, what we might call normal shapes, tend towards a bland featurelessness as we scale them up, if we take something like a circle the more we zoom in on it the more it will start to appear like a bland straight line and this would also be the case for other regular shapes such as squares, triangles and so on. The iterative functions behind fractals, in contrast, give us an infinite amount of structure and detail within their form. Irrespective of if we divide the shape into two or divide it into a million, each of these million little parts will itself have an infinite amount of detail and a form that resembles the whole.
Just as there can exist an infinite amount of form in a finite object there can with fractals also exist an infinite length within any finite length. This is best demonstrated by a fractal called the Koch curve that can be obtained by iterating a simple process of dividing a line and placing a triangle in its center, and then iterating on this to divide the lines on the triangle in a similar fashion. What we get when we do this is a path that will, in fact, have an infinite length to it. If we were to try and trace a path along this line, its infinite detail would prevent us from ever reaching the end. Thus, this finite length contains an infinite length within it.
The last thing we will note about fractals is that their scale-free property means that no scale is the “proper” frame of reference. Most linear systems that represent regular forms have features and structures within a limited range of scales. Thus, if we plot them we get a normal distribution, with the majority of features tending towards a mean, giving it some kind of “normal” frame of reference with respect to scale. These nonlinear fractals as we have noted, have scale invariance, meaning we will find features on all levels, resulting in there being no normal distribution to their occurrence and thus no normal frame of reference. The occurrence of features is instead distributed out as a power law. Big features exist but are very rare, whilst many small features also exist and this long tail does not drop off – because it is a fractal – it will go on infinitely. Thus, we can say that power laws are the algebraic expression of the scale-free structure to fractals. And just as there is no “normal” scale in fractals, there is no normal average to power law distributions.