Fractals as Symmetries under Scaling

Symmetries are both beautiful and powerful tools to explain the physics of our world. One of the most salient symmetries are invariances of physical laws. For instance, laws remain constant 1) over time, implying (through Noether's theorem) that energy is conserved 2) throughout space, implying conservation of momentum, and 3) under rotations, implying conservation of angular momentum. The Dutch physicist Lorentz also introduced invariance under "Lorentz transformations" which involve both space and time and which led Einstein to develop his theory of special relativity. This in turn led to the equivalence of energy and mass and the constancy of the speed of light.

Going one step further, Einstein also postulated another symmetry, namely that "gravity = acceleration" leading to the general theory of relativity (see earlier blog). In solid-state physics discrete rotational symmetries play a crucial role in describing chrystals such as snowflakes (see Figure).



In quantum mechanics an entirely different kind of symmetry states that particles of the same kind are fundamentally indistinguishable. This leads to all sort of quantum mechanical effects such as Bose-Einstein condensation, entanglement and much more. And the list goes on to even more abstract symmetries that transform the elementary particles into each other. This means that instead of viewing them as different they are rather different sides of a die. These so called gauge symmetries form the fundament of the "standard model" of physics and help explain the existence of e.g. fotons (quanta of light).

And then there is invariance under changing scale. In statistical mechanics, physical laws are the same on all scales at phase transitions where materials radically change their organization from, say, solid to liquid. The really exciting discovery was that the physics at these critical points is universal, i.e. irrespective of the details of the material that undergoes the transition, the physics is the same.

But scale invariance has also led to an entirely new branch of mathematics, namely that of fractal geometry. Fractals are objects designed to be self-similar across all scales. This type of invariance leads to counter-intuitive notions such as objects having fractional dimensions, say between a line and a plane. But above all, fractals are beautiful. (see Figures).



Why are fractals beautiful? I believe they are beautiful because they look incredibly complex, yet are extremely structured. So structured in fact that they can often be described by a few lines of code. Now, our brain is in the business of searching for structure in the world. Understanding the world is understanding its structure and it allows one to make predictions and that will help us survive and produce more offspring (see earlier blog). It seems that we are not too good in discovering the underlying structure in fractals. Can you tell me the equation that generated the fractals in the figures? Our brain simply did not evolve to deal with these type of geometric ojects. yet, our brain does recognize the enormous opportunity to explain structure in fractals but it is tortured by its inability to do so. Beauty in this sense is fascination, it draws your attention because it never succeeds in its ultimate mission: discover the structure in the world.



Fractals also seem to appear in nature. Perhaps not surprisingly, because simple genetic code can generate intricate geometric shapes that can be functional. A famous examples is your vascular system. Another beautiful example is a type of broccoli shown below. It looks genetically engineered, but I believe it is natural.