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.

Fright in Disneyland

On Febuary 20, birthday of my oldest daughter, we went to Disneyland. For my children the first time in their life. Every day, 9.30 pm, we heard the fireworks from where we live in Irvine (yes, past tense, because the economic downturn has caused Disneyland to temporarily abondon this feature), so we figured it was time for us to experience the ``happiest place on earth''.

Our first attraction was absolutely wonderful: the all new ``it's a small world after all''. The Netherlands had a display the same size as all of Africa which was of course nice (for us...). But let's not be cynical here, this attraction has a message with educational value, the puppets are simply beautiful and you leave the place with a positive feeling: the world is a benign, beautiful place.

Next on our list was "Thunder Mountain". This was a bit scarier, and my youngest daughter (7) asked me why everyone was raising their arms and screaming all the time. She concluded it must be because they want to show off that they are not afraid. She assured me that she was definitely afraid. All in all not much harm here.

But then it was time for "Priates of Caribbean". It came very highly recommended by one of the Disneyland employees and even the Pirates outside assured me that it was allright for small kids. They were debating the steepness of the drops the boat was making during the ride. However, that turned out to be beside the point. A brief excerpt:
-> A skeleton speared against a rock with a sword in his (her ?) chest.
-> Pirates selling a line of women to other pirates as their new brides (an euphemism for Pirates raping women).
-> Pirates drinking lots of alcohol (perhaps also drugs?).
-> Pirates burning down an entire village (in modern terms: a bombing raid).
-> Pirates torturing a man by submerging him in a waterwell (somehow the word Abu-Ghraib came to mind).
Now keep in mind, the pirates are the good guys here. This is not a realistic display of the aweful pirate terror of the old days. These are role models for our children. How appropriate do you think it would be if we would have a similar Disneyland attraction about the holocaust, or the genocide in Rwanda. If you think about this for a minute or two you quickly realize how bizarre this form of entertainment really is. To us, but certainly to children.

Somewhat shocked through this experience we went for Fantasy land and entered the "Snow White" ride. It can't get much more innocent than that you would tink. To my horror, it too consisted of scary witches and, yes, another corpse. Now we are talking a ride for the very smallest children. Is this really "the happiest place on earth"? Why this obsession with fear? In fact, almost all Disney movies are packed with fear. Why are we showing movies where the mother of main character (Bambi) gets brutally killed? What does this do to our kids?

I believe epic battles between good and evil have an enormous archetypical appeal to us. I am a huge fan of starwars myself. The symbolism can be found an all major religions in the form of the dualism God and Devil, Heaven and Hell, etc. That's allright, but not for the very young. As a society we are disgusted with pictures of nudity. Indeed, I have not seen a single naked breast in Disneyland. Yet, breasts will not shock young children, they grow up with it, its completely natural to them. Breasts are good: they are symbols of motherhood, protection, kindness. However, violence, skeletons, weapons, torture and fear are not natural to children. These things transmit the message that the world is an evil place. How twisted is our society really?


Let's end on a postive note. I am teaching an introduction to artificial intelligence class and for the first time I came face to face with Asimo on that Disneyland trip. Asimo is simply amazing. One constantly has the feeling that there is a little person inside that suit. And everytime you realize that this is not the case, it gets a little scary (a good kind of scary this time). What if Asimo decided not to listen to the boss, somewhat like a dog with big teeth. Asimo, walks, talks, runs and walks chairs, all very gracefully. It's truly amazing, a peek into the future. My youngest child is now Asimo fan. I am happy Asimo won over "Pirates of Caribbean" after all.