Symmetry Redux: Conservation Laws


One of the first topics I ever posted about here on Silhouette of Science is Symmetry. I was fascinated by the manifold ways, both obvious and subtle, that Symmetry pervades our world. In that post I touched upon the presence of Symmetry in processes biological, chemical and physical, and the striking absence of Symmetry in a select few instances. I appealed to cellular automata as an example of simplicity spawning complexity. And lastly I yearned for some deeper symmetry, or a more fundamental rule from which the apparent breaking of symmetries could be explained.

Continue reading “Symmetry Redux: Conservation Laws”

So Much for Symmetry

One of the first ideas that drew me to science was symmetry – the concept that everything has inherent balance, proportionality, and temporal harmony. I know it’s romantic to think that everything, from electrons to elephants, obeys the same universal laws; but for such a simple idea, it holds surprisingly true.

In Biology, plants and animals adhere to many different types of symmetry. Sunflowers and Nautilus shells spiral outward according to the golden ratio; trees and leaves display fractal geometry, in which the same pattern appears on many scales; and animals show bilateral symmetry, where left mirrors right.

Chemical bonding, too, is governed by symmetry – elements that have the same number of electrons in their outer (valence) shell, bond in the same way. The whole periodic table is based on this structure!

Much of Physics revolves around symmetry in the form of invariance, or lack of change based on coordinate system. It only seems right that in space up is no different than down! In fact, when drawing the diagrams that bear his name, Richard Feynman was so taken by symmetry, he concluded that a particle traveling forward in time is the same as its antiparticle traveling backward in time. In 1972, the Nobel Laureate PW Anderson went so far as to declare, “It is only slightly overstating the case to say that physics is the study of symmetry”!

But every once in a while nature throws a curveball our way. The narwal’s tusk is on its left jaw. And the wrybill is the only bird with its beak always bent to the right.

It’s almost frustrating that out of the millions of species on Earth, all but a few fit symmetries so nicely. What makes them different? What allows left-handed fermions to interact with the weak force when right-handed fermions cannot?

Chaos theory (pictured below), is the study of sensitivity to initial conditions. It says that simple systems, like the double rod pendulum, can have extremely complex results. With just two rods connected at a joint and hung as a pendulum, the double rod pendulum is one of the simplest mechanical systems. Yet it exhibits extremely complex behavior. It never retraces its earlier path, and never repeats a pattern. What’s more, changing the initial conditions even slightly leads to entirely unrecognizable results.

Maybe this sensitivity extends to other systems. In his book, A New Kind of Science, Stephen Wolfram studies cellular automata, or small grids of squares colored either black or white. He identifies 256 simple rules for deciding what color each square would be based on the squares around it, and runs each simulation on his computer. While the vast majority of the simulations produce simple, easily identifiable patterns, and fractals, a small fraction of the tests give different results.

Rule 110, although close to many of the other rules, produces results so complicated that it takes thousands of iterations, and millions of cells, for a pattern to emerge. Until then, it is essentially impossible to predict the behavior of a cell from the prior results.

The mathematician Horace Lamb once said of the chaotic nature of turbulence, “When I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic”.

Maybe chaotic systems can’t be predicted. Maybe their rules can’t be solved for with mathematics or computational power. But I’d like to think that our Universe is governed by simple, elementary laws; and that usually these laws produce simple results, but every once in a while produce an unrecognizable pattern which we perceive as disorder. Narwals and wrybirds break bilateral symmetry. But I’d like to believe that there’s some deeper, truer symmetry that we’re missing. As childish an ideal as symmetry is, it’s soothing to think that there is some underlying order.

Maybe nature isn’t throwing us a curveball; we might just be looking at turbulence the wrong way!