Physics for Everyone: Symmetry
Symmetry is considered the most important concept in all of physics. EVERYTHING in physics is built off the idea of it. And, to be true, without it, we could not even begin to conceive the most basic principles, theories, or formulations.
But just what do physicist mean when we talk about Symmetry?
It goes beyond the original Euclidean definition of a commensurable relationship. Symmetry also encompasses proportions, specifically those grounded in integers, as well as the more modern ideas of invariance and conservation. Essentially, it is balance – that nothing is lost without something gained, that two objects do not interact disproportionately. It is equivalence.
So, beyond setting down definition, which itself might be confusing, let’s start to talk about examples of symmetry – where does it occur, and just why is it so important? We see symmetry everywhere. In the laws of conservation of energy and momentum, in the way gravity, and all other forces, interact with objects, in the way bodies move, and things grow.
Simple visible examples of symmetry abound in the natural world. Crystals are a perfect example of symmetry – atoms and molecules of certain elements or compounds growing in a unilateral way to form near perfect shapes. We see it in the way organisms grow, especially animals. Most show some form of bilateral symmetry  where one side of their body reflects the other side; but some express a more radial symmetry – reflective sameness around a central point – like starfish and sea anemone.
The Greeks saw symmetry in geometry as the ultimate form of beauty. Circles, squares, and triangles were perfect to them because of their symmetry, and any shape that was not proportional itself *must* be made of several smaller symmetrical shapes.
We see proportionalism and conservation in the motion and interaction of bodies. When one strikes another, they (if they don’t stick together) rebound equivalently – the smaller, less massive one moving away with greater speed than the larger, more massive object.
A great example of this is the famous, and fun, Newton’s Cradle – the toy wherein we hang several metal balls from strings, placed one against the other in a line. Lift one up, let it go, and watch how, when it clacks into its neighbor, the energy is transferred through the line of balls to the one on the end, which then rises into the air almost to the level where the first start. And, where the cradle in a vacuum (and its struts not touching anything at all), that same ball would, in fact, rise to *exactly* the same level at which the first one was let go, and vice versa upon it’s drop and crash into its neighbor, and so on, infinitely, the reaction never stopping. Why? Because of conservation of energy and momentum. With no air resistance to slow down the balls (and nothing connected to the cradle to absorb energy through the strings and struts), the balls will continue to clack clack clack forever.
In the orbit of bodies, we see symmetry. It is well known that all planets (all celestial bodies, in fact) orbit in ellipses – a great figure with two foci determined by the center of mass of the objects involved in the orbit (ie, the Earth and the Sun), the initial motion of those objects, and the contributing tugs of nearby bodies (the latter two factors being why celestial bodies do not orbit in perfect circles).
We feel the effects of symmetry every day in the tug of gravity – an attractive force whose pull is proportionate to the masses of the two objects involved. Why does it seem all objects fall at the same rate (disregarding air resistance)? The answer to that is simply that the Earth is so much more massive than anything we find on its surface, that its mass simply overwhelms any difference in the mass between objects falling to its surface. So, for our perception, the objects fall at the same rate.
From a physicist’s standpoint, symmetry is found in mathematics. When one side of an equation matches the other side, we have symmetry (and, we find, we just about *always* find symmetry in nature). When something causes us to change one side of the equation, the other side must be equivalently changed. Take the basic equation called the Ideal Gas Law, which we use to determine pressure, heat, and volume. It is, famously, PV=nRT. Broken down, Pressure times volume equals the amount of the substance times a universal constant times the temperature of the substance. Effectively, this means that, if the pressure rises, so must either the temperature rise, or the volume shrink. Likewise, as the volume rises, so must either the temperature rise, or the pressure decrease. And if the temperature rises, then either the pressure or the volume must increase (and possibly both).
Perhaps one of the most fascinating symmetry, with the most profound results, was also, for a time, one of the most confounding. It is called the trifold symmetry CPT. Which stands for Charge Conguation, Parity, and Time Reversal. The concepts were first discovered in the late 1920s, but continued to baffle physicists until 1952, when Gerhart Lüders demonstrated the connection between the three. The ramifications of the trifold symmetry are astounding: one cannot change the charge of a particle without *both* reflecting it upon its axis *and* reversing its motion in time. And that last part of the symmetry is most definitely the most striking, for it demonstrates that, with no doubt, particles can move in more than one direction in time.
Conservation of energy is a symmetry that profoundly affects our daily lives. It’s responsible for our need for fuel for our cars, power plants to heat our homes and run our televisions, and even for our need to consume food daily. Simply put, it means that energy has to come from somewhere – it doesn’t just pop out of thin air, and, when used, it doesn’t simply just go away – it has to go somewhere. So when you want to move your car from the garage to work, it requires energy, which has to come from somewhere, and most typically that is the gasoline you put in your car. When you want to turn on the television, that energy came from a power plant miles away, which may have been creating energy out of burned coal, or perhaps water turning a turbine, or, even better, solar energy being trapped in collectors. Your personal energy to get up and go – to run, walk, talk, whatever it is you do – comes from the food you consume – your body converts a small portion of it into energy, and that just makes your muscles happy.
The universe is built on symmetry. Physics is constructed from it – nothing would exist without it, and we wouldn’t be able to even begin to fathom the workings of the universe in its absence. For physicists, it is simply the single most important concept there is.
