We regularly see symmetry throughout nature and in what we create. The wings of a butterfly, the face of a loved one, and even the triangular Whataburger building across the street from me right now all display this common motif. Symmetry often provides an important underlying structure of the laws of nature, and it characterizes our universe and its ability to be studied and understood.

The type of symmetry we are most familiar with is geometrical symmetry, or symmetry in shapes. A square, for example, can be cut multiple ways down its middle to produce two identical halves. Stated another way, the square’s shape is identical to the shape of its mirror image when oriented certain ways. You may remember finger painting on a piece of paper in elementary school and folding it in half before it dries in order to create an intricate symmetrical pattern. This same type of symmetry often shows up in biological systems. The right side of your body is roughly a mirror image of the left.

In a broader sense, symmetry is defined as the ability of some property of an object or law to stay the same when something is done to it. In the case of the square, its appearance does not change when we change our point of view by flipping it over its middle or rotating it through multiples of 90 degrees. Similarly, a perfect sphere remains the same any way you spin it about its center.

Just like these shapes and objects have symmetry, physical phenomena can also have symmetry. For example, the path of a ball thrown into the air creates a parabola, and this parabola is a mirror image of the one the ball would make if time were reversed. Therefore, the path is symmetric in regards to the direction of time. The laws of nature themselves also have certain symmetries. For example, the laws describing the physical world do not change depending on where you are in the universe. They are the same whether you are over here or over there. In addition, these laws do not change with time. The same laws that dictated the behavior of the early universe dictate physical behavior now. Finally, a rotated system in space obeys the same physical laws as a non-rotated system. Note that I said a rotated system, as opposed to one that is rotating (this will be relevant later on).

Often we take these facts for granted. Of course the laws of physics have the same affect on a system whether you have rotated it or not. In fact, you could think of rotating a system as if it were merely you having rotated your point of view around a stationary system. Likewise, it is intuitive to us that if you drop a ball under certain conditions, and then took the ball to a different place with identical conditions, the ball would drop in the exact same way.

It didn’t have to be this way, though. There is nothing logically impermissible about a universe with laws that change with time, position, and angular orientation. Thankfully, we live in a universe where its behavior can be studied through experimentation. If these symmetries of the physical laws did not exist, there would be no point in doing an experiment multiple times or in multiple places because the laws might change between trials. Conclusions could not be drawn. The scientific method would be pointless. (Not to mention there is the chance we wouldn’t exist in any form in a universe without symmetry.)

These symmetries also have crucial implications in regards to conservation laws. Emmy Noether, an early 20th century German mathematician (described by Einstein as the most important woman in the history of mathematics), proved the theorem that with every symmetry, there is a corresponding conserved quantity. For example, the fact that the physical laws are symmetrical with respect to position means that momentum is conserved. Likewise, we know energy is conserved because of the time-symmetry of the laws. (The proof of Noether’s Theorem requires knowledge of higher level calculus, so you will just have to trust me on this one.)

Symmetry also shows up in quantum mechanics and relativity. For example, the speed of light is the same with respect to you no matter how fast you are moving! This is called Lorentz invariance. The laws acting on a moving system are the same as those acting on a stationary system, but fast motion causes observed lengths and times to expand and contract in order to keep the speed of light constant. In particle physics, for every particle there exists a type of anti-particle with identical mass but opposite charge. The curious thing is that there is, for reasons still unknown, a much higher amount of matter in the universe than antimatter. We know this because when a particle meets its anti-particle, they annihilate each other, and look how much matter there is around you that isn’t annihilated! This is what is called a “broken symmetry.”

Let’s talk more about what is not symmetric. The physics of accelerating systems is not the same for systems moving at a constant velocity. This includes rotating systems, as rotation is a type of acceleration. For example, the fact that the Earth is spinning causes what are called the Coriolis and Cetrifugal forces. These forces, though small and often neglected in calculations, are responsible for physical phenomena that would not exist on Earth if it were not rotating. For example, the Coriolis force is largely responsible for typhoons, and the Centrifugal force is what makes one weigh slightly (less than one percent) less at the equator than at the poles of the Earth.

Another non-symmetry is that of scale. Richard Feynman, in a lecture of his on symmetry, used the example of a small model of a cathedral made of matchsticks. If the model cathedral were scaled up to the size of a real cathedral, it would collapse because of the weakness of the wood to stand the increased amount of gravitational force from the Earth. He points out that some might say that if the Earth is having an effect on the system, it should be included in the system and scaled up as well. “But then it is even worse,” he says, “because the gravitation is increased still more!”

So, what are we to make of the fact that our universe is often almost symmetric? Scientists are baffled when they discover unexpected broken symmetries, such as the matter and antimatter discrepancy and other broken symmetries in particle physics. We think of symmetry as perfection, and anything less than complete symmetry as unnerving and disappointing.

In the same lecture as referenced above, Feynman describes an intricate gate in Neiko, Japan that has carved into it elaborate details that are completely symmetrical except for one small design on one side that was purposefully carved upside down. The story is that it was made slightly off from perfect because the men who built it did not want to make God jealous of their perfection. Feynman lightheartedly goes on to say, “We might like to turn the idea around and think that the true explanation of the near symmetry of nature is this: that God made the laws only nearly symmetrical so that we should not be jealous of His perfection!”

The snowflake photograph was done by Alexey Kljatov.

Here are some links if you want to read more about symmetry!

Feynman’s Lecture

Symmetry Magazine

A good article

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