How many ways can you rotate or flip a square so it looks the same as when you started? Try it! Grab some scissors, cut out a square from a piece of paper, and see how many different ways you can flip or rotate it, leaving it looking the same. Well, you could rotate it 1 quarter of a full turn clockwise, or 1 quarter of a turn counterclockwise, you could also rotate it half a turn clockwise or counterclockwise (These actually bring the square to the same spot, so here we will think of them as the same), you could also flip the square about on the diagonals, or flip it horizontally or vertically. When I count these, I see that there are 3 rotations, and 4 flips that I can perform on the square that leave it looking the same. These are the symmetries of the square, and this actually reflects what symmetry fundamentally is: actions that you can perform on something that leaves it looking the same. For example, we say a line is symmetric because you can move along it and it doesn’t change, or a circle is symmetric because you can rotate in any way and it looks the same.
Yeah, this isn’t that exciting, but this idea of symmetry is actually more important than one might think. Normally when we think of symmetry, we might think of something like a painting, a face, or maybe a gemstone, related to the way something looks, as we may think that something symmetric is beautiful. But there is more to it, enough so that mathematicians have a field of study dedicated to the different types of symmetry and how they relate to one another, called Group Theory. But why should you care? Well, as it turns out, it is because of the idea of symmetry that a universe with predictable laws can exist at all! If you throw a ball, as long as it doesn’t bounce off of something, it will continue moving forward. Why? You may think that this is easy to answer, but that is just because we are used to a universe where objects keep moving forward after we throw them. Couldn’t our universe have some law that says that when we throw an object forward, it moves forward, and then suddenly changes direction and moves backward even though nothing pushes on it? Why do things keep moving when we push them?
Symmetry actually comes and saves the day. When we look out at the world we see that the way things act does not really change as you move from one spot to another. If you throw the ball into the air, it will act the same no matter where you are, in a field, on top of Baguette de France, falling down a flight of stairs, it doesn’t matter where, the universe doesn’t care. This is a symmetry, what physicists fancifully called “translational symmetry,” the way things act doesn’t change when you move to a different spot. This symmetry of the universe is what causes the ball to keep moving in a straight line when thrown. As the ball moves from one spot to another, symmetry requires that it keep acting the same, that is, moving forward. These types of symmetry lead to incredible results, as was shown by the genius mathematician, Emmy Noether. She proved that whenever the universe has a symmetry, there is something in the universe that must not change. For this symmetry in movement, the thing that doesn’t change is momentum, which is basically the tendency for an object to keep moving in a straight line. Who knew that something so simple as throwing a ball was connected to something as abstract and beautiful as symmetry?
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