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Seeing Things With The Power of Symmetry

Group theory studies the symmetrical patterns that underlie everything from Rubik’s cubes to plant biology. Seeing Things With The Power of Symmetry

As I gear up for another semester of virtual teaching, one of my invaluable pieces of equipment is a document camera. It is essentially a webcam on a long neck, which I point at my desk to show things I’m writing, drawing or making. The trouble is that if I bend the neck so that the camera faces down toward the desk, the image on the screen is upside down. To make the image appear the right way up, the stand and neck would need to be at the bottom of my page, which would get in the way when I write.

My webcam model is rather basic—that is, cheap—so there is no built-in way to rotate the image on the screen; it can only be flipped horizontally or vertically. However, some mental gymnastics reveals that if I combine those two flips the result is the 180 degree rotation I need. I am familiar with this from group theory, a branch of abstract mathematics that studies symmetry.

In everyday life, we typically say an object is symmetrical if we could draw a line down the middle and both sides match up when we imagine folding them over. This is called reflectional symmetry: A human face has it (more or less), but a hand doesn’t. Another type is rotational symmetry, where you can rotate an object and it still looks the same, like a windmill.

One of the insights of group theory is that symmetry can be thought of as an action rather than a property—flipping an object over or turning it around. It’s a small shift in perspective, but it means that we can think about combining symmetries by doing one of these moves and then another.

Understanding symmetry via group theory helps us boil down a situation to its fundamental building blocks.

Combining concepts to make new concepts is essentially the purpose of algebra, and rather than trying to visualize the objects it’s dealing with, group theory uses algebraic formulas and techniques. This is beneficial when our ability to form mental pictures of an object runs out, as when we are thinking about a mathematical object in four dimensions, or something with far too much symmetry to keep track of, like a Rubik’s Cube.

Understanding symmetry via group theory helps us boil down a situation to its fundamental building blocks. For example, by understanding the symmetries of a rectangle, I can start with the flipped reflections of the rectangular webcam image and combine them to produce the rotation I need. This search for fundamental building blocks is central to abstract mathematics.

Group theory has many applications because symmetry is so widespread in science. It’s found not just in shapes but also in number systems and systems of equations. Many plant structures in biology and molecular structures in chemistry depend heavily on symmetry.

One important application in physics is Einstein’s theory of relativity, which conceives our usual 3-dimensional space together with time as a 4-dimensional “spacetime.” Because space and time are related in somewhat counterintuitive ways, spacetime doesn’t behave quite like 3-dimensional space.

Of course, sometimes the best abstract mathematical approach is not the most practical. It might have been more helpful if my software had the “rotation” function built in along with the flips. But the abstract approach is better for developing mathematical theories that help us understand more complex phenomena in the world around us.

 

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