Group theory studies the
symmetrical patterns that underlie everything from Rubik’s cubes to plant
biology.
ILLUSTRATION: TOMASZ WALENTA
By
Eugenia Cheng
Feb. 4, 2021 12:55 pm ET
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Mathematician
Eugenia Cheng explores the uses of math beyond the classroom. Read more
columns here.
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.
