You could forgive mathematicians for being drawn to the monster group, an algebraic object so enormous and mysterious that it took them nearly a decade to prove it exists. Now, 30 years later, string theorists—physicists studying how all fundamental forces and particles might be explained by tiny strings vibrating in hidden dimensions—are looking to connect the monster to their physical questions. What is it about this collection of more than 1053 elements that excites both mathematicians and physicists? The study of algebraic groups like the monster helps make sense of the mathematical structures of symmetries, and hidden symmetries offer clues for building new physical theories. Group theory in many ways epitomizes mathematical abstraction, yet it underlies some of our most familiar mathematical experiences. Let’s explore the basics of symmetries and the algebra that illuminates their structure.

We are fond of saying things are symmetric, but what does that really mean? Intuitively we have a sense of symmetry as a kind of mirroring. Suppose we draw a vertical line through the middle of a square.

This line cuts the square into two equal parts, each of which is the mirror image of the other. This familiar example is called line symmetry. But there are other kinds of symmetry that have nothing to do with mirrors.

For example, the square also has rotational symmetry.

Here we see the process of rotating a square counterclockwise about its center point (the intersection of its diagonals). After it rotates 90 degrees (one quarter turn), it looks the same as before. It is this transformation of an object so that the result is indistinguishable from the original that defines a symmetry. The above rotation is one symmetry of the square, and our example of line symmetry can be thought of as another.

Let’s take a moment to define a few terms. We will call the original object the “pre-image” and the transformed object the “image,” and we will use the term “mapping” to refer to the process of transforming one object (a point, a segment, a square, etc.) into another. A symmetry requires that the transformation not alter the size or shape of the object. A transformation that meets this requirement is known as an “isometry,” or a rigid motion, and the fundamental isometries are reflection over a line, rotation about a point, and translation along a vector.