Math in Art

Group theory is a field of Mathematics that forms to generalize algebraic structures known as groups. Groups are just a set of objects (abstractions) that is equipped with a binary option—an operation that takes in two inputs and outputs another (i.e. Addition). However, groups don’t have to just describe numbers. They can, in fact, describe anything (so long as it holds the properties of closure, associativity, identity, and invertibility). One of the most common examples can be seen with the symmetry groups. Here’s an example of a symmetry group on a square:

Math in Art

This is important because it generalizes the notion of transformation. Notice how performing r1 (rotate 90°) after r2 (rotate 180°) is the same as performing r3 (rotate 270°). Or how performing r1 (rotate 90°) after fv (vertical reflection) is the same as performing fc (counter-diagonal reflection). Math has generalized every transformation as a composition of another transformation. But how would we visualize this more compactly? Surprisingly, multiplication tables, or more specifically group tables:

Math in Art

This seems daunting, but let’s look at the same examples before. First, let me say that we read the top first then the horizontal. So, if we have r2 (rotate 180°) and “multiply” it by r1 (rotate 90°), we get r3 (rotate 270°)! Now, if we “multiply” fv (vertical reflection) with r1 (rotate 90°) we get fc (counter-diagonal reflection). Notice how fv “times” r1 (fc) is not the same as r1 times fv (fd). This mean our group isn’t commutative. Now, for a nice visual, here’s a clip from a 3blue1brown video:

Rotate 90 counterclockwise (same as r3 because it is the same as rotating 270 clockwise) plus a horizontal flip (fh) is the same as a diagonal reflection (fd).
Rotate 90 counterclockwise (same as r3 because it is the same as rotating 270 clockwise) plus a horizontal flip (fh) is the same as a diagonal reflection (fd).

What does this have to do with art? Well, everything! Being able to generalize notions of symmetry, transformation, etc. into abstractions means we could generalize design itself! We’ve already started to do such a thing with music theory with an abelian group. Design could be no different. Proportion could be measured by different sizes of a group. Emphasis can be described by a group that changes point of view. Harmony can be described by a group of different scaled objects, where acting on them changes how each one moves until balance can be reached. So, all in all, generalizing design would be one of my main points of interest, and a topic I would be glad to explore further.