Symmetry Groups

Think of a spacial arrangement of geometrical elements. It could be a polygon, or a honeycomb lattice, or some other arrangement. A symmetry is a rigid transformation that preserves the arrangement.  Consider an equilateral triangle, for instance:

triangle symmetry 1

There are six “moves” that would preserve the triangle. To begin with, we could reflect it across any of the three axes of symmetry.

triangle symmetry 2

In addition, we could spin it through 120° around the center, clockwise or counterclockwise. And, finally, we could just leave it be.

Symmetries have the following property: if we perform two symmetries in succession, then the composition obtained is also a symmetry. For instance, if we reflect the triangle across two lines of symmetry, then this amounts to a 120° rotation. In general, if T and U are two symmetries, then we’ll use the notation U ∘ T to represent the symmetry obtained by performing first T and then U. We can think of this as a “multiplication” of symmetries. Be careful, though: the order matters. The symmetry U ∘ T is not necessarily the same as T ∘ U.

To make all this clearer, let’s consider another example. A square has four lines of symmetry:

square symmetry

Reflection across any of these lines preserves the shape. Label the four reflections as R1, R2, R3, and R4. We can also rotate the square clockwise around the center through 90°, or 180°, or 270°. If S is the symmetry that rotates the square through 90°, then S ∘ S = S2 is rotation through 180°, and S ∘ S ∘ S = S3 is rotation through 270°. And, finally, there’s the identity symmetry, which just leaves the square as it is. We’ll call that I. Notice that S4 = I, since S4 amounts to a 360° rotation.

So we have eight symmetries in all: I, S, S2, S3, R1, R2, R3, and R4. As mentioned above, we can view composition of symmetries as a kind of “multiplication” on this set. However, it isn’t commutative: the order matters. To see this, consider S and R1. Let’s see what S ∘ R1 does to the square:

square symmetry 2

This amounts to the transformation R4. Now let’s see what R1 ∘ S does:

square symmetry 3

This amounts to R2. So S ∘ R1 = R4 while R1 ∘ S = R2. It follows that S ∘ R1 ≠ R1 ∘ S.

What is true is that the composition of any two symmetries in our set is also in the set. The set of symmetries is “closed” under composition. In addition, each symmetry has its inverse, which carries the shape back to the original position. For instance, R1 is its own inverse, since R1 ∘ R1 = I. The same is true of all the reflections. On the other hand, the inverse of S is S3, since S ∘ S3 is a 360° rotation, which amounts to I. And, finally, the inverse of S2 is itself, since two 180° rotations amount to a 360° rotation. If we wanted to, we could make an 8 × 8 multiplication table for this set of symmetries. Each row and each column would contain each symmetry exactly once.

Another interesting thing to note is that we could use a pair of reflections to obtain the other symmetries. Take R1 and R2, for instance. Notice that S = R1 ∘ R2. We can compose S with itself to obtain the other rotations: S2 = R1 ∘ R2 ∘ R1 ∘ R2 and S3 = R1 ∘ R2 ∘ R1 ∘ R2 ∘ R1 ∘ R2. Finally, R3 = R2 ∘ R1 ∘ R2, while R4 = R1 ∘ R2 ∘ R1. So we’ve written every symmetry as a product of R1 and R2.

A group is a set of symmetry operations satisfying the following two requirements: (1) the composition of two operations in the group is also in the group, and (2) for each operation in the group, the inverse of the operation is also in the group. The group we’ve been discussing is the dihedral group D4. It has order 8, meaning that it contains 8 operations: I, S, S2, S3, R1, R2, R3, and R4. We say that D4 is generated by R1 and R2, and write D4 = 〈R1,R2〉.

In coming posts, we’ll see how the theory of groups can be used to classify ornamental symmetries.