1.2: Counting Symmetries

So how can we use mathematics to study symmetry? Well, the first thing we learn about in mathematics is counting, so perhaps we should try to count symmetries!

If we think of a perfectly symmetrical face, there are two symmetries: one from flipping left-to-right, and another from leaving the face alone. Some might argue that the face only has one symmetry, and leaving it alone doesn't count. In fact, we could argue all day about this point and make no progress until we became very precise about what is meant by 'a symmetry.'

An almost perfectly symmetrical face.

To settle the argument, we require a definition! This first definition of the text is intentionally very loose.

Definition 1.1.0: Symmetry

A symmetry of an object is a way of moving the object back onto itself without changing it.

In fact, doing nothing to an object is a way of moving it back onto itself. Thus, we will say that a symmetrical face has two symmetries.

Let's consider some more mathematical objects. A line segment always has two symmetries, just like a face. An equilateral triangle, though, has six symmetries: three rotations (including the rotation by \(0^{\circ}\)), and three rotations when flipped over. You can keep track of the various symmetries by labelling the corners of the triangle, and seeing where they end up after applying one of the symmetries. (See the illustration.)

The six symmetries of an equilateral triangle. The top row contains the three rotational symmetries, while the second row has the 'flipped' and rotated symmetries.

How many symmetries does a square have? How about an \(n\)-sided regular polygon?

We can also imagine an object which is symmetrical under some number of rotations, but which can't be flipped over. You can make such an object in many ways; one way is to take a square (or any other regular polygon) and then add an identical 'bump' just to one side of each corner. This object has rotational symmetry, but cannot be flipped.

In three dimensions, we have regular polyhedra. These are three dimensional objects with many symmetries! A tetrahedron has 24 symmetries, for example: twelve of these are rotations, and another 12 can be obtained by reflecting and then rotating.

There are five regular polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. How many symmetries does each one have? Try working it out directly for the smaller cases, then see if you can arrive at a formula for the polyhedra with more sides.

Of course, some objects have an infinite number of symmetries. A circle is a good example of this: every rotation is a symmetry, and there are infinitely many angles by which the circle may be rotated, all of which preserves its shape.

Thinking back to our regular tiling patterns, these also have infinitely many symmetries. All of them have translational symmetry, since you can translate the whole picture back onto itself. And you can translate in one direction as many times as you like, so there's at least one symmetry for every integer.

Contributors and Attributions

• Tom Denton (Fields Institute/York University in Toronto)