Response to a prompt from a non-mathematician friend.
What is symmetry? Usually we think of symmetry as how something looks — it’s the same on both sides of an axis, or you can fold an image along a crease but you only need one side to know how to complete the picture. You achieve this reflecting the half you see across the axis.
The word “symmetry” in the vernacular usually refers to this kind of reflective symmetry. In its origin, the word means something like “agreement in measure or form.” But mathematicians have deranged this word to turn it from a property that is into something you can do. This means instead of just observing the symmetry across an axis, we think of the symmetry as actually making the reflection happen. In the end, the resulting image is the same of course, but thinking of symmetries as actions is an important and useful step.
Now that symmetry is made into something kinetic, we can reconsider the forms symmetry can take. What is the fundamental result of applying a symmetry? Well, it’s really what we said already, ultimately the image or object to which the symmetry is applied appears the same. Taking this as our defining property allows us to include a few other kinds of symmetry we haven’t yet considered. In addition to reflective symmetry, there also rotational symmetry — like taking a wheel and spinning it on its axle — and translational symmetry.
Translational symmetry takes us immediately into the infinite. It is what it sounds like: you pick something up, slide it over, plunk it back down, and then it appears exactly the same as it did before. But I don’t mean it looks like the same object just in a different place. Rather, the whole montage looks exactly the same! Like if you had two photos of the scene where this happened side by side, a “before” and an “after,” they would look exactly the same. The thing is that no “finite” or “bounded” objects (as we are accustomed to) can have this symmetry. This is is because the bounded objects have extremities — a northernmost point, an southeasternmost point etc. If you take one of these and then drag further in that direction, you’ll always be cutting a new path, so the extremity can’t wind up back in a position occupied by the original object. Necessarily, the picture will look different.
But if you take something that extends infinitely — say an abstracted line, the continuum, our model of the real numbers — all of a sudden every point has somewhere back on the line to go when you slide. Here’s another funny thing that begins to emerge: the set of symmetries begins to look like the object it’s acting on. Considering the real line, every real number corresponds a symmetry of translation. That is, for a fixed real number , we can send each number in the line to and have the whole line slide over (to the left or the right depending on whether is negative or positive) and land back on top of itself. So the line has a line’s worth of symmetries! Does it have any others?
Well, again we have reflection. Pick a point in the line, and reflect the line about it. It’s not hard to see that everything goes back to somewhere else on the line. Moreover, points that were close together end up the same distance apart after applying the reflection. Symmetries that satisfy this property are called isometries, as they preserve the intrinsic geometry of the line.
So do we have another line’s worth of reflections to add to our list of symmetries? Well, you could say that if you want to think of these symmetries in isolation, as objects of a kind that are unable to interact with each other. But that would run counter to the notion we are developing of symmetries not as things that are but as things that do! A symmetry acts on the line, and once it has acted, another symmetry can act on the result and so on, kicking the points on the line this way and that.
To make the next point, let’s consider an example. Take the reflection through the number 1. Under this reflection, 1 stays put and 0 goes to 2. Maybe you can convince yourself that this is sufficient information to completely determine this symmetry. But here’s another way to do the same thing: reflect through 0, and then slide everything 2 to the right. With a bit more thought, you can probably convince yourself that the reflection through any point can be obtained similarly by a reflection through 0 followed by a slide, or a slide and and then a reflection through 0. What we say is that the reflections and translations altogether are generated by just the reflection through 0 together with the translations.