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.

OK this is gonna be a quick oddball post, but I’m so excited about this that I can’t not tell the world.

Pandemic life has found me, like many, drinking much more coffee made at home, and also exploring new hobbies and ventures. One of these new ventures is experimenting with mushroom cultivation. There are myriad techniques at all levels of sophistication for the many varieties of edible and medicinal mushrooms, but I was instantly excited when I learned that one can grow oyster mushrooms on used coffee grounds with virtually no technical set-up.

I mean for real, cheap coffee that turns into free food?! This is like a math graduate student’s dream.

So I saved my spent coffee grounds in the freezer for a few weeks, got some oyster mushroom spawn, then inoculated the grounds in a sanitized plastic container and waited. Among cultivated mushrooms, oysters are known for being particularly vigorous and adaptable, growing on anything from logs to old clothes to coffee and tea. The rapidity with which they colonized the grounds was astounding; already the next day the mass was glowing with patches of white mycelium waking up and stretching its legs. Within a week, a fluffy cushion of the stuff coated the whole surface. A few more weeks go by (maintaining high moisture in a cool, dark place) and voila!

One day the fruiting bodies start to “pin” and then they double in size for a few more days until they are ready to harvest. The next night, mushroom fried rice for dinner. : D

As the myco-aware are eager to tell you, this is a large untapped resource that society in general and the math community in particular has been sleeping on. In our department, another graduate student used to collect used grounds from the coffee machine for compost, but with very little effort every department could put its spent theorem-precursor to good use by setting up its own mushroom growing operation (myceliated substrate makes great compost after the fact).

Growing food is a radical community-building and liberatory act. The origin of the word university is the combination of uni (one) and versus (turned), “turned into one”: basically, community. Subsidizing graduate student grocery budgets is one thing, but whenever we get back to being able to sharing pots of coffee, this would be a great project for pushing on universities to be a little more like communities and less like bureaucracies.

I basically followed the instructions here, and I recommend the source book (in the link) to anyone interested in learning more.