Not long after I graduated with my master’s degree in mathematics, I came across the song Finite Simple Group (of Order Two). It’s an a cappella love song that consists almost entirely of puns from graduate-level math.
The number of people who will get all of the jokes is substantially lower than the number of people who have PhDs in math, though getting a master’s degree will allow you to get many of the jokes, at least from having heard other students talking about classes you didn’t take. (For example, some of these come from differential geometry, which I never took, but heard people talking about.)
Even the name is a mathematical pun. A klein bottle is a mathematical manifold somewhat akin to a mobius strip, except you cannot create one in three dimensions without a self-intersection. It can be created without a self-intersection in four dimensional space, hence why four is associated with “klein”. (There appear to be five people in Klein Four, but this is something of an implementation detail.)
This video was uploaded to YouTube 16 years ago (as of the time of writing), which was several years after it was recorded. The website kleinfour.com doesn’t seem to exist anymore. (Last time I checked it, it said that the members of kleinfour had long ago got their PhDs and went on to be professors at different universities, so were not in a position to make more music.)
I find it interesting to look at it now, after so much time has passed. As I recall it came out roughly at the time that I was in grad school, and it felt incredibly relatable. Grad students in math do an enormous amount of work; it’s not even that they’re assigned so much as that they are there because they live and breath math. That creates a lot of pressure of its own, quite apart from grades and assignments (which create their own pressure), and so they sometimes need outlets where they do something else. I can remember late at night taking a break with some other grad students I was working with to do some experiments dropping paper helicopters down the area next to the central stairs and seeing what designs too the longest to reach the bottom. (Of the kinds we tried, it was spirals with a small central weight.)
This song is well done; it is composed well and the singer sings well. The puns were generally not strained. I had hoped that they would go on to do other things, but it was not to be. I have no idea whether the members of the group ever think about the Klein Four anymore; sometimes it’s easier to forget work that one has done than it is to forget art by someone else that one really appreciated. (Sometimes not.) The fact that no one has bothered to keep up the Klein Four website makes me suspect that all of its members have moved on.
If they have, though, there is still this video up on YouTube. Video has the curious property that it never changes. If you watch it a hundred times, it is the same moment in history, every time. You change as you watch it, of course, but I think that, properly considered, this gives a bit of insight into that strange aspect of how this life works. The past is real, but we cannot get at it, and yet it still matters. I believe it was Saint Augustine who first proposed the image of our existence being shattered over the moments of time and God, at the end of time, gathering up those shattered fragments and putting them together into a coherent whole.
Ironically, it is the past mattering that is why one should not dwell on the past. Because we will, after the end of days, be reunited with the past, we need to put our efforts into the present—because it will become part of that past that will eventually become an eternal present. That is, of course, harder to see when the goodness of the moment is less clear than it was at some of its clearest moments in the past, but that’s only a defect in our sight.