Thinking about the Monster
Disclaimer: I’m an amateur mathematician. I wrote this mainly as a way of clarifying my own understanding. I encourage you to research this on your own if you find it interesting!
There are these things called groups in the branch of math called group theory (which consists of studying symmetries of objects). Groups are a sort of structure. A good example of a group is the group that describes the symmetries of a square: it tells you how you can move the square around and end up with the same looking shape (you “preserve its structure”, if you want to be a bit more technical). The interesting thing is that we formalize the movements (rotations and reflections) so that we can play with them the same way we do other algebra: as an example, you might use “r*s” to describe the action of rotating the square by 90 degrees, then flipping it.
I was reading about this subset of groups called finite simple groups, which are like prime numbers for other finite groups. You can’t break them down any further than they already are. For example, if you want to talk about the symmetry of a 15-sided shape, you can instead describe all of its movements by combining different movements of triangles and pentagons, just like how 15 is the product of primes 3 and 5. Any finite group (a group that doesn’t have infinitely many elements) can be constructed using combinations of finite simple groups. Don’t worry about how, but just know that we are interested in these because if we can identify all of them, we can figure how to build any other finite group by combining them somehow. Or something.
But anyway. For a long time people were trying to classify all the finite simple groups, and they succeeded in 2004. The proof isn’t something localized in one place; rather, it was the combined efforts of many mathematicians over many years. One of the finite simple groups is called the Monster group, and the fact that it exists really hurts my brain.
All of the other groups are pretty crazy as it is. Normal shapes can be displayed in 2 or 3 dimensions. Most of these groups describe abstract objects that can only “exist” in much higher dimensions (I put quotes around “exist” because the existence is something we can only ever think about, not create physically). But the Monster, this group takes it to a new level. This one has such a ridiculous number of elements that I can’t even come close to comprehending anything about it.
It has 808017424794512875886459904961710757005754368000000000 elements. This means you can move this weird, abstract object in that many ways and still preserve its structure. To put this in perspective, the square symmetry group I was talking about earlier has 8 elements. And the number of elements in the Monster is an exact number. And it’s unique, and a prime building block of other groups, despite its ridiculous size. You can’t break it down any further. It must exist; it’s a fact of nature. We made it up, but it follows all the rules and therefore, it exists.
The part that really blows my mind is that the smallest dimension you can construct the Monster in is dimension 196883. What the fuck.
(Additional note if you want more mind blowing: the dimensions that you can construct the Monster group in just happen to be values that are extremely close (close enough for it not to be a coincidence) to the first terms of some totally unrelated functions in a different branch of math. It’s just patterns and they’re all interrelated in absurd ways.)
Most of this information comes from the book Symmetry: A Journey into the Patterns of Nature by Marcus du Sautoy, and various bits of poking around Wikipedia. Thanks to Leah for suggestions on ordering and examples.
