⭐️skip the reveal, just take me to the thing⭐️
It’s in there somewhere…
I flunked pre-cal in high school. It was murderously tedious and I wasn’t innately curious enough to grasp the “why” part. Why are triangles interesting? Why are polynomials interesting? Why is pi interesting?
Later that same year, my friend Stephen and I hit upon the idea that, since pi had infinite digits, and since those digits were non-repeating and random, that pi must, therefore, contain everything in a Library of Babel sense. The number 996699? It’s digits 13430–13435. My social security number? Starting at digit 4,539,613,003. The entire works of Shakespeare, written in modern English, encoded in binary? It’s in there somewhere.
A stunning thing about pure math is how quickly we run up against incomprehensible beauty when just poking around systems and asking very basic questions. Need to count things? Invent the natural numbers. Need to measure things? Invent the rationals. Need triangles? Invent the square root… (you see where this is going?)
Need to identify the sides of the pentagram? Well good luck, now we have “a ratio of incommensurable magnitudes” (Hippasus? He perished at sea for that impiety). These are the irrationals. They make up the vast bulk of what we call the “real” numbers and they are terrifying.
After college I worked down the hall from Gregory Chaitin who had devised a far more rigorous claim about the irrationals than mine about pi: Anything that we can write down as a numeric sequence must exist somewhere in the reals. This includes mundane things, like, the real number that encodes literally every episode of the “Real World, Season 5" in 4k video (in addition to everything else on Netflix), but also real numbers that encode things like: the answer to every possible yes/no question that has been or will ever be asked, the entire evolutionary arc of a (countably infinite) universe, past, present and future and an oracle which tells you whether any finite Turing Machine will halt. (I highly recommend picking up a copy of MetaMath)
What about symmetry? Human beings have one(ish) symmetry (bilateral), starfish have… five? infinity? one? well anyway a different kind.
Group theory aims to clarify and systematize what we mean by symmetry. Humans and starfish are simple enough, but how many symmetries does a Rubik’s Cube have?
If you start down this path, then it turns out that in a very precise sense there is a biggest thing. A thing that encodes the most symmetries, and nothing else in our universe can possibly be more uniquely symmetric. It’s called the monster group, and it is somehow related to string theory.
I find this fact bizarre and comforting, knowing it’s out there, perfectly symmetric, floating around in the mathematical sea like the mother of all Rubik’s cubes.
Structure and Randomness
Many mathematical systems contain an edge where order tips over and becomes disorder: Terence Tao calls this structure and randomness.
God may not play dice with the universe, but something strange is going on with the prime numbers. — Paul Erdos
There are exactly 26 “sporadic” groups, of which the monster is the largest. These function as a sort of “prime numbers for symmetry.” But why 26? ¯\_(ツ)_/¯
Btw have you ever played the “name the largest number” game? Like you get into a competition with your brother where you (an idiot) say “a billion” and then your brother (a genius) says “a billion and one”? Turns out it’s phenomenally deeply interesting.
A visual flavor of this phenomenon comes from tiling: you have a large floor needs tiling and for economic reasons you don’t want to have to make that many different shapes of tiles. Seems generally pretty mundane?
And yet, tiling admits deep exploration, both mathematically and artistically (& computationally), and there remain tremendous unanswered questions about what types of tilings are even possible.
Your typical bathroom or floor tilings repeat the same structures over and over, but Islamic artists in 15th century discovered a class of aperiodic tilings, that never repeat, forever, constantly reconfiguring themselves in new and devious ways.
In the 1960s, the logician Hao Wang linked aperiodic tilings to Turing completeness. Unaware of the prior work from the 15th century, however, he conjectured that aperiodic tilings should not exist as they would imply computational undecidability of his system. His student Robert Berger later disproved this conjecture by constructing a 20,426 tile aperiodic counterexample.
In the 1970’s Roger Penrose re-discovered a much simpler aperiodic tiling with reflection and five-fold rotational symmetry. The “Penrose tiling” is probably the best-known example of aperiodic tilings. You’ve probably seen them around, or read about them in Neal Stephenson books.
There are many, many more species of aperiodic tilings and they are all strangely beautiful.
I made a tiny web app for exploring a particular flavor of aperiodic tilings with statistical circular symmetry. You can check it out here:
(you’ll need a relatively modern web browser / phone)
There are many variations. If you find one you like, click the permalink and share it in the comments!
I’m enormously grateful to the maintainers of the Tilings Encyclopedia, without which I would not have made it much beyond the Penrose tilings.
Built with Box::new(🔺) using Rust 🦀 and WebAssembly🕸.