The Second Dumbest Way to Solve a Maze

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2025-01-14 17:30:07

You’re probably familiar with The Dumbest Way To Solve A Maze: a method involving the simulation of gas diffusion through the maze’s hallways. I’d like to introduce what is likely the second dumbest way to solve a maze. While this method isn’t inherently dumb (in fact, it involves math that is far from ‘dumb’), it’s so impractical for actual use that it’s simply ‘dumb’ to apply it to maze-solving.

The Riemann mapping theorem states that for a non-empty, simply connected open subset $U \subset \mathbb {C}$ there exists a biholomorphic function $f\colon U \to \mathbb D$, where $\mathbb D$ is the open unit disk. We call this function Riemann mapping 1.

For those of you who slept over Complex Analysis classes in elementary school, here’s a simplified consequence of the theorem: if you have have a set $U$ in the plane with no holes, you can map that set onto a disk $\mathbb D$. And when I say map, I mean that disk can be a complete map of the set $U$: each point in $U$ corresponds uniquely to a point in $\mathbb D$, and vice versa.

Think of a map of the world. We use a piece of paper to represent the planet Earth, and when we point to a location on the map, we’re referring to a specific place on Earth. It is same with sets $U$ and $\mathbb D$. If we pick a point in $U$ (red dot on illustration above), Riemann map gives us a corresponding point in $\mathbb D$ (illustrated with violet dot), and therefore we can think of $U$ as a map of $\mathbb D$. But it works other way around (because $f$ is biholomorphic), and we can understand $\mathbb D$ as a map of $U$.