Weyl's tile argument

Hey there, little buddy! Do you know what a tile is? It's a little piece of something, like a piece of a puzzle. And do you know what a floor is? It's what you walk on in a room or building.

Now, imagine we have a bunch of tiles and we want to cover a whole floor with them. But we want to make sure that every single point on the floor is covered by a tile, with no gaps or overlaps. This is what mathematicians call a "tiling problem."

A guy named Weyl came up with a way to solve this problem for a certain kind of floor. It's called a "quasicrystal," which is a really fancy word for a floor that has a pattern that never repeats itself, like a kaleidoscope. These kinds of floors are very tricky to tile because there's no simple way to figure out their shape.

Weyl's argument goes something like this: Imagine we have a quasicrystal floor, and we're trying to tile it with square tiles. But no matter how we arrange the tiles, there are always going to be little gaps that can't be filled. This is because the shape of the floor is just too weird!

But here's the cool part: Weyl realized that if we used tiles that weren't square, but instead were shaped like little rhombuses (which are like diamonds), we could fill in all the gaps and cover the entire floor! This is because the rhombuses can fit together in a way that squares can't.

So Weyl's tile argument is basically a way of proving that even a really complicated floor can be fully tiled if we use the right kind of tiles. It's all about finding the right shape to fit the space!