Infinite Tessellation of a Two Dimensional Plane
This happens a lot. I was watching students at work in my great colleague Carol’s class, and I got to thinking about tiling a plane.
Students were doing the classic MC Escher style tessellation, where you start from a 10 x 10 square, and with cuts, reconstruct a shape that will rotate, translate, or reflect across the entire plane.
It puzzles some kids what happens at the edges. There are bits of shapes left over at the edges of this very small plane. This can be confusing. Luckily, every school has tiled floors, and there are always helpfully cut pieces of tiles used to finish the edges of the room (they don’t just leave holes in the floor!)
But clearly, a finite plane like a piece of paper must finish somewhere. What about an infinite plane? Could we keep sliding our shape that will tessellate over the plane forever? Or will some hang off the “edges”, at some point?
Infinity messes you up like that. If we think about numbers, we can always have some number n, and we can always then have some number n+1. This is true for natural numbers, to start with. So a new number will always “fit”.
But 2D space can be full at some point, right?
Here is a screencapture of an infinite Minecraft world with a few things placed on it. Theoretically you could keep pushing in one direction forever (laying blocks the whole way). The limit is how much of “infinity” is actually programmed into the game. Perhaps with an infinitely recursive programming loop, the flatness of this plane truly does go on forever.
So Cantor proved that more numbers would always fit into the set of numbers. I don’t know how to prove that you can always fit one more shape into an infinite 2D plane.
Define the 2D plane as some rectangle, with some length, l, and some width, w.
I can always add 1, or 2, or 10, or infinity to l, and I can do the same for w. So to make our shapes fit, assume they are 10 x 10 in area, and imagine them reverting back to squares.
I can always expand this rectangle by l+10, and w+10. The plane will keep moving, and opening up wider and longer, to fit our shapes. There are no limits, when you are tiling an infinite plane.
Below is a question you can ask kids. Obviously a paper has thickness. It just seems flat. To an ant, though, it wouldn’t seem flat at all.
Let’s consider dimensions for a moment.
A point is dimensionless. To think that this universe may have started from a zero dimensional point is…astounding.
A line is one dimensional. A one dimensional being would be standing on a line, which goes to infinity in either direction. She could not see anything to her left or right. She could see a point, ahead, or behind, if it was there, on the line. “Standing” is a misnomer here-there is no “3D”, in “1D” space.
A two dimensional being would live on a plane. She could see lines on the plane, but not up or down. There is no 3D space, in 2D space. A 2D being is perfectly positioned to see lines and shapes in that plane though.
Three dimensional beings, like us, can observe an entire plane, like a piece of paper, or a tiled floor. We can see 2D spaces very well, but we can’t see ALL of our 3D world. We have no vantage point from above to see all of it.
A four dimensional being would see a bunch of 3D spaces at once. The movie Interstellar does a good job at showing this-but it includes time as a dimension. Still, there is a vantage point shown on different 3D spaces at the same time. (Like you could look upon a bunch of pieces of paper at the same time).
In five dimensions? All bets are off. 5D beings watch over 4D beings who watch over 3D beings, and so on…
But we were concerned with tiling a plane before this digression. You could start tiling an infinite plane, and keep tiling forever. There would always be room for more shapes. Infinity is comforting, compelling, and mindblowing. Escher, in the infinite arms of the universe, is out there somewhere, and he is…still tessellating. And he will be, for all time.
