A change of setting was the first thing they did. Would you like to tile two-dimensional space? A lattice is an infinite array of points arranged in a grid. If you have a proper tiling, you can define a tile as a finite set of points on the grid and then make copies of that set of points and slide them around.
There are tilings that are possible in lattices but not in continuous space, which is different from proving the periodic tiling conjecture for high-dimensional lattices. They're related. They were going to come up with a counterexample that they could modify to work in the continuous case as well.
They were able to find two tiles in a very high-dimensional space. The tiles are able to fill the space they occupy occasionally. Greenfeld said that this wasn't enough. tiling by two tiles is more flexible than tiling by a single tile. It would take another year and a half for them to come up with a real counterexample.
They created a new language and reworked their problem as an equation. All possible ways to tile a high-dimensional space were represented in this equation. It is difficult to describe things with a single equation. Sometimes you need more than one equation to describe a complex set in space.
They changed the question they were trying to answer. They came up with a system of equations where each equation had a different constraint. This allowed them to ask a question about tiles that all cover the same space using the same set of translations.
In two dimensions, you can tile the plane with a square by sliding it up, down, left or right. The same set of shifts can be used to tile the plane in other shapes, such as a square with a bump added to the right edge and removed from the left edge.
If you take a square, a puzzle piece, and other tiles that use the same set of shifts, you can build one tile that uses a single set of translations to cover three-dimensional space. They would need to do this in more dimensions.
Adding more dimensions didn't hurt us since we were working in high dimensions. It gave them the flexibility to find a solution that worked for them.
The mathematicians wanted to change their tiling problem into a series of lower-dimensional tiling equations. Higher-dimensional tile construction would be dictated by those equations.
The system of tiling equations was thought of as a computer program by Greenfeld and Tao. Each of the logic circuits that are built up are not very interesting. You can combine them and make a circuit that can communicate on the internet.
He said that they began to see their problem as a programming problem. The program as a whole would guarantee that any tilings fitting all the criteria must be aperiodic because each of their commands would be different.
What kind of properties were needed to make that happen? A tile in one layer of the sandwich may be shaped in a way that only allows certain movements. The list of constraints would have to be carefully built so that they wouldn't be so restrictive as to preclude any solutions at all.
The game here is to build the correct level of constraint.
Greenfeld and Tao wanted to program a grid with an infinite number of rows and a large number of columns. The mathematicians wanted to fill every row and diagonal with a certain sequence of digits that correspond to the constraints they could describe with tiling equations. The pair discovered that the solution to the tiling equations was also aperiodic. There is only one solution to the puzzle and it is funny. It took a long time to locate that.
Izabella aba is a mathematician at the University of British Columbia. This is a different method of using that structure.
Greenfeld and Tao created a completely elementary object and lifted it up so that it looked more complicated.
They created a high-dimensional aperiodic tile first in the continuous setting and then in the discrete setting. They have a tile that is full of twists and holes. The tile is a bad one, said Tao. We didn't try to make this tile pretty. They don't know the dimensions of the space it lives in, but they know it's massive. Everything is explicit and computable because of the constructive proof. We didn't check it because it's far from optimal.
The mathematicians believe they can find tiles in lower dimensions. Some of the more technical parts of their construction involve working in special spaces that are very close to being two-dimensional. She doesn't think they'll find a three-dimensional tile, but she thinks it's possible that a 4D one could.
Iosevich said that they did this in the most embarrassing way possible.
The work marks a new way to build tiles that could be applied to other tiling-related questions. That will allow mathematicians to push further at the limits of complexity. Higher-dimensional geometry seems to be an emerging principle. It's true that the intuition we get from two and three dimensions can be deceiving.
Questions not just about the boundaries of human intuition but about the boundaries of mathematical reasoning are tapped into. Kurt Gdel showed in the 1930s that any logical system that is sufficient for basic math is incomplete. There are a lot of undecidable statements in mathematics.
It is also full of problems that cannot be solved in a short time. Problems about tilings can be undecidable according to mathematicians. It is possible to prove that it is impossible to figure out if they tile a given space or not. In principle, the only way to do it would be to lay tiles next to each other.
Richard Kenyon is a mathematician at Yale University. It is not the first example of this situation where a mathematical theory is incomplete but it is the most down-to-earth one.
A general statement about pairs of high-dimensional tiles is undecidable because it proves that nobody will ever be able to figure out if certain pairs of tiles can be made to completely cover the space they occupy.
A statement about a single tile could beundecidable. It has been known since the 1960s that it would be possible to determine if any tile could cover the plane.
The opposite is not always correct. That doesn't mean that an undecidable one does.
Greenfeld and Tao are going to use some of the techniques they developed for their recent result. We believe that the language we created should be able to create an undecidable puzzle. We won't be able to prove that tile space or tile space isn't tile space.
To prove that a statement is undecidable, mathematicians usually show that it is equivalent to another question that is already undecidable. If this tiling problem turns out to be undecidable as well, it can serve as one more tool for demonstrating undecidability in other contexts.
The result of Greenfeld and Tao is a warning. Thematicians like clean statements. It's not a fact that all interesting statements in mathematics need to be pretty and that they need to work out the way we want them to.
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