Dirty geometry that violated the age-old conjecture of tiling
If you think of the square as a jigsaw puzzle piece and the other mosaics use the same set of transformations, they can be stacked like cold sandwich slices to make a mosaic that uses a transformation set to cover 3D space. Greenfeld and Tao should do this on a larger scale. Since we’re working on higher dimensions, adding another dimension doesn’t hurt our work too much, says Tao. In this way, more flexibility can be achieved to reach a good solution.
Mathematicians wanted to reverse this sandwiching procedure and rewrite the equation of the high-dimensional tiling problem as a set of tiling equations in lower dimensions. These equations later determine the mosaic structure in high dimensions.
Greenfeld and Tao likened their tiling system of equations to a computer program: each line of code or equation is an instruction, and a combination of instructions can produce a program that achieves a specific goal. According to Tao, logic circuits are made of AND and OR gates, each of which is not attractive on its own; But you can stack them up to get a circuit that draws a sine wave or makes an internet connection.
So they considered the problem as a kind of programming problem. Each command is equivalent to a different feature that is necessary to reach the final tiling; So the program generally guarantees that the tiling must be non-periodic.
Then the question was raised as to what features are necessary to implement tiling equations. For example, a mosaic in a layer of a sandwich may be configured to allow only certain types of movements. For this reason, mathematicians applied their constraints precisely so as not to prevent all solutions. According to Greenfeld, the main challenge here is to achieve the right level of constraint to encrypt the correct puzzle.
The puzzle that Greenfeld and Tao sought to program with their mosaic equations was actually a grid with an infinite number of rows and a large but finite number of columns. These two mathematicians wanted each line and each diagonal to be filled with specific sequences of numbers that are equivalent to the types of constraints described by tiling equations. They likened the network to a giant Sudoku puzzle. The two mathematicians then discovered that the sequences were non-periodic; It means that the solution of the system related to the mosaic equations was also non-periodic. According to Tao, in principle, there is only one solution to this puzzle, and it is interesting that “almost” is periodic, not “quite”.
As Yousevich says, Greenfeld and Tao actually took a very basic object and took it to a point where things look more complicated. To do this, they constructed a high-dimensional non-periodic mosaic first in a discrete setting and then in a continuous context. Their mosaic is so complicated and full of holes that it hardly covers the space. In fact, it is an irregular mosaic. Tao says he made no attempt to beautify the mosaic. Later, he and Greenfeld did not calculate the mosaic space either, but simply know that this space is 2 to the power of 100 to the power of 100. If you try to write this number on the pages of all the books in the world, you will still run out of paper. Their proof is a kind of structural proof, in such a way that everything is obvious and computable, but still far from optimal.
In fact, mathematicians think they can find non-periodic mosaics in even lower dimensions. The reason for this mentality was that some parts of their structure included special spaces that were close to two-dimensional space; But Greenfeld believed that he had found a three-dimensional tile and that there might be a four-dimensional tile as well.