Terence Tao, an Australian-American mathematician of Chinese ethnicity, Fields Medal winner and professor of mathematics at the University of California, Los Angeles, announced on his personal blog in September that he had overturned the “periodic tiling conjecture”. According to arXiv, a free distribution service and an open-access archive, the paper co-authored by Tao and his collaborator Rachel Greenfeld was uploaded in the early morning of November 29.
The full version of the paper named “A counterexample to the periodic tiling conjecture” consists of 48 pages. The co-author of this paper is Rachel Greenfeld, former Hedrick Assistant Professor at the Department of Mathematics of UCLA mentored by Terence Tao, and now a member at the School of Mathematics of the Institute for Advanced Study, Princeton.
The periodic tiling conjecture was put forward in two papers in 1987 and 1996. This conjecture holds that in a plane, there is no single geometric figure that can cover the whole plane aperiodic. Periodic and aperiodic are two methods to cover the plane respectively. Periodic tiling is a very regular method, that is, by repeatedly copying, translating, and moving a certain pattern, the whole plane can be regularly paved.
The aperiodic tiling method is not that simple. The most typical example is “Penrose tiling” proposed by Nobel Prize winner Roger Penrose. He designed a thin quadrilateral and a fat quadrilateral, which can cover the whole plane with these two figures. However, there is no specific rule about how these two figures are distributed.
Therefore, the periodic tiling conjecture is that there is no geometric figure that can cover the whole plane aperiodically by itself. This conjecture has been proved in two-dimensional space, so some mathematicians think that it can also be extended to three-dimensional or even higher-dimensional space.
But now, this conjecture has been refuted by Terence Tao and Rachel Greenfeld as it applies to a higher dimensional space. After this paper came out, Alex Iosevich, a mathematician at the University of Rochester, ridiculed the method used to refute the conjecture, “They not only overturned this conjecture, but also overturned it in an extremely humiliating way.”
While failing to prove the periodic tiling conjecture in three-dimensional space, Rachel Greenfeld and Tao began to think about whether this conjecture is problematic in high dimensions. So, they both began to look for counterexamples.
In August 2021, they approached the target for the first time by founding two tiles that can achieve aperiodic filling in ultra-high dimensional space, but not one. On September 19, 2022, Tao said in his blog post that they created a “tiling language”, which used tiling equations to describe non-periodic functions. This proof is strongly associated with the type of reasoning needed to solve Sudoku puzzles, so they used some Sudoku-like terms in their arguments to provide intuitive and visual effects.
In a blog post on November 29, Tao wrote, “The original strategy was to build a tiling language that was capable of encoding a certain ‘p-adic Sudoku puzzle’, and then show that the latter type of puzzle had only non-periodic solutions if p was a sufficiently large prime. As it turns out, the second half of this strategy worked out, but there was an issue in the first part.”
Tao and Greenfield regarded their equation system as a computer program. Each line of code or equation is a command, and these commands can be combined to generate a program to achieve a specific goal. In the end, they found a target tile in a very high-dimensional space which had not yet been calculated in detail.
In addition, Tao said that using their newly created language should create an undecipherable puzzle. “There may be some tiles, and we can never prove whether it can cover the space where it is,” said Tao. Therefore, if the periodic tiling conjecture is proven to be indecisive, it can be used as a new tool to prove the indecisiveness of other problems.