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The periodic secret shop conjecture, which has been a difficult problem in mathematics for many years, has been broken by Tao Zhe Xuan and Rachel Greenfeld.

The "periodic secret shop conjecture" in geometry was overturned by Tao Zhe Xuan.

A few years ago, mathematicians proved that no matter how complex or ingenious the secret shop you came up with, it would be impossible to design a secret shop that could only non-periodically cover the entire plane if you could only use translation for a single one.

Mathematicians speculate that this result also applies to high-dimensional space.

This assumption is called periodic secret shop conjecture.

But now, Tao Zhe-Xuan and others have overturned this conjecture by constructing a secret shop that can fill high-dimensional space aperiodic, but not periodically.

Paper address: https://arxiv.org/ abs / 2211.15847

What is the periodic secret shop conjecture? The secret shop problem can be said to be the oldest and most classical problem in geometry.

The so-called "secret shop" refers to the mosaic of plane graphics.

In other words, it is spliced with plane figures of exactly the same shape and size, so that they can be laid together without leaving gaps or overlap.

In the dense shop problem, it is easy to cover a space with squares, triangles or hexagons.

In the 1960s, however, mathematician Robert Berger discovered an interesting set of coverings that could completely cover a plane, but only in a way that would never be repeated.

As the first group of aperiodic coverings, it consists of 20426 planar graphics. Of course, Berger soon reduced it to 104.

After that, mathematicians worked hard to reduce this number.

Today, the most famous is the aperiodic compact shop discovered by Penrose in the 1970s, which can cover a plane with only two shapes: kites and darts.

How to come up with a secret shop that does not repeat? It's not that hard.

This can be done by adjusting many repeated periodic coverings.

For example, in an infinite square shaped like a chessboard, we move each row so that it deviates significantly from the above line.

The trick is to find a set of tessellations that can cover the entire plane, but in a non-repetitive way.

Now that Penrose has reduced the number of secret shop graphics to two, is it possible that such a cleverly shaped figure can also form a secret shop?

The answer is yes, but only if you can rotate and reverse the graph.

However, if it is stipulated that the rotation of the figure is not allowed, it is impossible not to leave gaps.

A few years ago, mathematician Siddhartha Bhattacharya proved that no matter how complex and subtle the dense tile figure is, it is impossible to design a dense tile that can only cover the entire plane aperiodic if it is stipulated that only a single displacement or translation can be used.

That is, if you impose enough restrictions on a shape to fill the space, you can force a periodic pattern to appear.

Paper address: https://arxiv.org/ abs / 1602.05738

Mathematicians speculate that Bhattacharya's two-dimensional results are also applicable to high-dimensional spaces.

They speculate that just as there are no aperiodic two-dimensional graphics, there are no suitable three-dimensional (or more complex) graphics, which can be extended to any large dimension.

This assumption is called periodic periodic tiling conjecture.

This conjecture was broken by Tao Zhe Xuan and others, but they were wrong.

In a preprint released last month, Tao Zhe Xuan and Rachel Greenfeld finally overturned this conjecture.

"as long as you have two secret shops, they can be combined into very complicated things. Tao Zhe Xuan said that the difference is that they do not use the way mathematicians usually expect.

Their approach is to build a dense shop that can fill high-dimensional space aperiodic, but not periodically.

In this regard, other mathematicians infer that their conclusions may be correct in all dimensions.

Mathematician Mihalis Kolountzakis said, "this is a pleasant surprise. I hope this conjecture is correct in all dimensions." However, in a sufficiently high dimension, I am afraid it will not go too far if I only rely on intuition. "

The work of Tao Zhe-Xuan and others has not only broken through the boundary between the possible and impossible in geometry, but also extended to the problem beyond geometry-the limit of logic itself.

In 2019, Rachel Greenfeld came to UCLA as a postdoctoral researcher.

Prior to Rachel Greenfeld, Tao Zhe Xuan and she had been working independently on another issue related to translational tilings. Subsequently, the two men turned their attention to the periodic secret shop conjecture.

Previously, this conjecture has been known in one and two dimensions, and they are trying to prove it in three dimensions-- if you can move a three-dimensional version of a shape to cover the entire three-dimensional space, then there must be a way to spread the entire three-dimensional space periodically.

Tao Zhe Xuan and Rachel Greenfeld have made some progress, and through some techniques, they have re-proved this conjecture in two dimensions.

However, when they tried to apply the same technique to three-dimensional space, they hit a brick wall.

Tao Zhixuan said: "at some point, we feel very depressed, so we have to think: well, maybe we can't prove this conjecture in a higher dimension for a reason." We should start looking for counterexamples. "

They began to comb through the literature in all non-cyclical areas.

They start with the first document in history, a collection of more than 20000 coverings published in 1964, which can cover the plane by translation, but only aperiodic.

From here, they embark on new techniques to build a single aperiodic covert shop.

Their idea is to change the environment.

If you want to lay out a two-dimensional space, instead of trying to lay a continuous plane, consider laying a two-dimensional grid-that is, an array of infinite points arranged in the grid.

The cloak can be defined as a set of finite points on the grid. If you have a suitable secret shop, you can accurately cover every point in the mesh by copying a limited set of points and sliding them around.

Proving the "discrete" periodic closeness conjecture of high-dimensional grids is slightly different from proving the continuous version of this conjecture, because some coverages are possible in the lattice, but impossible in continuous space.

However, these two conjectures are related. Tao Zhe-Xuan and Greenfeld hope to present a discrete counterexample and then modify the proof to make it applicable to continuous situations.

In the summer of 2021, they finally approached their goal-they found two secret bunks in an ultra-high-dimensional space.

These two covert shops can fill the space in which they are located, but they are not periodic.

Paper address: https://arxiv.org/ abs / 2108.07902

"that's not enough," Greenfeld said. "the two are very close, but two secret shops are more unstable and much less rigid than one. "

It will take them another year and a half to build a real counterexample for periodic covert conjecture.

Secret shop sandwiches they started by creating a new language-rewriting the problem to be solved in a special equation.

What they need to solve is the unknown "variables" in this equation, which represent all possible ways to cover high-dimensional space.

"however, it is very difficult for you to describe things with an equation," Tao said. "sometimes you need multiple equations to describe a very complex set of spaces. "

As a result, Tao Zhe Xuan and Greenfeld reconstructed the problem they were trying to solve.

They realized that they could design a set of equations instead, in which each equation would have different constraints on its solution.

In this way, they can decompose the problem into a problem about many different coverings-in this case, all coverings cover a given space with the same set of translation.

For example, in a two-dimensional space, you can close the plane one unit at a time by sliding a square up, down, left, or right.

But other shapes can also use exactly the same set of displacements to lay the plane: for example, a bulge is added to the right edge of a square and the left edge is removed, just like a jigsaw puzzle.

If we stack them with a square, a jigsaw puzzle, and other blocks that use the same set of displacements, like cold cuts in sandwiches, we can build a block space that covers three dimensions with a single set of translations.

Tao Zhe Xuan and Greenfeld need to do this in more dimensions.

"because in any case, we all work in a high dimension, so adding a dimension won't have a bad effect on us," Tao said.

Instead, an additional dimension is added to provide them with additional flexibility.

According to the secret shop arrangement of children's toy research, Tao Zhe-Xuan tried to reverse the construction process of this sandwich, rewriting the single equation and high-dimensional secret shop problem into a series of lower-dimensional secret shop equations.

These equations determine the appearance of the high-dimensional dense shop structure after that.

As Tao Zhe Xuan said before: "as long as you have two secret shops, they can be combined into very complex things." "

Tao Zhe-Xuan and Greenfeld treat equation systems as computer programs: each line of code or equation is a command that can be combined to generate a program that achieves a specific goal.

Tao Zhe Xuan said: "Logic circuits are made up of very basic objects, and these and or gates are not very interesting." "

"but you can stack them together to create a circuit that can draw sine waves or communicate over the Internet. "

"so we began to think of this problem as a programming problem. "

Each command is a different attribute that their final secret shop needs to meet, so the whole program needs to ensure that any secret shop that meets all criteria must be aperiodic.

In this way, the question becomes: in all the secret shop equations, what attributes need to be encoded to achieve this?

For example, the shape of a closet in the first layer of a sandwich may allow only certain types of movement.

Tao Zhe Xuan and Greenfeld must carefully establish a list of constraints-so that it is not strict enough to exclude any solution, but enough restrictions to exclude all periodic solutions.

"the key to the game is to build the right level of constraints to encode the right puzzle. "Greenfeld said.

Infinite Sudoku Tao Zhe Xuan and Greenfeld hope that the jigsaw puzzle programmed with their cloak equations is a grid with infinite rows and a large number of finite columns.

For the two, this is a huge Sudoku puzzle: fill each line and diagonal of the puzzle with a specific sequence of numbers that correspond to the limitations they can describe with the closeness equation.

The two then found an aperiodic sequence-which means that the solution of the system of related cryptographic equations is also aperiodic.

"there is basically only one solution to this puzzle. Interestingly, this is an almost periodic solution (almost periodic), "Tao Zhe Xuan said." "it took us a long time to find out. "

Izabella aba, a mathematician at the University of British Columbia, said: "almost periodic functions are not a new concept in mathematics, but this is indeed an innovative use of almost periodic functions. "

As Iosevich said, Tao Zhe Xuan and Greenfeld "created a basic object and elevated it to an extremely complex level." "

According to the almost periodic function, Tao Zhe-Xuan and Greenfeld constructed a high-dimensional aperiodic plane graph in discrete and continuous scenes.

The plane figure of the design is very complex, full of twists and turns and holes, so that there is almost no dense space.

Tao Zhe Xuan and Greenfeld do not calculate the dimensions of the space in which they live. All they know is that it is big, about 2 to 100 to the 100 (or 3 followed by 199 zeros)!

"our proof is constructive, so everything is clear and calculable," Greenfeld said. "but because it's very, very far from the best, we didn't verify it. "

They believe that aperiodic planar graphics can be found in lower dimensions because some of the more technical parts of their construction need to be done in a conceptually "very close to two-dimensional" space.

She doesn't think they will find a three-dimensional plane, but she says four-dimensional graphics may exist. So, Iosevich says, they not only refuted the periodic secret shop conjecture, but also "did it in the most face-beating way." "

The next step: try the research of incomplete theory Tao Zhe Xuan and Greenfeld, which provides a new method for constructing aperiodic covert shop, which they think can be used to refute other conjectures related to covert shop.

This work touches not only on human intuition, but also on the boundaries of mathematical reasoning.

In the 1930s, mathematician Kurt G ö del put forward the famous Godel's incompleteness theory.

Godel proposed that "self-consistency" and "completeness" in the natural number system cannot have both: some propositions can neither be proved nor falsified in the system. Can only give up one, save the other, a little bit of fish and bear paw can not have both meaning.

Similarly, there are many computationally unsolvable problems in mathematics, that is, problems that no algorithm can solve in a limited time.

Mathematicians discovered in the 1960s that questions about secret shops may also be unsolvable.

In other words, for some sets of graphics, it can be proved that it is impossible to figure out whether they can complete the secret shop in a limited time.

"it's a very simple question, but it's still beyond the scope of mathematics," said Richard Kenyon, a mathematician at Yale University. "this is not the first example of a mathematical theory that is unsolvable or incomplete, but it is indeed the most approachable theory. "

Last year, Tao Zhe-Xuan and Greenfeld found that the general proposition about high-dimensional cryptographic pairs is unsolvable: they proved that no one can be sure whether planar graphics can be covertly tiled (whether periodic or aperiodic).

Is the proposition about a single plane figure also unsolvable?

Since the 1960s, it has been known that if the periodic secret shop conjecture is true, then people can determine whether any given graph can achieve secret shop.

But on the contrary, this is not necessarily the case. Just because there are aperiodic graphics does not mean that the problem is unsolvable.

This is what Tao Zhe Xuan and Greenfeld want to figure out next.

"We think it is very reasonable that the language we create should be able to create an uncertain problem," Tao said. Therefore, there may be some secret shops, and we can never prove that it is a closed space or a non-closed space. "

In order to prove that a proposition is unsolvable, mathematicians usually prove that it is equivalent to another proposition that is known to be unsolvable.

Therefore, if this secret shop problem also proves to be unsolvable, then it can be used as a reference tool to prove other propositions-this meaning goes far beyond the scope of secret shop.

At the same time, the results of Tao Zhe Xuan and Greenfeld are, to some extent, a reminder to researchers.

"mathematicians like simple propositions," Iosevich said.

Unfortunately, not all interesting mathematical propositions can do this. More often, we need our research to achieve the desired results. "

Reference:

Https://www.quantamagazine.org/nasty-geometry-breaks-decades-old-tiling-conjecture-20221215/

Https://terrytao.wordpress.com/2022/09/19/a-counterexample-to-the-periodic-tiling-conjecture/

Https://baike.baidu.com/item/%E5%AF%86%E9%93%BA/5106336

This article comes from the official account of Wechat: Xin Zhiyuan (ID:AI_era), author: editorial Department

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