Mathematicians cut figures in search of parts of equations

A new work on the problem of “equal alignment” explains when it is possible to cut one figure and assemble another from it

If you have two flat pieces of paper and scissors, can you cut one piece and rearrange the pieces to get the other? If you can, then these two figures are “scissor congruent” [ equal ].



However, mathematicians are interested in whether it is possible to detect such a relationship in figures without using scissors? In other words, do these figures have such characteristics that could be measured in advance and determine whether they are congruent?



For two-dimensional figures, the answer is simple. You just need to measure their area; if they match, then the figures are scissor congruent.



But for figures in higher dimensions - for example, for a three-dimensional ball or an eleven-dimensional donut that is impossible to imagine - the question of cutting and reassembling in a different form becomes much more complicated. And despite centuries of trying, mathematicians could not determine the characteristics that confirm the equal composition for most figures of higher dimension.



However, this fall two mathematicians made the most significant breakthrough in solving this problem in several decades. In a paper presented at the University of Chicago on October 6, Jonathan Campbell of Duke University and Inna Zakharevich of Cornell University took a significant step toward proving scissor congruence for shapes of any dimension.



But not only. Like most important math problems, equanimity is a rabbit hole: a humble statement that draws mathematicians into the deep hole of complex mathematics. In an attempt to understand scissor congruence, Campbell and Zakharevich may have shown a new way to reason about a completely different area of ​​this science: algebraic equations.

First cut

Equal alignment may seem like a simple task. More than 2000 years ago, Euclid realized that two two-dimensional figures of the same area can be rearranged from one to another. It is reasonable to assume that figures of higher dimensions of the same volume can be redone similarly.



But in 1900, David Hilbert suggested that this task is actually not so simple.



In that year, speaking at the International Mathematical Congress in Paris, he identified 23 open problems , which, in his opinion, will guide mathematical thought in the coming century. The third of them was related to scissor congruence [equal composition of equal polyhedrons]. Hilbert suggested that not all three-dimensional figures of the same volume are congruent - and challenged mathematicians by proposing to find a pair of figures proving this.



A year after the speech, Hilbert’s student, Max Dan , did just that. Such a term seemed to mathematicians suspicious. “Some people think that Hilbert included this problem in the list only because his student had already solved it,” Zakharevich said.



Whether it was a conspiracy or not, the result of Dan turned the mathematicians' idea of ​​equal representation upside down. He proved that a tetrahedron of a single volume is not equal to a cube of the same volume. No matter how you cut the first, you can never put together the pieces of the second.



In addition to demonstrating that volume equality is not enough to determine equal composition, Den proposed a new way to measure shapes. He proved that any three-dimensional figures, equal to each other, must have the same volume, and also coincide to a new extent.



Dan concentrated on the inner corners between the two faces of the three-dimensional figure. For example, inside a cube, all faces meet at right angles. But in more complex forms, the angles are different and have different importance. The angles between the longer edges more influence the shape of the figure than the angles between the shorter edges, so Den assigned the corners a weight based on the lengths of the edges that form them. He combined this information into a complex formula that yielded a single number - the “Den invariant” - for a given figure.





Mathematicians want to know when a figure can be cut and assemble another from it.

Two-dimensional figures are equally spaced if they have the same area.

Three-dimensional figures are equally composed if they have the same volume and Dehn invariant.

The cube and tetrahedron are not equally composed - they have the same volume, but different Den invariants.

Shapes can be cut into pieces, and graphs of equations can be cut into subgraphs. Mathematicians are looking for an analogue of the Dehn invariant, which shows that two equations consist of identical pieces.



Deng proved that any three-dimensional figures equidistant to each other must have the same volume and the Deng invariant. But he could not answer a more complex question: if the three-dimensional figures have the same volume and the Dan invariant, does this mean that they are necessarily equal? Jean-Pierre Sidler finally proved this in 1965. Three years later, Björg Jessen showed that these same two characteristics determine the equidimensionality in four dimensions.



The results of Sidler and Jessen were serious steps forward, but mathematicians are a greedy people: is there enough volume and Dehn's invariant to determine the equal composition of figures in all dimensions? Are these measurements sufficient in geometrical spaces other than Euclidean — in spherical geometry (imagine the latitude and longitude on the surface of the Earth) or the saddle-shaped universe of hyperbolic geometry?



At the end of the 20th century, the mathematician Alexander Borisovich Goncharov proposed an approach that, in his opinion, could solve the whole problem once and for all - and at the same time relate the equalness with a completely different field of mathematics.

Strange connections

Math is full of unexpected connections. Zakharevich says that doing math is like stumbling upon something strange in nature and trying to understand why it is.



“If you meet a ring of mushrooms in a forest and don’t know how mushrooms grow, you’ll think about how they know how to grow around? - she said. “The reason is that mushrooms have a mycelium growing underground.”



In 1996, Goncharov formulated a set of hypotheses suggesting the existence of a mathematical structure, also hidden under the surface. If this structure exists, it will be able to explain why some mathematical phenomena - including equal composition - work that way.



One hypothesis states that the volume of the figure and its Dehn invariant is sufficient to determine the equal composition of figures of any dimension and in any space.



“Goncharov said that the same principles that apply in three dimensions apply to all,” said Charles Weibel of Rutgers University.



But Goncharov, now employed at Yale, also predicted that this hidden structure would explain much more than that. He said that equal alignment is a more universal concept, and that it is applicable not only to cutting geometric shapes, but also to cutting shapes generated by solutions of algebraic equations - for example, the graph of the equation x ² + y ² + z ² = 1. And the information needed to classify by equal composition reflects the information needed to classify algebraic equations — such that equations of the same class are made up of identical pieces.



The connection was shocking, as if a principle suitable for systematizing animals would somehow allow you to systematize chemical elements as well. Many mathematicians find this idea as strange as it seems at first glance.



“This is completely mysterious. At first glance, these things should not be connected at all, ”said Campbell.

Slicing equations

To understand how geometrical figures and algebraic equations can be similar, it will first be useful to understand how solutions of equations can be divided into parts. To do this, let's go back to our previous example and draw a graph of the equation x ² + y ² + z ² = 1.



It will be a sphere. However, this surface is not only a collection of solutions to this equation: it is also a collection of many smaller graphs, or subgraphs, of solutions of other equations. For example, on the surface of a sphere, you can draw a circle in the manner of the earth's equator. This is one subgraph representing solutions of the algebraic equation x ² + y ² = 1. Or you can isolate a single point on the north pole of the sphere corresponding to the equation z = 1. By studying the various subgraphs that can be drawn within a larger graph - something like its constituent parts - you can find out some properties of a larger chart.



For more than 50 years, mathematicians have developed the theory of subgraphs of algebraic equations. Just as ordinary matter consists of atoms, so, according to mathematicians, algebraic equations consist of fundamental parts called “motives”. The term comes from the French word motif, denoting the basic elements of the melody.





Inna Zakharevich from Cornell University



“Motives are fundamental components. They will talk about everything that algebraic equations consist of, like a melody, consists of various components, ”said Zakharevich. A sphere, for example, consists of circles, points, and planes. Each of them consists of components (manifested as a result of mathematical actions on them), and so on, lower and lower, until we come to the motives, the alleged foundation of algebraic equations.



Mathematicians need to classify algebraic equations according to their motives in order to obtain a complete and systematic picture of equations belonging to the most important mathematical objects. This is a difficult and unfinished task. But in 1996, Goncharov suggested that sorting figures by equal composition and sorting algebraic equations by motive are two sides of one task - that is, classification of one will give you a principle by which the other can be classified.



He suggested that this connection is based on the analogue of the Dehn invariant. Only instead of appearing from the simplest geometric calculations, this analogue should arise from a similar calculation of the motives of algebraic equations (“motive coproduct ”).



“The idea is that the Dan invariant problem is parallel to another motive problem,” Weibel said.



But in order to discover such a connection, mathematicians first need to prove that the Dehn invariant really sorts the figures into equally composed groups. Den himself showed that any equidistant three-dimensional figures have equal volumes and the Den invariant. However, Den, and everyone else after him, did not refute the possibility that there are certain figures of higher dimensions of the same volume and with the same Dan invariant, which are not equally equal. In their new work, Campbell and Zakharevich tried to permanently close this opportunity.

Two for the price of one

In June 2018, Campbell and Zakharevich worked for three weeks at the Advanced Research Institute in Princeton, New Jersey. They had long been interested in equal treatment, but Zakharevich believed that Goncharov’s hypotheses were too complex to be dealt with in such a short time. But Campbell still wanted to try, and Zakharevich did not have to persuade for a long time.



“Jonathan said: 'We have three weeks, let's try to approach this and see what happened, by the end of the first,” said Zakharevich. Two weeks later, they developed many key ideas that underlie their new work.



In the work, they conduct a counter-intuitive thought experiment. To understand it, imagine that you have a hotel with many rooms. You need to arrange all the equal figures with each other in the same room. We do not know how to determine that the figures are equally spaced - this is the root of the problem. However, for our thought experiment, let's imagine that this is possible. Or, as Zakharevich says, "We will pretend that there is a certain omniscient person who knows whether two figures are equal or not."



Having sorted the figures by room, we verify that all figures in the same room have the same volume and the same Dan invariant. It is also important to verify that all figures of the same volume and with the same Den invariant were in the right room - that figures that had fallen off the collective were not hanging around in the hotel bar. The goal of a thought experiment is to prove the existence of an ideal, one-to-one correspondence between groups of equal figures and groups of figures having the same volume and the same Dan invariant. The existence of such a correspondence will prove that only the volume and the Dan invariant will really be enough for you to determine the equal composition of the figures.



Goncharov predicted the existence of such a correspondence, and Campbell and Zakharevich proved its presence - under one condition. Correspondence exists if another unproven result related to the Beilinson hypotheses is true.



Goncharov’s two hypotheses — the classification of equal figures by volume and the Dehn invariant, as well as the classification of algebraic equations by the analogue of the Dehn invariant — are not fully proved by Campbell and Zakharevich. However, their work nevertheless provides mathematicians with a clearer idea of ​​how to prove them all: if you can prove Beilinson’s hypotheses, then thanks to the work of Campbell and Zakharevich, you’ll also get equal balance in addition.



“Their work really rethinks this task,” Weibel said. “When you connect two hypotheses in this way, it sheds light on the structure of the object being studied.”



Campbell and Zakharevich are now working with yet another mathematician, Daniil Rudenko from the University of Chicago, trying to determine the relationship between the cutting of figures and analysis into parts of the equations proposed by Goncharov. Rudenko had already made some progress in this direction. Now, together with Campbell and Zakharevich, he hopes to move much further.



“I think we have every chance to make significant progress. Maybe this way it even turns out to prove Goncharov’s hypotheses, ”Rudenko said.