Exercises in pure mathematics led to the creation of a large-scale theory about the structure of the world
Sometime in the middle of the summer of 2016, Hungarian mathematician Gabor Domokos stepped onto the porch of the house of Douglas Jerolmak , a geophysicist from Philadelphia. Domokosh had travel suitcases with him, a severe cold and a burning secret.
A little later, two men walked along the gravel driveway in the back yard where Jerolmak's wife kept a taco cart. Crushed limestone crunched under their feet. Domokosh pointed at his feet.
"How many facets does each of these stones have?" - he asked. Then he grinned. "What if I told you that there are usually six of them?" And then he asked an even more general question that he hoped would permanently reside in his colleague's brain. What if the world is made of cubes?
Jerolmak objected at first: maybe houses are built of bricks, but the Earth is made of stones. And the shape of the stones is obviously different. Mica crumbles into scales, crystals break along rigidly defined axes. However, Domokosh argued that pure mathematics alone implies that any stones that break at random will generate shapes with an average of six faces and eight vertices. If we take the average for all of them, it will tend to some ideal cube. Domokosh said he proved it mathematically. Now he needed Jerolmak to help him show that this also happens in nature.
“It was a clear, geometric prediction, born of nature, and without any physics,” said Jerolmak, a professor at the University of Pennsylvania. "How the hell did nature allow this at all?"
Over the next few years, the couple explored their geometric idea, exploring everything from microscopic fragments of rocks to outcrops of geological rocks, planetary surfaces and even Plato's Timaeus dialogue . All this covered the project with a touch of mysticism. One of the greatest philosophers around 360 BC mapped five platonic solidswith five "elements" of the universe: earth, air, fire, water and stellar matter. By coincidence and / or foresight, Plato matched the cubes, which are best stacked, with the ground. “And I thought - okay, now we have already slightly entered the territory of metaphysics,” said Jerolmak.
Gabor Domokos and Douglas Jerolmak
However, they continued to find medium cuboids in nature, as well as several forms that did not look like cubes, but obey the same theory. As a result, they created a new mathematical platform: a descriptive language that expresses how things fall apart. Published this year, their joint work with the title resembled an esoteric tome from the Harry Potter series: Plato's Cube and the Natural Geometry of Fragmentation.
Several geophysicists contacted by the journal say the same mathematical platform can be used for other tasks, such as studying the erosion of rock faults or preventing dangerous landslides. "This is very interesting," said geomorphologist Mikael Attal of the University of Edinburgh, one of two reviewers of this work. Another reviewer, geophysicist David Furbisch of Vanderbilt University, said, "This kind of work leaves me wondering if I can somehow take advantage of these ideas?"
All possible faults
Long before his visit to Philadelphia, Domokosh had a more harmless mathematical question.
Let's say you smashed something into many pieces. Now you have a mosaic - a set of shapes that can be put together without overlaps or breaks, like the floor in an ancient Roman bath. Also, assume that all shapes are convex.
At first, Domokosh wondered if it was possible only by means of geometry to predict what figures on average such a mosaic would consist of. Then he wanted to learn how to describe all the other possible sets of such figures.
In two dimensions, you do not need to break anything into pieces to study this issue. Take a piece of paper. Cut it at random by dividing the sheet in two. Then make one cut at each of these polygons. Repeat the process several times. Calculate the average number of vertices for each piece of paper.
For a person studying geometry, finding the answer to this question will not be so difficult. “I put a case of beer, that I can help you get this formula in a couple of hours,” said Domokosh. On average, the pieces should have four vertices and four sides, and their average shape will be rectangular.
The same problem can be viewed in three dimensions. About 50 years ago, a Russian nuclear physicist, Nobel Peace Prize laureate, and later a dissident, Andrei Dmitrievich Sakharov pondered the same problem when he was cutting cabbage with his wife. How many vertices will each of the resulting pieces have on average? Sakharov handed this task over to the legendary Soviet mathematician Vladimir Igorevich Arnold and his student. However, they did not find a complete solution, and their attempts were largely forgotten.
Moeraki boulders in New Zealand
Domokosh, who did not know about their work, wrote a proof, the answer to which was cubes. But he wanted to check its correctness. He decided that if the answer to this problem already exists, it should be hidden in the unfathomable work of the German mathematician Wolfgang Weil and Rolf Schneider - 80-year-old titan of the field geometry [in the original does not include the name - apparently refers to the book " Stochastic and integral geometry "/ approx. per.]. Domokosh is a professional mathematician, but the text of the book was too heavy even for him.
“I found a person who agreed to read the part of the book I needed and translate it back into human,” Domokosh said. He found a theorem there for any number of dimensions. She confirmed that cubes do appear in the answer in three dimensions.
Now Domokosh has found averaged figures that are obtained by cutting a flat surface or a three-dimensional brick. A more general question emerged. Domokosh realized that he could also develop a mathematical description of not only average figures, but also potentially any: what set of figures, in principle, can be obtained by dividing an object?
Recall that the figures obtained after the disintegration of the object are a mosaic. They can be put together without overlaps or breaks. The rectangles that we cut the sheet into can easily be made up so that they fill the 2D mosaic. Hexagons are also capable of this - in the idealized case of a set, which mathematicians call " Voronoi diagram ". But the plane cannot be paved with pentagons or octagons.
Geometry of Mars. To analyze the surface - in this case, the honeycomb-like surface of a Mars crater - researchers mark all the tops and sides. They count the number of vertices for each of the cells and the number of cells for which each of the vertices is common.
To correctly classify the mosaics, Domokosh began to describe them with two numbers. The first is the average number of vertices per cell. The second is the average number of different cells for which there is one common vertex. So, for example, in a mosaic of hexagonal tiles, each tile has six vertices. And each vertex is common to three hexagons.
In mosaics, only certain combinations of these two parameters work, which gives a small range of figures into which something can, in principle, disintegrate.
Again, this range is fairly easy to find in two dimensions, but much more difficult in three. In three-dimensional space, cubes fit together very well, but there are other types of shapes, including those that form three-dimensional versions of the Voronoi diagram. In order not to overcomplicate the problem, Domokosh limited himself to a mosaic of regular convex cells with common vertices. As a result, he and the mathematician Zsolt Langi came up with a new hypothesis by sketching a curve that fits all possible three-dimensional mosaics. They published the work in Experimental Mathematics magazine, “and then I sent it all to Rolf Schneider, our deity,” Domokos said.
Space of cubes. In three dimensions, most stones are broken up into cubes with eight vertices per cell. A map of admissible mosaics of convex shapes with regular cells that have common vertices fits into a narrow strip. The area of cuboid shapes is highlighted in red.
Vertical: the number of vertices per cell
Horizontal: the number of common cells at each vertex
“I asked him if I need to explain how I came to such a hypothesis, but he said that he knew about it,” Domokosh laughs. "It was a hundred times more important to me than the acceptance of an article by any magazine in the world."
More importantly, Domokosh now had a platform. Mathematics provided a way to classify all the ways to partition surfaces and blocks. And geometry predicted that if you break a flat surface by accident, it will split into something like rectangles. In three dimensions, splitting will result in something like cubes.
But for all this to matter to someone other than a small group of mathematicians, Domokosh had to prove that the real world also obeys these rules.
From geometry to geology
By the time Domokosh was in Philadelphia in 2016, he had already achieved something in solving the problem in relation to the real world. Together with colleagues from the Budapest University of Technology and Economics, they collected fragments of dolomite that broke away from the Harmashkhatar-hegy rock, located in Budapest. For several days, a laboratory worker, without any prejudice about cubes, diligently counted the number of faces and vertices of hundreds of pieces. What average score did he get? Six faces, eight peaks. Domokos, together with Janos Törok, a computer simulator, and Ferenc Kun, an expert in fragmentation physics, discovered that medium cuboids appeared in other types of rocks, such as gypsum and limestone.
Armed with mathematics and early physical evidence, Domokosh pitched his idea to an overwhelmed Jerolmak. “He hypnotized me and everything else just disappeared for a while,” Jerolmak said.
Their alliance was not new. Many years ago, Domokosh gained fame by proving the existence of the Gömböts.- a funny three-dimensional figure, stubbornly turning over into a certain position of balance. To find out if the Gömböts could exist in reality, he attracted Jerolmak, who helped to apply this concept to explain the round shape of pebbles on Earth and Mars [Vladimir Arnold put his hand here, for the first time raising the question of the existence of such bodies / approx. per.]. Now Domokosh again asked for help to turn some theoretical mathematical concepts into tangible stone.
Gömbötz is a convex three-dimensional homogeneous figure with exactly one point of stable equilibrium and one point of unstable
The couple agreed on a new plan. To prove the existence of Platonic cubes in nature, they needed to show more than just a random coincidence of geometry and a handful of pebbles. They needed to look at all the rocks, and then sketch out a convincing theory of how abstract mathematics could infiltrate a messy geophysics, and then into an even more messy reality.
At first, “everything seemed to work,” Jerolmak said. Domokosh's mathematics predicted that the fragments of stones should, on average, be cubes. An increasing number of real fragments seemed to fit into this theory. However, Jerolmak soon realized that in order to prove the theory, it was necessary to deal with exceptions to the rules.
After all, the same geometry makes it possible to describe many other mosaic patterns, the existence of which is allowed in both two and three dimensions. Djerolmak could immediately name several types of real stones, not similar to rectangles and cubes, which could still fit into this more extensive classification.
Perhaps these examples would completely refute the theory of the cubic world. Or, perhaps more interestingly, they would only appear on special occasions from which geologists could learn new lessons. “I said I know it doesn't work everywhere, and I need to know why,” Jerolmak said.
Over the next few years, Jerolmak and his team, working on both sides of the Atlantic, began marking out exactly where real-life examples of pieces of stones fell on the Domokosh platform. Examining essentially two-dimensional surfaces - cracked permafrost in Alaska, dolomite outcrops, cracks in a granite block - they found polygons that, on average, had four sides and four vertices, just like cut paper. Each of these geological phenomena seemed to manifest themselves where the rocks simply cracked. In this area, Domokosh's prediction came true.
Universe of tiles. All possible convex tiles that completely cover the plane can be plotted against the average number of vertices on a tile (y-axis) and the average number of cells dividing one vertex (x-axis). Real world examples:
6 - pavement of giants , 7 - permafrost in Alaska, 8 - dried mud, 9 - granite surface.
But there was one type of flat surface that lived up to Jerolmac's hopes: it was an exception with its own history. Dirt-covered flat surfaces dry out, crack, get wet, tighten, and then crack again. The cells on such surfaces have, on average, six sides and six vertices - an approximately hexagonal Voronoi diagram. A similar appearance has a rocky surface that appeared after the solidification of lava, which solidifies from the surface and down.
Interestingly, it is these systems that are formed under the influence of other forces that squeeze them out, instead of pushing them in. Geometry reveals geological features. Jerolmak and Domokosh believed that such a Voronoi diagram, albeit quite rare, could also appear on a much larger scale than they had previously studied.
The Voronoi diagram divides the plane into separate sections, each of which consists of all the points closest to the starting point.
Counting the crust
During development, the team met in Budapest and spent three frantic days frantically trying to include more real-life examples in the model. Djerolmak soon brought up a new pattern on the computer screen: a mosaic of the Earth's tectonic plates. The plates sit on the lithosphere, an almost two-dimensional skin on the planet's surface. The pattern looked familiar, and Jerolmak called others to admire it. “We were all shocked,” he said.
At first glance, it seems that the plane drawings tend to the Voronoi diagram, and not to the square grid. And then the team made the calculations. In an ideal Voronoi mosaic of hexagons on a plane, each cell should have six vertices. Real tectonic plates had an average of 5.77 peaks.
At this point, the geophysicist could already celebrate the victory. But mathematics did not suit me. “Doug's mood was rising. He worked as if he were a regular, - said Domokosh. "And the next day I was upset because I was thinking about this breakup."
In the evening Domokosh went home, still devoured by this difference. He wrote down all the numbers again. And suddenly a revelation descended upon him. The hexagon mosaic can pave the plane. But the Earth isn't flat - at least outside of some of the controversial corners on YouTube. Imagine a soccer ball made up of pentagons and hexagons. Domokosh processed the data taking into account the spherical surface and found that on the ball, the Voronoi mosaic cells should have an average of 5.77 vertices.
This idea helped researchers solve one of the important and open questions in geophysics: how are tectonic plates of the Earth formed? Some believe that these plates are a by-product of convection currents moving deep in the mantle. Their opponents believe that the earth's crust is a separate system. It expanded, became brittle and broke. Matching the slabs to the Voronoi diagram, which resembles a mud crust, may support the second theory, Jerolmak said. “It also gave me a sense of how important that job was,” Attal said. "Phenomenal."
Crucial moment
In three dimensions, there were very few exceptions to the cube rule. And they, too, could be explained by simulating unusual forces pushing outward. One distinctly non-cubic formation is on the coast of Northern Ireland, where waves crash against tens of thousands of basalt columns. In Irish it is called Clochán na bhFomhórach , a road of stones for supernatural beings. In English it is called "the bridge of the giants ".
It is important that these columns and other similar volcanic formations are hexagonal. However, judging from Tyrok's simulations, mosaics similar to this pavement are simply three-dimensional structures that grew from the two-dimensional base of Voronoi diagrams after the volcanic rock cooled.
Bridge of Giants in Northern Ireland
The team argues that if you take the big picture, most of the mosaics of the cracked stone can be classified using Platonic rectangles, two-dimensional Voronoi diagrams, and all together - Platonic cubes in three dimensions. Each of the patterns can tell its own geological story. And, yes, given some peculiarities, we can say that the world is made of cubes.
"They've duly validated their model against reality," said Martha-Carey Epps , a science scientist at the University of North Carolina. "My initial skepticism has faded."
“The mathematics tells us that if we crush stones, whatever we want, by accident or on purpose, we still have a limited set of options,” Furbish said. "Isn't that smart?"
Perhaps you will be able to take, for example, a real place consisting of fractured rock, count the vertices and edges, and then draw a conclusion about the geological processes going on there.
“For some places, we have data that allows us to look at this question from this angle,” said Roman Dibayas , a geomorphologist at Pennsylvania State University. “It would be cool if we could draw conclusions from things that are not obvious, like the giants' pavement - just hitting a stone with a hammer and seeing what the shards look like.”
Jerolmak, who at first believed that the connection with the Platonic solids could be accidental, now accepted this hypothesis. After all, in the end, the Greek philosopher believed that the correct geometric forms are necessary for the knowledge of the Universe, although they themselves are invisible to the eye, and appear only in the form of distorted shadows.
"This is literally the most obvious example you can think of. The statistical average of all these observations is a cube, said Jerolmak. "But such a cube cannot be found."