The ancient philosopher Plato conjectured that the universe was made up of particular geometric shapes; The earth of cubes. The results of a multidisciplinary research team found the truth in Plato’s belief.
The researchers found that the ancient Greek philosopher was doing something.
Plato, the Greek philosopher who lived in the 5th century BC. C., believed that the universe was made of five types of matter: earth, air, fire, water and the cosmos. Each one was described with a particular geometry, a platonic shape. For earth, that shape was the cube.
Science has constantly moved beyond Plato’s conjectures, seeking instead atom like the building block of the universe. However, Plato appears to have been on to something, according to the researchers.
“It turns out that Plato’s conception of the earth element made of cubes is literally the average statistical model for real earth. And that’s just amazing. ” Douglas Jerolmack
In a new article in the procedures of the National Academy of Sciences, a team from the University of Pennsylvania, the Budapest University of Technology and Economics, and the University of Debrecen use mathematics, geology, and physics to demonstrate that the average shape of rocks on Earth is a cube.
“Plato is widely recognized as the first person to develop the concept of an atom, the idea that matter is made up of some indivisible component on the smallest scale,” says Douglas Jerolmack, geophysicist at the Earth Department of the School of Arts and Penn Sciences. and Environmental Sciences and in the Department of Mechanical Engineering and Applied Mechanics of the Faculty of Engineering and Applied Sciences. “But that understanding was only conceptual; Nothing in our modern understanding of atoms derives from what Plato told us.
“The interesting thing here is that what we find with the rock, or the earth, is that there is more than a conceptual lineage in Plato. It turns out that Plato’s conception of the earth element made of cubes is literally the statistical average model for real earth. And that’s just amazing. “
The group’s discovery began with geometric models developed by mathematician Gábor Domokos of the Budapest University of Technology and Economics, whose work predicted that natural rocks would fragment into cubic shapes.
“This document is the result of three years of serious thought and work, but it is a central idea,” says Domokos. “If you take a three-dimensional polyhedral shape, cut it randomly into two pieces, and then cut it over and over again, you will get a lot of different polyhedral shapes. But in an average sense, the resulting shape of the fragments is a cube. “
The research team measured and analyzed the fragmentation patterns of rocks they collected, as well as previously assembled data sets. Credit: Courtesy of Gablor Domokos and Douglas Jerolmack.
Domokos drew two Hungarian theoretical physicists to the circle: Ferenc Kun, an expert in fragmentation, and János Török, an expert in statistical and computational models. After discussing the potential of the discovery, says Jerolmack, Hungarian researchers brought their findings to Jerolmack to work together on geophysical issues; in other words, “How does nature allow this to happen?”
“When we brought this to Doug, he said, ‘This is a mistake or this is big,'” Domokos recalls. “We are working backwards to understand the physics that results in these shapes.”
Fundamentally, the question they answered is what shapes are created when rocks are broken into pieces. Surprisingly, they discovered that the central mathematical conjecture unites geological processes not only on Earth but also around the solar system.
“Fragmentation is this ubiquitous process that is grinding planetary materials,” says Jerolmack. “The solar system is full of ice and rocks that break constantly. This work gives us a signature of that process that we have never seen before. “
Part of this understanding is that the components that come off a previously solid object must fit without gaps, like a dropped plate about to break. It turns out that the only so-called Platonic shapes — polyhedrons with sides of equal length — that fit without gaps are cubes.
“One thing we have speculated in our group is that Plato possibly looked at a rock outcrop and after unconsciously processing or analyzing the image in his mind, he surmised that the average shape is something like a cube,” says Jerolmack.
“Plato was very sensitive to geometry,” adds Domokos. According to tradition, the phrase “Let no one ignorant of geometry enter” was engraved on the door of Plato’s Academy. “His insights, backed by his extensive thinking about science, may have led him to this idea about cubes,” says Domokos.
To test whether their mathematical models were true in nature, the team measured a wide variety of rocks, hundreds they collected and thousands more of previously collected data sets. It doesn’t matter if the rocks had naturally worn away from a large outcrop or if they had been blown up by humans, the team found a good fit to the cubic average.
However, there are special rock formations that seem to break the cubic “rule”. The Giant’s Causeway in Northern Ireland, with its towering vertical columns, is one example, formed by the unusual basalt cooling process. These formations, although rare, are still encompassed by the mathematical conception of team fragmentation; they are only explained by unusual processes at work.
“The world is a messy place,” says Jerolmack. “Nine times out of 10, if a rock breaks, squeezes, or cuts, and these forces usually occur together, you end up with fragments that are, on average, cubic shapes. It is only if you have a very special stress condition that you get something else. The land just doesn’t do this often. “
The fracture patterns that scientists identified can be found not only on Earth, but around the solar system, including the mosaic surface of Jupiter’s moon Europa. Credit: NASA / JPL-Caltech / SETI Institute
The researchers also explored fragmentation in two dimensions, or on thin surfaces that function as two-dimensional shapes, with a depth that is significantly less than width and length. There, the fracture patterns are different, although the core concept of dividing polygons and arriving at predictable average shapes still holds.
“It turns out that in two dimensions you are equally likely to get a rectangle or a hexagon in nature,” says Jerolmack. “They are not true hexagons, but they are the statistical equivalent in a geometric sense. You can think of it as cracking paint; a force is acting to separate the paint equally from different sides, creating a hexagonal shape when it cracks. “
In nature, examples of these two-dimensional fracture patterns can be found in ice sheets, dry mud, or even the Earth’s crust, the depth of which is far greater than its lateral extent, allowing it to function as a de facto material two-dimensional factor. . The Earth’s crust was previously known to fracture in this way, but the group’s observations support the idea that the fragmentation pattern results from plate tectonics.
Identifying these patterns in rock can help predict phenomena such as rock fall hazards or the probability and location of fluid flows, such as oil or water, in rocks.
For researchers, finding what appears to be a fundamental rule of nature arising from age-old ideas has been an intense but satisfying experience.
“There are a lot of grains of sand, pebbles and asteroids out there, and they all evolve by chipping universally,” says Domokos, who is also a co-inventor of the Gömböc, the first known convex shape with the minimal number, only two, of points of static balance. Chipping by collisions gradually eliminates the equilibrium points, but the shapes do not become a Gömböc; The latter appears as an unattainable end point of this natural process.
The current result shows that the starting point can be a similarly iconic geometric shape: the cube with its 26 balance points. “The fact that pure geometry provides these supports for a pervasive natural process makes me happy,” he says.
“When you pick up a rock in the wild, it’s not a perfect cube, but each one is kind of a statistical cube shadow,” adds Jerolmack. Remember Plato’s allegory of the cave. He postulated an idealized form that was essential to understanding the universe, but all we see is distorted shadows of that perfect form. “
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Reference: “Plato’s Cube and the natural geometry of fragmentation” by Gábor Domokos, Douglas J. Jerolmack, Ferenc Kun and János Török, July 17, 2020, procedures of the National Academy of Sciences.
DOI: 10.1073 / pnas.2001037117
Douglas Jerolmack is a professor in the Department of Earth and Environmental Sciences at the College of Arts and Sciences and in the Department of Mechanical Engineering and Applied Mechanics at the College of Engineering and Applied Sciences at the University of Pennsylvania.
Gábor Domokos is professor and director of the MTA-BME Morphodynamics Research Group at the Budapest University of Technology and Economics.
Ferenc Kun is a professor in the Department of Theoretical Physics at the University of Debrecen.
János Török is an associate professor in the Department of Theoretical Physics at the Budapest University of Technology and Economics.
