Plato, the Greek philosopher who lived in the 5th century BC. C., believed that the universe was made up of five types of matter: earth, air, fire, water and cosmos. Each was described with a particular geometry, a Platonic shape. For the earth, that shape was the cube.
Science has constantly moved beyond Plato’s conjectures, looking instead at the atom as the building block of the universe. However, Plato seems to have been onto something, researchers have found.
In a new document 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 uses mathematics, geology, and physics to show that the average shape of rocks on Earth is a cube.
“Plato is widely recognized as the first person to develop the concept of the atom, the idea that matter is made up of some indivisible component on the smallest scale.” says Douglas Jerolmack, a geophysicist in the Department of Earth and Environmental Sciences at Penn College of Arts and Sciences and 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 rock, or earth, is that there is more than a conceptual lineage going back to Plato. It turns out that Plato’s conception that the earth element is made up of cubes is literally the average statistical model of the real earth. And that’s just mind blowing. “
The group’s find 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 article is the result of three years of serious thinking and work, but it returns to a central idea,” says Domokos. “If you take a three-dimensional polyhedral shape, cut it randomly into two pieces, and then cut these pieces over and over again, you get a lot of different polyhedral shapes. But in a middle sense, the resulting shape of the fragments is a cube. “
Domokos attracted two Hungarian theoretical physicists to the circuit: Ferenc Kun, an expert on fragmentation, and János Török, an expert on statistical and computational modeling. After discussing the potential of the discovery, Jerolmack says, the Hungarian researchers brought their find to Jerolmack to work together on geophysical questions; in other words, “How does nature allow this to happen?”
“When we told Doug about this, he said, ‘This is a mistake or this is big,'” recalls Domokos. “We work 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 found that the central mathematical conjecture ties together geological processes not only on Earth but also throughout the solar system.
“Fragmentation is this omnipresent process that is crushing planetary materials,” says Jerolmack. “The solar system is full of ice and rocks that are constantly breaking apart. 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 together without any gaps, like a falling plate about to break. It turns out that the only one of the so-called Platonic shapes – polyhedra with equal length sides – that fit without spaces are cubes.
“One thing we have speculated in our group is that, quite possibly Plato looked at a rocky outcrop and after unconsciously processing or analyzing the image in his mind, he conjectured 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 up by his broad 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 were dynamized 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 soaring vertical columns, is one example, formed by the unusual cooling process of basalt. These formations, although rare, are still included in the team’s mathematical conception of fragmentation; they are simply explained by unusual processes at work.
A mesmerizing and messy world
“The world is a messy place,” says Jerolmack. “Nine times out of 10, if a rock separates, squeezes or shears, and these forces generally occur together, you end up with fragments that are, on average, cubic in shape. It is only if you have a very special stress condition that you get something else. The earth doesn’t do this often. “
The researchers also explored fragmentation in two dimensions, or thin surfaces that function as two-dimensional shapes, with a depth that is significantly less than the width and length. There, the fracture patterns are different, although the central concept of dividing polygons and arriving at predictable average shapes still holds.
“It turns out that in two dimensions you are almost 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 cracked 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 in the earth’s crust, the depth of which is much greater than its lateral extent, allowing it to function as a two-dimensional de facto. material. The earth’s crust was previously known to fracture in this way, but the group’s observations support the idea that the fragmentation pattern is the result of plate tectonics.
Identifying these patterns in the rock can help predict phenomena such as rockfall hazards or the likelihood and location of fluid flows, such as oil or water, in the rocks.
For researchers, finding what appears to be a fundamental rule of nature that emerges from ancient knowledge has been an intense but satisfying experience.
“There are many grains of sand, pebbles, and asteroids, and they all evolve by splintering in a universal way,” says Domokos, who is also a co-inventor of the Gömböc, the first known convex shape with the minimum number – just two – of points of static equilibrium. Collision chipping gradually eliminates the balance 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 an equally iconic geometric shape: the cube with its 26 balance points. “The fact that pure geometry provides these supports for an omnipresent natural process makes me happy,” he says.
“When you collect a rock in the wild, it’s not a perfect cube, but each one is kind of a statistical shadow of a cube,” adds Jerolmack. Remember Plato’s allegory of the cave. He postulated an idealized shape that was essential to understanding the universe, but all we see are distorted shadows of that perfect shape. “
Source: University of Pennsylvania
SHARE this trailer with your friends on social media …