Perhaps the Cubists were right. Researchers have found that when everything from icebergs to rocks breaks apart, their pieces tend to resemble cubes. The finding suggests a universal rule of fragmentation at scales ranging from the microscopic to the planetary.
“It’s a very beautiful combination of pure mathematics, materials science, and geology,” says Sujit Datta, a chemical and biological engineer at Princeton University who was not involved in the work.
The finding builds on the previous work of mathematician Gábor Domokos of Budapest University of Technology and Economics, who in 2006 helped prove the existence of the gömböc, a gemstone-like shape that has only one stable balance point. Set a gömböc down on a table, and it will always come to rest in the exact same position, unlike, say, a cylinder, which can rest on its end or its side. In subsequent work, Domokos and his colleagues found that entities such as pebbles washing downriver and sand grains blowing in the wind tend to erode toward gömböcish shapes without ever achieving that ideal. “The gömböc is part of nature, but only as a dream,” Domokos says.
He and his team then turned to the other side of this process—how rocks themselves are born. They started their study “fragmenting” an abstract cube in a computer simulation by slicing it with 50 two-dimensional planes inserted at random angles. The planes cut the cube into 600,000 fragments, which were, on average, cubic themselves—meaning that, on average, the fragments had six sides that were quadrangles, although any individual fragment need not be a cube. The result led the researchers to suspect that cubes might be a common feature of fragmentation.
The researchers tried to confirm this hunch using real-world measurements. They headed to an outcrop of the mineral dolomite on the mountain Hármashatárhegy in Budapest, Hungary, and counted the number of vertices in cracks in the stone face. Most of these cracks formed squarish shapes, which is one of the faces of a cube, regardless of if they had been weathered naturally or had been created by humans dynamiting the mountain.
Finally, the team created more-powerful supercomputer simulations modeling the breakup of 3D materials under idealized conditions—like a rock being pulled equally in all directions. Such cases formed polyhedral pieces that were, in an average sense, cubes, the researchers report this week in Proceedings of the National Academy of Sciences.
Skeptics might point out that many things in the natural world don’t fragment into cubes. Minerals such as mica, for instance, come off in flakes, while basaltic formations including the Giant’s Causeway in Northern Ireland break into hexagonal columns.
That’s because real materials are not like the idealized forms found in the team’s simulations, says Douglas Jerolmack, a geophysicist at the University of Pennsylvania and co-author of the paper. They usually contain interior structures or properties that favor noncubic breakages. For example, mica flakes because it is weaker in one direction than in the perpendicular directions. “But in a statistical averaged sense, rocks are born as something that’s a vague shadow of a cube,” Jerolmack says. The findings, he adds, could help hydrologists predict fluid flow through cracks in the ground for oil extraction, or help geologists calculate the sizes of hazardous rocks breaking off cliff faces.
Some find the study a bit difficult to parse, however. “You need to have this abstract theoretical view of earth surface processes to really dig into what this can mean,” says Anne Voigtländer, a geologist at the GFZ German Research Centre for Geosciences. “It’s sometimes hard for geologists to understand the value of it, or to see where it applies.”
Jerolmack agrees that, in some sense, the result is more philosophical than scientific. He notes that his team took inspiration from the Greek philosopher Plato, who related each of the four classical elements—earth, air, fire, and water—to a regular polyhedron, coincidentally linking earth with the cube. But Plato is more remembered for his allegory of the cave, in which he speculated about certain idealized and eternal forms, of which only garbled versions existed in the real world. “With the naked eye you see distorted images—the fragments,” Domokos says. “But in order to see the ideal, you have to use your mind.”