Perhaps the Cubists were right. Researchers have found that when everything from icebergs to rocks breaks, its pieces tend to resemble cubes. The finding suggests a universal rule of fragmentation at scales ranging from the microscopic to the planetary.
"It is a very beautiful combination of pure mathematics, materials science and geology," says Sujit Datta, a chemical and biological engineer from Princeton University who was not involved in the work.
The finding builds on previous work by mathematician Gábor Domokos of the Budapest University of Technology and Economics, who in 2006 helped demonstrate the existence of the gömböc, a gemstone-like shape that only has a stable equilibrium point. Place a gömböc on a table, and it will always rest in exactly the same position, unlike, for example, a cylinder, which can rest on its end or side. In later work, Domokos and colleagues discovered that entities such as pebbles that wash downstream and grains of sand that blow in the wind tend to erode into gömböcish forms without achieving that ideal. "The gömböc is part of nature, but only as a dream," says Domokos.
He and his team turned to the other side of this process: how rocks are born. They began their study by "fragmenting" an abstract cube in a computer simulation by cutting it with 50 two-dimensional planes inserted at random angles. The planes cut the cube into 600,000 shards, which were, on average, cubic, meaning that, on average, the shards had six sides that were quadrangles, although any single shard need not be a cube. The result led researchers to suspect that cubes might be a common feature of fragmentation.
The researchers tried to confirm this hunch using real-world measurements. They made their way to an outcrop of the mineral dolomite on the Hármashatárhegy Mountain in Budapest, Hungary, and counted the number of vertexes in cracks in the stone face. Most of these cracks formed square shapes, which is one of the faces of a cube, regardless of whether they had been naturally weathered or created by humans who dynamited the mountain.
Finally, the team created more powerful supercomputer simulations that model the breaking of 3D materials under ideal conditions, like a rock that is pulled equally in all directions. Such cases formed polyhedral pieces that, in an average sense, were cubes, the researchers reported this week in procedures of the National Academy of Sciences.
Skeptics might point out that many things in the natural world don't break up into cubes. Minerals like mica, for example, flake off, while basaltic formations, including the Giant's Causeway in Northern Ireland, break into hexagonal columns.
This is because the actual materials are not like the idealized shapes found in the team's simulations, says Douglas Jerolmack, a geophysicist at the University of Pennsylvania and co-author of the article. They usually contain internal structures or properties that favor non-cubic breaks. For example, mica flakes because it is weaker in one direction than it is in perpendicular directions. "But in an average statistical sense, rocks are born as something that is a vague shadow of a cube," says Jerolmack. The findings, he adds, could help hydrologists predict fluid flow through cracks in the ground for oil extraction, or help geologists calculate the size of dangerous rocks that break down cliff walls.
However, some find the study a little difficult to analyze. "It is necessary to have this abstract theoretical vision of the processes of the earth's surface to delve into what this can mean," says Anne Voigtländer, geologist at GFZ German Geoscience Research Center. "Sometimes it is difficult for geologists to understand its value or see where it applies."
Jerolmack agrees that, in a sense, the result is more philosophical than scientific. He notes that his team was inspired by the Greek philosopher Plato, who linked each of the four classical elements: earth, air, fire, and water, with a regular polyhedron, which coincidentally joins earth to cube. But Plato is best remembered for his allegory of the cave, in which he speculated on certain idealized and eternal forms, of which only confusing versions existed in the real world. "At a glance you see distorted images, the fragments," says Domokos. "But to see the ideal, you have to use your mind."
