Geometry Reveals How the World Is Assembled From Cubes

But from mathematics alone, Domokos argued, any rocks that broke randomly would crack into shapes that have, on average, six faces and eight vertices. “I was like, oh, OK, now we’re getting a little bit metaphysical,” Jerolmack said.Get Quanta Magazine delivered to your inboxGábor Domokos (left) and Douglas Jerolmack had previously collaborated on a project that spanned mathematics and geophysics.Courtesy of Gábor Domokos; Eric Sucar, University of PennsylvaniaBut they kept finding cuboid averages in nature, plus a few non-cubes that could be explained with the same theories. When their paper was published earlier this year, it came titled like a particularly esoteric Harry Potter novel: “Plato’s Cube and the Natural Geometry of Fragmentation.”Several geophysicists contacted by Quanta say the same mathematical framework might also help with problems like understanding erosion from cracked cliff faces, or preventing hazardous rock slides. That confirmed that cubes were indeed the 3D answer.Now Domokos had the average shapes produced by splitting a flat surface or a three-dimensional block. Geometry also predicted that if you fragmented a flat surface randomly, it would break into rough rectangles, and if you did the same in three dimensions, it would produce rough cubes.But for any of this to matter to anyone other than a few mathematicians, Domokos had to prove that these same rules manifest themselves in the real world.By the time Domokos swung through Philadelphia in 2016, he had already made some progress on the real-world problem. Working with János Török, a specialist in computer simulations, and Ferenc Kun, an expert on fragmentation physics, Domokos found that cuboid averages showed up in rock types like gypsum and limestone as well.With the math and the early physical evidence, Domokos pitched his idea to a stunned Jerolmack. Domokos’ mathematics had predicted that rock shards should average out to cubes. Off the top of his head, Jerolmack could picture a few real-world fractured rocks that didn’t look like rectangles or cubes at all but could still be classified into this larger space.Perhaps these examples would sink the cube-world theory entirely. “I said I know that it doesn’t work everywhere, and I need to know why,” Jerolmack said.Over the next few years, working on both sides of the Atlantic, Jerolmack and the rest of the team started plotting where real examples of broken rocks fell within Domokos’ framework. Domokos crunched the numbers for the surface of a sphere and found that on a globe, Voronoi mosaic cells should average 5.77 vertices.This insight might help researchers answer a major open question in geophysics: How did Earth’s tectonic plates form? But Török’s simulations produced Giant’s Causeway-like mosaics as three-dimensional structures that had simply grown up from a two-dimensional Voronoi base, itself produced when volcanic rock cooled.Zooming out, the team argues, you could classify most real fractured-rock mosaics using just Platonic rectangles, 2D Voronoi patterns, and then — overwhelmingly — Platonic cubes in three dimensions. “How clever is that?”Specifically, perhaps you could take a real fractured field site, count up things like vertices and faces, and then be able to infer something about the geological circumstances responsible.“We have places where we have data we can think about in this way,” said Roman DiBiase, a geomorphologist at Pennsylvania State University.