Geometry reveals how the world is made of cubes

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At first, “everything seemed to work,” Jerolmack said. Domokos’ mathematics predicted that the rock fragments should turn into cubes. A growing number of rock shards seem happy to comply. But Jerolmack soon realized that proving the theory would also require confronting cases of rule violations.

After all, the same geometry provided a vocabulary to describe the many other mosaic patterns that could exist in two and three dimensions. From the top of his head, Jerolmack could imagine a few fractured real-world rocks that didn’t look like rectangles or cubes at all, but could still be categorized into that larger space.

Perhaps these examples would completely sink the theory of the cubic world. More promising, perhaps they would only arise under separate circumstances and lead to separate lessons for geologists. “I said I knew it wasn’t working everywhere, and I needed to know why,” Jerolmack said.

Over the next several years, working on both sides of the Atlantic, Jerolmack and the rest of the team began to trace where real examples of broken rock were found in the Domokos setting. When the team studied mostly two-dimensional surface systems – cracking permafrost in Alaska, outcrop of dolomite, and exposed cracks in a block of granite – they found polygons with an average of four sides and four vertices, as did the sliced ​​sheet of paper. . Each of these geologic cases seemed to appear where the rocks had simply fractured. Here, Domokos’ predictions held firm.

Illustration: Samuel Velasco / Quanta Magazine; Based on graphics from doi.org/10.1073/pnas.2001037117; point images: Lindy Buckley; Matthew L. Druckenmiller; Hannes Grobe; Courtesy of John Turkish

Another type of fractured slab, meanwhile, turned out to be what Jerolmack had hoped for: an exception with its own story to tell. Mudflats that dry, crack, get wet, heal and then crack again have cells averaging six sides and six vertices, following the roughly hexagonal Voronoi model. Rock made from cooling lava, which solidifies downward from the surface, can take on a similar appearance.

Tellingly, these systems tended to form under a different kind of stress – when forces were pulling outward on a rock instead of pushing it. Geometry revealed geology. And Jerolmack and Domokos believed that this pattern of Voronoi, while relatively rare, could also occur at scales much larger than they had previously considered.

A Voronoi diagram separates a plane into individual regions, or cells, so that each cell consists of all the points closest to a “seed” starting point.Illustration: Fred Scharmen

Count the crust

Halfway through the project, the team met in Budapest and spent three lightning days sprinting to incorporate more natural examples. Soon Jerolmack created a new model on his computer: the mosaic of how Earth’s tectonic plates fit together. The plates are confined to the lithosphere, an almost two-dimensional skin on the surface of the planet. The pattern sounded familiar to him and Jerolmack called the others. “We were like, oh wow,” he says.

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