Geometry Reveals How the World Is Made of Cubes
November 29, 2020
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At first, “everything seemed to work,” Jerolmack said. Domokos’ mathematics had predicted that rock shards should average out to cubes. An increasing number of actual rock shards seemed happy to comply. But Jerolmack soon realized that proving the theory would require confronting rule-breaking cases, too.

After all, the same geometry offered a vocabulary to describe the many other mosaic patterns that could exist in both two and three dimensions. 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. More promisingly, perhaps they would arise only in distinct circumstances and carry separate lessons for geologists. “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. When the team investigated surface systems that are essentially two-dimensional—cracking permafrost in Alaska, a dolomite outcrop, and the exposed cracks of a granite block—they found polygons averaging four sides and four vertices, just like the sliced-up sheet of paper. Each of these geological cases seemed to appear where rocks had simply fractured. Here Domokos’ predictions held up.

Illustration: Samuel Velasco/Quanta Magazine; Based on graphics from; spot images: Lindy Buckley; Matthew L. Druckenmiller; Hannes Grobe; Courtesy of János Török

Another type of fractured slab, meanwhile, proved to be what Jerolmack had hoped for: an exception with its own distinct story to tell. Mud flats that dry, crack, get wet, heal and then crack again have cells averaging six sides and six vertices, following the roughly hexagonal Voronoi pattern. 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 type of stress—when forces pulled outward on a rock instead of pushing it in. The geometry revealed the geology. And Jerolmack and Domokos thought this Voronoi pattern, even if it was relatively rare, might also occur on scales far larger than they had previously considered.

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

Counting the Crust

Midway through the project, the team met in Budapest and spent three whirlwind days sprinting to incorporate more natural examples. Soon Jerolmack pulled up a new pattern on his computer: the mosaic of how Earth’s tectonic plates fit together. Plates are confined to the lithosphere, a nearly two-dimensional skin on the surface of the planet. The pattern looked familiar, and Jerolmack called the others over. “We were like, oh wow,” he said.

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