Geometry Reveals How the World Is Made of Cubes

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 doi.org/10.1073/pnas.2001037117; 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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Biological pattern-forming systems characterized better through geometry than simulations

bird flock
Like the collective motions of bird flocks, the patterns result from the concerted interactions of many individual particles without a central coordinator. Credit: CC0 Public Domain

Ludwig-Maximilians-Universitaet (LMU) in Munich physicists have introduced a new method that allows biological pattern-forming systems to be systematically characterized with the aid of mathematical analysis. The trick lies in the use of geometry to characterize the dynamics.


Many vital processes that take place in biological cells depend on the formation of self-organizing molecular patterns. For example, defined spatial distributions of specific proteins regulate cell division, cell migration and cell growth. These patterns result from the concerted interactions of many individual macromolecules. Like the collective motions of bird flocks, these processes do not need a central coordinator. Hitherto, mathematical modeling of protein pattern formation in cells has been carried out largely by means of elaborate computer-based simulations. Now, LMU physicists led by Professor Erwin Frey report the development of a new method which provides for the systematic mathematical analysis of pattern formation processes, and uncovers the their underlying physical principles. The new approach is described and validated in a paper that appears in the journal Physical Review X.

The study focuses on what are called ‘mass-conserving’ systems, in which the interactions affect the states of the particles involved, but do not alter the total number of particles present in the system. This condition is fulfilled in systems in which proteins can switch between different conformational states that allow them to bind to a cell membrane or to form different multicomponent complexes, for example. Owing to the complexity of the nonlinear dynamics in these systems, pattern formation has so far been studied with the aid of time-consuming numerical simulations. “Now we can understand the salient features of pattern formation independently of simulations using simple calculations and geometrical constructions,” explains Fridtjof Brauns, lead author of the new paper. “The theory that we present in this report essentially provides a bridge between the mathematical models and the collective behavior of the system’s components.”

The key insight that led to the theory was the recognition that alterations in the local number density of particles will also shift the positions of local chemical equilibria. These shifts in turn generate concentration gradients that drive the diffusive motions of the particles. The authors capture this dynamic interplay with the aid of geometrical structures that characterize the global dynamics in a multidimensional ‘phase space.” The collective properties of systems can be directly derived from the topological relationships between these geometric constructs, because these objects have concrete physical meanings—as representations of the trajectories of shifting chemical equilibria, for instance.

“This is the reason why our geometrical description allows us to understand why the patterns we observe in cells arise. In other words, they reveal the physical mechanisms that determine the interplay between the molecular species involved,” says Frey. “Furthermore, the fundamental elements of our theory can be generalized to deal with a wide range of systems, which in turn paves the way to

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