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Plato’s cube is nature’s preferred choice for fragmentation.

Plato’s cube is nature’s preferred choice for fragmentation.

Plato is widely recognized as the first to create the concept of an atom, or the idea that matter is made up of indivisible components at the smallest scale. The Greek philosopher proposed the primordial forms of the universe’s structure, arguing that the universe was made up of five different forms of matter: earth, air, fire, water, and ether. Each was described by a certain geometry, a platonic form, Earth having assigned the cube. That premise now contains a fundamental reality, according to researchers. By studying their shapes and fragmentation patterns, they discovered that the average of all the shapes and fragmentation patterns of various rocks is a cube.

A group of researchers from the University of Pennsylvania, Budapest University of Technology and Economics, and the University of Debrecen combine arithmetic, geology, and physics to prove it:

We apply the theory of convex mosaics to show that the average geometry of natural two-dimensional (2D) fragments, from mud cracks to Earth’s tectonic plates, has two attractors: “Platonic” quadrangles and “Voronoi” hexagons. In three dimensions (3D), the Platonic attractor is dominant: remarkably, the average shape of natural rock fragments is cuboid.

Fragmentation across planets and scales. AD show planetary surfaces and rings. (A) Saturn’s rings composed of ice (Inset). Image credit: NASA/JPL-Caltech/Space Science Institute. (B) Jupiter’s moon Europa showing cracked planetary shell. Image credit: NASA/JPL-Caltech/SETI Institute. (C) Polygonal cracks on Pluto. Image credit: NASA/JHUAPL/SwRI. (D) Surface of the asteroid Bennu. Image credit: NASA/Goddard/University of Arizona. EH show example processes forming fragments on Earth. (E) Iceberg calving. Image credit: Australian Antarctic Division/Ian Phillips. (F) Rock falls. Reproduced from ref. 58, which is licensed under CC BY 4.0. (G) Volcanic eruptions that produce pyroclastic flows, forming breccia deposits (Inset). Image credit: US Geological Survey/Peter Lipman. Inset image credit: Siim Sepp (http://www.sandatlas.org/). (H) Mine blasting. Image credit: Sarolta Bodor (photographer).
Examples of fragments and fracture lines. (A) Natural fragments approximated by convex polyhedra. (B1) Granite wall showing global cracks. (B2) Approximation of fragmentation pattern by regular primitive mosaic (black lines) and its irregular version with secondary cracks (red lines).
Mosaics in 2D. (Left) Symbolic plane with geometrically admissible domain shaded gray. Patterns 1 to 7 marked with black circles are deterministic periodic patterns. Patterns 8 to 12 are geometric simulations of random mosaics: 8, regular primitive; 9 and 10, advanced (irregular) primitive; 11, Poisson–Voronoi; and 12, Poisson–Delaunay. (Right) Red squares (13 to 21) correspond to analyzed images of natural 2D mosaics.

The group’s discovery began with geometric models established by Budapest University of Technology and Economics mathematician Gábor Domokos, whose work indicated that natural rocks would fracture into cubic forms.

“If you take a three-dimensional polyhedral shape, slice it randomly into two fragments and then slice these fragments again and again, you get a vast number of different polyhedral shapes. But in an average sense, the resulting shape of the fragments is a cube,” said Domokos.

They, thus, solved the issue of what forms are formed when rocks shatter into fragments. Furthermore, the basic mathematical hypothesis connects geological processes not only on Earth but also across the solar system.

Part of this knowledge is that the components that emerge from a previously solid item must fit together without gaps, much like a dropped dish on the point of shattering. Cubes, it turns out, are the only so-called platonic shapes (polyhedra with equal-length sides) that fit together without gaps.

To see if their mathematical models held true in nature, the researchers measured hundreds of pebbles they gathered and thousands more from previously obtained information. The scientists discovered an excellent match to the cubic average whether the boulders had naturally weathered from a huge outcropping or had been dynamited away by humans. There are models that break this rule, such as The Giant’s Causeway in Northern Ireland, with its soaring vertical columns, which is one example, formed by the unusual process of cooling basalt.

The researchers also looked into fragmentation in two dimensions, or on thin surfaces that behave as two-dimensional forms but have a depth that is much less than the breadth and length. The fracture patterns are different there, but the basic idea of dividing polygons and arriving at predictable average forms remains.

“When you pick up a rock in nature, it’s not a perfect cube, but each one is a kind of statistical shadow of a cube,” adds Jerolmack. “It calls to mind Plato’s allegory of the cave. He posited an idealized form that was essential for understanding the universe, but all we see are distorted shadows of that perfect form.”

Serial reproduction reveals the geometry of visuospatial representations, Thomas A. Langlois, Nori Jacoby, Jordan W. Suchow, and Thomas L. Griffiths

Published: March 2021
DOI: https://doi.org/10.1073/pnas.2012938118

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