The paper is here:
Gábor Domokos, Douglas J. Jerolmack. Plato’s cube and the natural geometry of fragmentation. PNAS (2020)
https://www.pnas.org/doi/10.1073/pnas.2001037117
Abstract:
Plato envisioned Earth’s building blocks as cubes, a shape rarely found in nature. The solar system is littered, however, with distorted polyhedra—shards of rock and ice produced by ubiquitous fragmentation. 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. When viewed through the lens of convex mosaics, natural fragments are indeed geometric shadows of Plato’s forms. Simulations show that generic binary breakup drives all mosaics toward the Platonic attractor, explaining the ubiquity of cuboid averages. Deviations from binary fracture produce more exotic patterns that are genetically linked to the formative stress field. We compute the universal pattern generator establishing this link, for 2D and 3D fragmentation.
Voronoi diagrams I see have few if any hexagons (use your favorite mathematical reference or image search). Is that idea that if the points are distributed equidistant in 'alternating' ranks [0], then the diagram is hexagons?
Also, what is "binary breakup" and "binary fracture"?
[0] Alternating ranks: I mean something like the following (is there a better name?):
The dots need to be the vertices of equilateral triangles for the Voronoi diagram to be hexagons, the above is a rectangular grid rotated 45 degrees.
You can overlay a regular hexagonal tessellation over a regular triangular tessellation to see this.
In context, binary breakup and binary fracture apppear to mean a splitting ofa whole into two parts along a given line or plane