Plato's cube and the natural geometry of fragmentation

TL;DR

This paper explores the geometry of natural fragments, revealing that 2D and 3D natural shards tend to cluster around specific geometric shapes, with cuboids being dominant in 3D, linking natural fragmentation patterns to Plato's geometric ideals.

Contribution

The study applies convex mosaic theory to demonstrate that natural fragmentation patterns are attracted to specific geometric forms, especially cuboids in 3D, and introduces a universal pattern generator for these shapes.

Findings

Natural 2D fragments cluster around quadrangles and hexagons.

3D fragments predominantly resemble cuboids.

Binary breakup drives mosaics toward Platonic shapes.

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 2D fragments, from mud cracks to Earth's tectonic plates, has two attractors: "Platonic" quadrangles and "Voronoi" hexagons. In 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.