Cubes in Nature: Geometry, Fragmentation, and the Architecture of Rocks

On a mild autumn day in 2016, Gábor Domokos’s arrival at Douglas Jerolmack’s laboratory in Philadelphia sparked a collaboration that bridged pure mathematics and planetary geology. Domokos, a professor of mathematics at the University of Szeged, had long pursued a question that, while simple to state, eluded a concise solution: when a solid is fractured randomly, what shapes do the resulting pieces assume on average? Drawing inspiration from classical problems in dissection, he proposed a hypothesis that the average fragment is essentially a cube—each piece possessing six facets and eight vertices. The first hint came from a casual sidewalk conversation during which Domokos pointed to a pile of gravel and proposed that their average number of sides might hover around six. He extended this idea to three‑dimensional bodies, arguing that any random brittle fracture will produce a mosaic of convex cells that can be represented by a distribution over the Platonic solids. Because a cube is the only convex polyhedron that can tile space without overlap or gaps, it emerged naturally as the statistical attractor of random fragmentation. To move beyond hypothesis, Domokos needed a general mathematical theory capable of predicting not only averages but also the full space of possible mosaics. He began by formalizing the idea of a *mosaic*: a partition of a volume into convex cells that fit together perfectly. He then introduced two key parameters for any mosaic: (1) the average number of vertices per cell and (2) the average number of cells meeting at a vertex. In two dimensions, these parameters reduce to the familiar fact that random cuts of a sheet of paper produce polygons with an average of four sides. Extending to three dimensions, Domokos and collaborator Zsolt Lángi conjectured a curve that bounds all admissible pairs (average–vertex count, average–edge‑meeting count) for convex, face‑regular mosaics. The breakthrough came when Domokos uncovered a previously inaccessible theorem in the works of German geometrists Wolfgang Weil and Rolf Schneider—an elegant proof that any sequence of random hyperplane cuts in n‑dimensional space yields an average cell shape equivalent to an n‑cube. The theorem, which Domokos had translated into accessible language, provided the rigorous foundation for his cube hypothesis. Empirical validation required a shift from the abstract to the tangible. Domokos brought the mathematics to the field by collecting thousands of rock shards from a dolomite cliff on Hármashatárhegy in Budapest. Using digital imaging, he counted facets and vertices on each grain, finding averages that matched the predicted six faces and eight vertices. Subsequent investigations by Jerolmack’s team—spanning gypsum, limestone, permafrost fractures, granite blocks, and mud flats—reinforced the cube law in a wide variety of settings. Not all natural mosaics adhere to the cube rule. When the dominant stress is extensional rather than compressional, the fragmentation pattern shifts toward a two‑dimensional Voronoi tessellation characterized by hexagonal cells. Jerolmack’s observations of drying mudflats, cooling lava, and the basalt columns of North Ireland’s Giant’s Causeway demonstrate that this hexagonal mode is a distinct geometric attractor, one that arises from outward‑pushing forces. By mapping measured vertex and facet statistics onto Domokos–Lángi’s theoretical curve, each natural pattern can be classified according to its underlying stress regime. The implications extend beyond rock fragmentation. One striking example is the arrangement of Earth’s tectonic plates. Treating the lithosphere as an almost two‑dimensional manifold, Jerolmack and colleagues plotted the vertices and edges of 202 tectonic polygons. The resulting average of 5.77 vertices per plate closely matches the theoretical average for a sphere composed of hexagonal Voronoi cells, suggesting that the global plate pattern may reflect the same geometrical constraints that govern smaller‑scale geological mosaics. What makes the cube law remarkable is its universality: regardless of the mechanism—impact, thermal contraction, mechanical failure—or scale, the random breaking of a brittle body will produce fragments whose statistical geometry converges on the cube. Conversely, systematic deviations from this average—such as the prevalence of hexagons—provide a diagnostic tool for identifying underlying geological processes and stress histories. As Dr. Mikaël Attal of Edinburgh remarked, "The mathematics is telling us that when we begin to fracture rocks, however we do it, there is only a certain set of possibilities." Future work will focus on refining the predictive models for non‑convex or anisotropic materials and exploring the limits of the cube law in non‑Euclidean contexts. Nevertheless, the partnership between Domokos and Jerolmack has opened a new chapter in understanding how the geometry of fragmentation can illuminate the history of the planetary surface, from the micro‑shards in a laboratory beaker to the vast plates that shape continents.