A new mathematical equation has uncovered a surprising commonality: the dramatic shatter of a dropped vase, the delicate crumble of a crushed sugar cube, and the swift pop of an exploding bubble all break apart in strikingly similar ways.

A French scientist has made a significant discovery, uncovering a mathematical equation that precisely models the size distribution of fragments created when something shatters. This groundbreaking formula, detailed in a study published November 26 in the journal Physical Review Letters, demonstrates remarkable universality, applying to a diverse range of materials including solids, liquids, and even gas bubbles.

While cracks often spread through objects in seemingly unpredictable ways, scientific research has uncovered a surprising order: the size distribution of the resulting fragments exhibits remarkable consistency, regardless of the material’s composition. This means a predictable statistical ratio of larger to smaller pieces can consistently be observed after an object breaks. For scientists, this intriguing uniformity strongly suggests an underlying, universal principle at work in the very process of fragmentation.

A groundbreaking study from Emmanuel Villermaux, a physicist at France’s Aix-Marseille University, has shifted the focus in fragmentation research. Instead of analyzing the mechanics of how objects break, Villermaux delved into the characteristics of the fragments themselves.

His new research introduces the principle of “maximal randomness,” theorizing that objects undergoing fragmentation tend towards the most disordered and chaotic possible outcome. In essence, Villermaux argues that the most probable pattern of breakage is the one that maximizes entropy, or the overall level of disorder within the system.

Even the seemingly random process of an object shattering operates within defined physical parameters. To explain these inherent limitations, physicist Villermaux introduced a pivotal conservation law, which he and his colleagues first uncovered in 2015. This fundamental principle establishes precise physical constraints governing the spatial distribution and density of fragments that emerge during an object’s disintegration.

Leveraging the integration of two core principles, Villermaux successfully derived a mathematical equation that precisely models the pattern of fragment sizes resulting from any shattered object. To validate this breakthrough, he rigorously compared the equation’s theoretical predictions against years of accumulated fragmentation data collected from a remarkably diverse array of materials.

This extensive dataset included common items like glass and spaghetti, as well as complex phenomena such as liquid droplets and gas bubbles. It further extended to pressing environmental concerns like plastic fragments found in the ocean, and even offered archaeological insights from flakes produced by early stone tools. Remarkably, in every instance, the empirical data from these varied sources consistently and accurately matched the equation’s projected size distribution.

To empirically test his equation, Villermaux devised a series of experiments involving sugar cubes and heavy objects. He meticulously observed how the brittle structures fragmented upon impact, analyzing the resulting patterns.

Speaking to *New Scientist*, Villermaux revealed the origins of this unique methodology: it began as “a summer project with my daughters.” He further explained that these initial observations were made “a long time ago when my children were still young,” but he revisited the data recently, finding them compelling for their ability to illustrate his current theoretical work.

Despite its significance, the newly identified law is not universally applicable. Researchers emphasize two key exclusions: it does not govern situations lacking inherent randomness, such as the predictable division of a smooth liquid stream into uniformly sized droplets. Nor does it account for conditions where the resulting fragments interact with each other, a characteristic behavior observed in materials like certain plastics.

The intricate process of how materials break apart, known as fragmentation, carries significant real-world benefits, according to Ferenc Kun, a physicist at Hungary’s University of Debrecen. In an interview with New Scientist, Kun explained that deeper insights into this phenomenon could prove crucial for optimizing energy usage in industrial ore shattering and enhancing preparedness for hazardous rockfalls.

Speaking to New Scientist, Villermaux indicated that subsequent research would focus on precisely determining the absolute minimum dimensions a fragment could possibly attain.

In an accompanying viewpoint article, Kun also theorized that the diverse shapes of fragments might exhibit a comparable underlying relationship.