Understanding the natural geometry of crystals has long fascinated scientists, especially when studying the behavior of materials at different temperatures and pressures. A major problem in the field is that the shapes formed when energy is minimized always bend outwards – what scientists call convex, meaning no part of the surface cave is facing inwards. This problem becomes more interesting when you look at shapes in three dimensions, and the situation becomes more complicated.
Dr. Emanuel Indrei from Kennesaw State University and Dr. Aram Karakhanyan from the University of Edinburgh responded to this challenge by studying well-known mathematical problems related to crystal formation. Their findings, published in the journal Mathematics, explore whether crystals formed by energy balance (i.e. finding the most effective shape of a given mass) will have convex shapes when following certain general rules.
Their study centers on a detailed mathematical demonstration (step-by-step proof) showing that shapes using the least energy do protrude in three dimensions under specific conditions. Dr. Indrei and Dr. Karakhanyan studied the situation where the troops involved were pushed evenly outward and the total energy remained within the set limit. They found that all the best shapes were convex, or at least formed the shape of smaller materials. They came to this conclusion using known results on the stability obtained by Dr. Indrei, recently published in the journal Calculation of Changes and Partial Differential Equations, which implies the resistance of shapes to changes and mathematical tools to deal with how energy changes are related to shapes.
Their results are important because they help clarify which types of force and energy patterns guarantee convex crystal shape. If the pulling force is the same in all directions and the potential energy increases with distance from the center (called radial symmetry), their findings suggest that the convex shape will always result. As the researchers explain: “Our theorem implies a convexity of a large number of potentials; our argument also includes non-convex potentials.”
A particularly interesting part of their work involves a way to test convexity by looking at the curves or curves of shapes. The researchers found that under the assumption of regularity of energy, if the crystal becomes flat at some point, it must be flat anywhere nearby, meaning that the shape cannot bend in some parts and elsewhere. This provides a useful tool for predicting when and when a crystal loses its outward curve and a clearer understanding of shape consistency.
Summarizing their research, Dr. Indrei and Dr. Karakhanyan point out the importance of a relatively large material with a uniform outward curvature and resistance to small changes. When these factors are present, the resulting shape is not only convex, but also does not easily lose its form. Their findings show that the shape of the crystals follows basic rules that are more ordered than they appear. “Our new idea for the three-dimensional Almgren problem is to utilize the stability theorem… and the first change in the free energy PDE with new maximum principle methods,” the researchers said.
Here, PDE refers to partial differential equations, an equation commonly used to describe physical quantities such as the change of energy or heat in space and time. The biggest principle is mathematical rules, which help predict the behavior of a function based on its boundaries.
This study marks an important step in understanding how crystals form shapes when energy is minimized. It continues to use mathematics to explain the long tradition of the physical world, a tradition that traces back to pioneers such as Gibbs and Curie. This new study could help guide future theoretical research and practical efforts to model and design materials with specific shapes and properties.
Journal Reference
Indrei, E. , Karakhanyan, A. “Three-dimensional shapes of crystals.” Mathematics, 2025; 13 (614). doi:
Indrei, E. “On the equilibrium shape of the crystal.” Calculation. var. Partial. different. wait. 2024, 63, 97. Doi:
About the Author
Emanuel Indrei He is an assistant professor of mathematics at Kennesaw State University. He received his PhD dissertation and was selected as the Frank Gerth III Dissertation Award. He was a 2012 NSF EAPSI Fellow, a postdoctoral fellow at the Australian National University, a Huneke postdoctoral scholar at the Institute of Mathematical Sciences at CA Berkeley, and a Pire postdoctoral fellow at Carnegie Mellon University. The main topics in his research are nonlinear PDE, free boundary problems, and geometric and functional inequality. Over the past few years, he has proved the non-transfer point conjecture, solved the Almgren problem in two dimensions (also on one dimension), and made progress for the first eigenvalue of Laplacian on the Polya-Szego conjecture.
Aram Karakhanyan He is an associate professor of mathematics at the University of Edinburgh, where he explores nonlinear partial differential equations and geometric analysis. His research covers capillary and K-Surfaces, Monge-ampère equations, reflector surfaces, phase transitions and free boundary problems. It is worth noting that he solved the near-field reflector problem (listed in Yau’s 100 public challenges) and gained an in-depth understanding of obstacle problems and nonlinear elasticity. His contribution extends to homogenization theory, studying the regularity of the minimizer under complex constraints. Karakhanyan has received years of multi-year grants, including the EPSRC scholarship and the Polonez Award, where he leads interdisciplinary teams to address analytical challenges. He regularly works internationally and directs graduate students at the forefront of mathematical analysis.
