Crystallography of honeycomb formation under geometric frustration

Honeybees are known to collectively construct highly regular hexagonal structures of the honeycomb in a distributive manner. As they naturally build comb on tree branches or cavities, they are often faced with various geometric constraints, resulting in non-regular hexagons of different sizes with topological defects. In this work, we study how bees collectively adapt to their environment to regulate the honeycomb. We 3D-print experimental frames with imprinted foundations and gaps to study the features of the comb built under various geometric frustrations such as different sizes, orientations, and shapes. We then use X-ray tomography and computer vision techniques to compare comb built under different given cell sizes and study how bees transition between them, as well as the comb built to connect the vertical, horizontal and curved-shaped gaps with various orientations. We find that the structure of the honeycomb under various frustrations show clear evidence of recurring patterns and can be replicated through a computational model of crystallographic lattice formation, in which the minimized potential is a variation of the Lennard-Jones potential that only considers first-neighbor interactions according to a Delaunay triangulation. Our model provides a description for the emergence of the global patterns in the honeycomb structure using local rules and information. It also extends the application of the Lennard-Jones model beyond the physical domain to encompass biological systems, thereby demonstrating its universality.