How do honeybees handle geometric frustration?

Built by thousands of bees, the wax-made honeycomb is essential to the survival of a colony. And because the comb is so expensive—the bees must consume nearly four kilograms of honey to secrete less than half a kilogram of wax—they must minimize its ratio of wax to storage space. A honeycomb’s hexagonal tessellation naturally minimizes the length of the boundary per unit area of storage space. But as the bees build their nests in preexisting tree cavities, they must deal with geometric constraints that force them to combine cells of different shapes and sizes, producing nonregular hexagons and topological defects in the comb. To date, the mechanisms that control the construction of geometrically constrained honeycombs remain unclear.

Under the direction of the biophysicist Orit Peleg and the aerospace engineer Francisco López Jiménez , Golnar Gharooni Fard—a University of Colorado Boulder doctoral student—has now studied how the bees adapt to that wild environment. To mimic the geometric constraints, Gharooni Fard made experimental frames using three-dimensional printing that precisely controlled the sources of geometric frustration—the tilt angle (A) and shifts (L and h) in the horizontal and vertical axes—imposed on the hexagonal lattice, as shown in the first figure below. She introduced the constraints only to discrete segments of the frame, each one separated by gaps. That frame geometry prevented the bees from simply extending the hexagonal foundations to fill in the gaps.

After a series of experiments on 10 hives, the researchers quantified the bees’ strategies to overcome the mismatches in the lattice planes. After taking pictures of the fully constructed frames, Gharooni Fard and her colleagues used computer-vision techniques to identify individual comb cells. Armed with those images, they reconstructed the structure of the comb, which reveals the nonregularity of the cell shapes built within the gap, as shown in the figure below. Inspired by the similarities between the grain boundaries in the reconstructed combs and those in graphene, the researchers developed a crystallography-based algorithm to position the cell centers at locations in the lattice that minimize a variant of the Lennard-Jones potential.

The researchers found quantitative agreement between the results of their experiments and the model. For example, topological defects (cells with more or less than six neighbors) appear as a consequence of several geometric frustrations, and the researchers plotted a positive correlation between the tilt angle of two hexagonal lattices and the density of the defects. Unsurprisingly, when there was no tilt between lattices, defects were rare, and the bees consistently built regular hexagons to merge them. The agreement between experiments and simulations also revealed the effectiveness of using the tools of crystallography to understand honeycomb global structures as emerging from the local interaction between cells and their surroundings. (G. Gharooni Fard et al., Proc. Natl. Acad. Sci. USA 119, e2205043119, 2022 .)