Crystallography of honeycomb formation under geometric frustration

Abstract

As honeybees build their nests in preexisting tree cavities, they must deal with the presence of geometric constraints, resulting in nonregular hexagons and topological defects in the comb. In this work, we study how bees adapt to their environment in order to regulate the comb structure. Specifically, we identify the irregularities in honeycomb structure in the presence of various geometric frustrations. We 3D-print experimental frames with a variety of constraints imposed on the imprinted foundations. The combs constructed by the bees show clear evidence of recurring patterns in response to specific geometric frustrations on these starter frames. Furthermore, using an experimental-modeling framework, we demonstrate that these patterns can be successfully modeled and replicated through a simulated annealing process, in which the minimized potential is a variation of the Lennard-Jones potential that considers only first-neighbor interactions according to a Delaunay triangulation. Our simulation results not only confirm the connection between honeycomb structures and other crystal systems such as graphene, but also show that irregularities in the honeycomb structure can be explained as the result of analogous interactions between cells and their immediate surroundings, leading to emergent global order. Additionally, our computational model can be used as a first step to describe specific strategies that bees use to effectively solve geometric mismatches while minimizing cost of comb building.

Honey bees and social wasps reach convergent architectural solutions to nest-building problems

“…Furthermore, we would expect 5-7-sided cell pairs where the 5-sided cell is a deformed worker cell and the 7-sided cell is a deformed reproductive cell. This agrees with current and previous observations [ 7 , 30 ].…”

“…To relate the cell center representation to cells, we use the centers to generate a Voronoi partition to decompose the plane into polygonal areas, i.e., polygons that have the properties that all the points inside a polygon are closest to the cell center that is inside the partition, and all the edges are equidistant from 2 cell centers. Prior studies on irregular nest construction found that the Voronoi partition generated by cell centers is in good agreement to the observed cell walls [30]. To derive a bound on the number of necessary non-hexagonal cells, we use the fact that the graph generated by connecting centers whose Voronoi partition polygons share an edge is a Delaunay triangulation, i.e., a graph where any circumscribing circle of a triangle contains no other vertices.…”

“…Pairs of 5- and 7-sided cells are known to exist in A . mellifera and are even found in graphene, an abiotic hexagonal lattice, which suggests a basic geometric reason for this pairing [ 7 , 15 , 17 , 29 , 30 ]. Here, we show that paired 5- and 7-sided cells are an architectural feature shared across all species with size dimorphism.…”

“…Given these observations, we then reasoned how non-hexagonal cells would arise if the builders acted optimally when transitioning between patches of worker and reproductive cells. The mathematical argument is based on studies of cells in A. mellifera, where the number of walls for each cell depends on the position of optimally spaced cell centers [7,30]. These cells are well represented by the polygons in a Voronoi partition, and in the following, we leverage the fact that in this construction the neighborhood graph of cell centers is a Delaunay triangulation, i.e., the circumscribing circle of each triangle in the graph contains no other centers (see Methods).…”

Adaptive Cell Size, Merging, Tilting, and Layering in Honeybee Comb Construction

“…at the boundaries, while conforming to the space and resources available (13). The resulting structure inevitably contains non-regular hexagons and topological defects with various cell sizes, to adapt to environmental constraints (14)(15)(16)(17). Adjusting the sizes of hexagons during honeycomb construction may serve additional purposes, often dictated by the colony's requirements.…”