Structural Physics of Bee Honeycomb

Honeybee combs have aroused interest in the ability of honeybees to form regular hexagonal geometric constructs since ancient times. Here we use a real space technique based on the pair distribution function (PDF) and radial distribution function (RDF), and a reciprocal space method utilizing the Debye-Waller Factor (DWF) to quantify the order for a range of honeycombs made by \textit{Apis mellifera}. The PDFs and RDFs are fit with a series of Gaussian curves. We characterize the order in the honeycomb using a real space order parameter, OP$_{3}$, to describe the order in the combs and a two-dimensional Fourier transform from which a Debye-Waller order parameter, \textbf{\textit{u}}, is derived. Both OP$_{3}$ and \textbf{\textit{u}} take values from [0, 1] where the value one represents perfect order. The analyzed combs have values of OP$_{3}$ from 0.33 to 0.60 and values of \textbf{\textit{u}} from 0.83 to 0.98. RDF fits of honeycomb histograms show that naturally made comb can be crystalline in a 2D ordered structural sense, yet is more `liquid-like' than cells made on `foundation' wax. We show that with the assistance of man-made foundation wax, honeybees can manufacture highly ordered arrays of hexagonal cells.