Abstract
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 Apis mellifera ligustica. 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, u, is derived. Both OP 3 and 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 u from 0.59 to 0.69. 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. This is the first description of honeycomb utilizing the Debye-Waller Factor, and provides a complete analysis of the order in comb from a real-space order parameter and a reciprocal space order parameter. It is noted that the techniques used are general in nature and could be applied to any digital photograph of an ordered array.
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Acknowledgments
The honeycomb used in this study was purchased from Sawyer’s Apiaries, Swanton, OH, 43558, USA. F. H. Kaatz thanks Sarah Long for useful discussions. We thank the anonymous referees for comments that improved the quality of the paper. The authors declare that they have no competing financial interests.
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Kaatz, F.H., Bultheel, A. & Egami, T. Order parameters from image analysis: a honeycomb example. Naturwissenschaften 95, 1033–1040 (2008). https://doi.org/10.1007/s00114-008-0418-4
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DOI: https://doi.org/10.1007/s00114-008-0418-4