title: Order parameters from image analysis: a honeycomb example.
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.
authors.affiliation: Department of Mathematics and Life/Natural Sciences, Owens Community College, Toledo, OH 43699, USA. fhkaatz@yahoo.com
authors.forename: Forrest H
authors.initials: FH
authors.lastname: Kaatz
authors.forename: Adhemar
authors.initials: A
authors.lastname: Bultheel
authors.forename: Takeshi
authors.initials: T
authors.lastname: Egami
mesh_terms: D000818:Animals; D001516:Bees / Q000502:physiology*; D003198:Computer Simulation; D017753:Ecosystem; D006722:Honey; D006799:Housing, Animal; D009425:Nesting Behavior