## Abstract

The paper presents, and discusses, four candidate explanations of the structure, and construction, of the bees’ honeycomb. So far, philosophers have used one of these four explanations, based on the mathematical Honeycomb Conjecture, while the other three candidate explanations have been ignored. I use the four cases to resolve a dispute between Pincock (Mathematics and Scientific Representation, Oxford University Press, Oxford, 2012) and Baker (Synthese, 2015) about the Honeycomb Conjecture explanation. Finally, I find that the two explanations focusing on the construction mechanism are more promising than those focusing exclusively on the resulting, optimal structure. The main reason for this is that optimal structures do not uniquely determine the relevant optimization leading to the optimal structure.

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## Notes

See the entry ‘The evolution of honeycomb’ of the Darwin Correspondence Project, https://www.darwinproject.ac.uk/the-evolution-of-honey-comb, last retrieved June 23, 2015, and references therein.

Not only certain kinds of bees (such as

*Apis mellifera*), but also hornets (such as*Vespa orientalis*), and social wasps, build combs of high regularity.I will drop the qualifier “candidate” from here on. It should be noted that all four proposals are not (fully) accepted as actual explanations in science.

Fejes Tóth’s formulation of

*honeycomb*-type structures is but one possibility. In principle, there might be other relevant kinds of honeycombs. For example, honeycombs with different global structures, say, cells between two concentric cylinders, might be taken into consideration.The width

*w*is the distance between the two planes. The choice of width determines the depth of cells: if the width is large compared to the volume, i.e., if \(w \gg \root 3 \of {v}\), which is true for actual honeycombs, then the cells will be deep orthogonal to the planes; otherwise, they will be shallow. Fejes Tóth also considers the optimization problem which lets the width of cells vary; this is the*second isoperimetric problem*. He shows that the bees’ solution is not optimal with respect to this problem, but notes that it might be biologically sensible to let the cells be of a certain, fixed depth: a certain depth might be necessary for breeding purposes. Thus, the biologically relevant problem is the first isoperimetric problem.Fejes Tóth’s is only a relative optimality result. The optimal solutions to both isoperimetric problems seems to be unknown.

For example, the optimal solution to the first isoperimetric problem formulated by Fejes Tóth is still open. Even problems that seem relatively easy are unsolved. For example, the optimal way of stacking cylinders in two layers is unknown; the same is true for the general “tin can stacking problem” in three dimensions; see Hales (2000, p. 440).

The Kepler problem was solved by Hales, but the Kelvin problem remains open. For a long time, the candidate solution was a structure proposed by Kelvin, but in 1994, Denis Weaire and Robert Phelan found a counterexample to Kelvin’s structure, now called Weaire-Phelan structure.

The experiment does not constitutes a mathematical answer to the mathematical dry-foam and the wet-foam problems. A mathematical proof is missing in both cases.

Hypotheses linking the bees’ honeycomb to soap bubbles via an equilibrium process have been around for some time; see e.g. Klarreich (2000, p. 159). The biological results reported below were unknown to Weaire & Phelan in 1994.

Pirk et al. also found that the cell bottoms, or closings, are not in fact formed by three rhombi (see Fig. 2a for the rhombic design), but spherical. This finding about the cell bases, however, has been contested by Hepburn et al. (2007). They examined moulds taken from newly constructed as well as from old cells, and found that while old cells indeed have spherical bases, newly-constructed cells have rhombic bases. Pirk et al. made the mistake of taking moulds exclusively from old cells. This finding is an important part of the bigger picture: if the cell bases were spherical, this would indicate that not all parts of cells were constructed by thermal equilibrium; the finding by Hepburn et al. shows that the thermal construction can explain the structure of the whole cell.

The idea that the primary optimization might concern packing has been around for some time; see Pirk et al. (2004) for some useful references.

Other parameters besides optimized packing and tiling are relevant to comb construction. For example, the structure has to be sufficiently stable; see Weaire and Hutzler (1999, p. 167). The actual honeycomb might not reach the point of structural transition realized in the Weaire-Phelan experiment because stability stands in the way.

The biological reason for this variation is that the cells are built for breeding different kinds of hornets. Breeding drones and queens requires larger cells than breeding smaller working hornets.

Note that if the

*explanandum*is the lateral symmetry of cells, it could be thought that the HC is relevant to the explanation. We will see below that it is not.See Pierce (1989, sections 6.5; 7.1) for a detailed account of the relevant models.

The following is based on Kadmon et al. (2009).

Note that if the boundary is hexagonal, this problem does not have closed-form analytic solutions and has to be solved numerically. The numerical procedure yields eigenvalues; see Kadmon et al. (2009, Table 1).

Representation theory studies symmetries by examining how a group acts on linear vector spaces. An irreducible representation is the fundamental building block of this action. The determination of symmetry works as follows. According to representation theory, the number

*p*of components of an irreducible representation has to be at least as big as the number*s*of distinct eigenvalues of an operator associated with a system, if the system has the symmetry group of this representation (\(s \le p\)) (this is an application of Schur’s lemma). If the number of distinct eigenvalues*s*is bigger than the number*p*of the components of an irreducible representation, this means that the system does not have the symmetry group of this representation. If, however,*s*matches*p*, we have evidence that this is the right symmetry group. In the current situation, the connection between frequencies and symmetry is as follows. Assume we observe*s*distinct frequencies. Each frequency corresponds to an eigenvalue. If two frequencies of the cell coincide, we have one eigenvalue associated with a lateral mode, i.e., \(s = 1\), which means that we have evidence for \(p=1\), such that the two-dimensional space has an irreducible representation with one component, i.e., the irreducible representation itself is two-dimensional. Thus, the cell has a symmetry with a two-dimensional irreducible representation. A useful account of the mathematics can be found in Sternberg (1994, Ch. 2–3).This fact has been known since antiquity; its first statement is usually attributed to Pappus; see Hales (2001).

Numerical calculations provide the eigenvalues of the physical model with a hexagonal cross section; this makes it possible to get approximations of the acoustic frequencies that may be used in comb construction. Kadmon et al. also address the other idealizations.

There is now an extensive philosophical literature on idealizations; see Frigg and Hartmann (2012) for a useful discussion of idealizations in the context of scientific models.

An anonymous referee has pointed out that the HC might play an explanatory role in the Acoustic Resonance Hypothesis after all: The HC shows that using the optimal regular polygon is sufficient, because it is also the global optimum. I agree. However, this is an explanatory relation between two mathematical results, not between the

*explanandum*in question, which is an empirical structure, and a mathematical result. An explanatory relation between the two mathematical results does not imply that the HC explains anything about empirical comb structures; for the latter, additional empirical considerations come into play.

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## Acknowledgments

I thank Claus Beisbart, Michael Esfeld, Hannes Leitgeb, Thomas Müller, Tilman Sauer, Raphael Scholl, and participants of the research colloquium AG Müller in Konstanz for comments on the paper at various stages of the writing process.

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This work was partially supported by the Swiss National Science Foundation, grant numbers 100011-124462/1 and 100018-140201/1, as well as by the Templeton World Charity Foundation through grant TWCF0078/AB4.

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Räz, T. The silent hexagon: explaining comb structures.
*Synthese* **194**, 1703–1724 (2017). https://doi.org/10.1007/s11229-016-1014-3

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DOI: https://doi.org/10.1007/s11229-016-1014-3