Summary: The orientation of cells in wild honeycomb appears to be independent of gravity. This article proposes the structural basis for this. 

honeycomb
Figure 1. The hexagonal form of the honeycomb (from a bee keeper’s hive)

The hexagonal honeycomb (Figure 1) is one of the most well-studied natural structures and the design is one of the most commonly used patterns in engineering across a range of scales and applications (see the comprehensive 69-page review by Zhang et al., 2001). It is well appreciated now (and proven mathematically) that the hexagonal honeycomb is the most efficient use of material when it comes to partitioning two-dimensional space into regular areas (Hales, 2001). It has also been shown recently that the hexagonal shape is an emergent property from constructing circular cells in close proximity (Karihaloo et. al, 2013), something surmised by Thompson (1917) and others previously. In this post, I wish to comment on another advantage of the bee’s hexagonal honeycomb that is rarely, if ever discussed in a biological or mathematical context, though it is well known to the engineering community, and that is the fact that the hexagonal honeycomb is transversely isotropic (more on this later). I speculate this is an important prerequisite that enables honeybee nests to be constructed from a wide range of substrate structures with varying topologies, some examples of which are shown in Figure 2.

honeybeesubstrates
Figure 2. Honeycomb nests are constructed from a wide range of substrates at varying angles (photo credits from left: Papyierre1, Bernard Dupont, ImkerijHaarlem – Creative Commons)

Structural Functions of the Honeycomb

A honeybee’s nest has to accomplish several functions – it is primarily a place for rearing brood, and for storing honey, nectar and pollen and it does so with minimal expenditure of construction material. From a structural standpoint, it is likely that the most significant load the honeybee nest has to withstand is the gravity load resulting from the weight of the nest, comprising the wax, the nest’s contents and the honeybees themselves. As little as 1 lb of wax is allocated in the storage of approximately 22 lbs of honey, with cell walls measuring as little as 73 microns in thickness (Winston, 1987).

Gravitational loads point downward towards the center of the earth at all times, regardless of the location of the honeycomb. While the comb is constructed with cells running close to the horizontal, with a slight upward tilt to prevent its contents spilling out, it would also stand to reason that bees would orient their cells in-plane as well, to optimize the distribution of material in accordance with this directionality. In other words, no matter the inclination of the substrate on which the honeycomb is constructed, all honeycombs should have the same, optimal design orientation, perhaps with the non-diagonal edge running along the vertical gravitational axis, as shown in Figure 1. This is not, however, what researchers have found.

Gravity Independence (and Bees in Space)

In a simple yet elegant experiment, Pratt (2000) coaxed honeybees to construct honeycomb from substrates positioned at different angles and concluded that there was a “clear dependence of cell orientation on substrate orientation, and complete independence from the direction of gravity.” Combs were built from substrates at four different orientations, two of which are shown in Figure 3 – combs from all 4 substrates in the study had the corner vertex pointing towards the substrate, not in the direction of gravity.

honeycomb-gravity
Figure 3. Gravity independence of honeycomb orientation, re-drawn based on Pratt (2000)

Shumakova et. al (2006) backed up this work with experiments on 12 more colonies and concluded that the “regular cell pattern appeared to depend on the starting position on the top bar, not on gravity” and in fact found combs were constructed in a wide range of orientations, with only a 10-20% preference by bees for the vertical orientation shown in Figures 1 and 3.

A more dramatic study preceding the two aforementioned ones was conducted in the 1980s, when NASA sent bees into zero-gravity aboard a Shuttle Mission to assess, among other things, whether the bees would construct comb, and if so, what its properties would be (Vandenberg et. al, 1985). They found that the bees survived in zero-g and constructed about 200 square centimeter of comb and that the cells “did not angle consistently downward when built at zero-g“.

These three studies point to the independence of comb cell orientation to gravity. This despite evidence that honeybee workers have the ability to use hair plates on the base of their necks to detect gravitational orientation (Lindauer and Nedel, 1957). In other words, honeybees have an awareness of the direction of gravity, which they use for other purposes, but seemingly do not let it influence the orientation of their comb. How are they able to get away with this, given that resisting gravity loads is a key design requirement of nest construction from a structural standpoint?

Transverse Isotropy

The central argument of this post is that honeybees can ignore gravity in their orientation of comb cells because the deformation response of the hexagonal honeycomb is transversely isotropic. A transversely isotropic material is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy. In other words, within this plane, the material properties are the same in all directions (see Figure 4). With regard to the honeycomb, it thus does not matter how the comb is oriented relative to gravity, the deformation arising from this gravity load is the same in any direction since the plane of isotropy always runs.

transisotropy
Figure 4. Isotropic materials have the identical response independent of the direction of loading – in the above case, modulus E1 = E2

There are only 3 regular shapes that can partition two-dimensional space into equal areas: triangles, squares and hexagons. Of these, the hexagonal and the triangular honeycombs share this property of transverse isotropy. The square shape creates a honeycomb structure that is strongly anisotropic. This is best indicated with a polar plot, shown in Figure 5, which is redrawn from one of my favorite figures in Gibson and Ashby’s book on Cellular Solids (2nd Edition). E1* and E2* represent the effective moduli of the honeycombs in the 1- and 2-directions respectively. The values plotted are for the specific instance of the ratio of the thickness (t) over length (l) being 0.2.

polar_honeycomb
Figure 5. Polar diagram comparing the in-plane relative effective stiffness of hexagonal, square and triangle honeycomb structures. The triangular and hexagonal shapes are isotropic, the square is not – redrawn from Gibson & Ashby (1997) – approximate scale only

In Conclusion

There are many reasons why honeybees build their comb with hexagonal cells: it emerges naturally from a closed packing of circles, it minimizes material and it may even help with communication.  If you buy the argument above, we can add to this list the fact that it allows the in-plane orientation of the honeycomb to be gravity independent with no negative consequences, on account of the transverse isotropy of hexagonal cells. Whether these all just convenient coincidences or an indication of a deeper meaning of the nature of the hexagon, I am not sure. Perhaps Winnie-the-Pooh was right all along, when he said, “you never can tell with bees.”

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