The extraordinary regularity of honeycomb is one of the natural wonders that has most excited minds. For example, the mathematician Pappus of Alexandria had noticed soon Antiquity that these hexagonal structures allowed the bees to develop the maximum cells for a minimum of wax. In this day of return, that's enough to make jealous crabs in math. How these little beasts without two grams of brains they can develop as optimal as they are elegant buildings? Two thousand years later, the false explanations were always die hard ... A little geometry to start (I promise: no equation!)
You are a bee and you must make a maximum of cells of wax for receiving larvae. To not be jealous, all the cells must have the same surface. What format did you interest in choosing to consume the least possible wax? If you lack imagination and you only use polygons, you only have three choices: the equilateral triangle, square and hexagon. All other polygons will leave "holes" between them when you stick it to each other. Use less wax can look back to the form which gives the largest area for a given perimeter. Now more sides a polygon has, the more surface is large (at constant perimeter): It seems logical, since its shape approximates more and more that of a circle that is precisely the figure of largest area. Of the three possible
polygons (triangle, square, hexagon), so this is the hexagon that can make the most of cells with less wax. Skeptics can check this with the formula it offers the best surface on the perimeter.
Well all this reasoning is pretty good but there is no reason to consider only polygons, dammit! Why not more funny shapes, like those Escher imaginative example:
Could it be that some of these figures are used to "pave" the most economical plan? It feels good not, but it incredibly hard to prove. This conjecture honeycomb which enshrines the regular hexagon as the champion of the paving had to wait more than 2300 years before being rigorously demonstrated in 1999!
divine work or effect of natural selection?
How not see evidence of God's intervention in this beautiful optimality? "The bees, and inspiration by the divine will, are capable of applying blindly the finest math," wrote the scientist Fontenelle the seventeenth century . For Kepler bees "are endowed with a soul and thus capable of making the geometry". Even Jean-Henri Fabre , the pope of the modern entomology, referred in their goals all rational attempts to explain: "In his books, he wrote , the Creative Power always geometrizes ( ...) Plato said. There is really solving the problem of Wasps. " Even today, honeycombs inspire all sorts of mystical explanations ( here or there example).
The argument was so powerful that divine Darwin himself has long studied the subject in the Origin of Species . He was annoyed because apart from these hexagonal structures, not found in wasps and bees that cylindrical cavities more or less coarse, especially for solitary species. How could she bee "learn" to make hexagons from cylinders? Obviously he has sought the response of the side of natural selection (p. 304 ):
Thus, in my opinion, the most surprising of all known instincts, that of the bee, can be explained by the action of natural selection. Selection Nature has taken advantage of the slight modifications, and many successive suffered by the instincts of a more simple then it has gradually brought the bee to describe more fully and more regularly spheres placed in two rows at equal distances, and widen and raise the flat walls on the lines of intersection. It goes without saying that the bees do not know more than they describe their spheres at a distance from each other, they know what it is that the various sides of a hexagonal prism of his or lozenges basis. The determinant of the action of natural selection was building strong cells, the shape and capability to contain larvae, achieved with the minimum expenditure of wax and work. The swarm individual who has built the most perfect cell with any work and any expenditure is made into honey wax is the most successful, and sent its newly acquired economical instincts to successive swarms in turn also have had more chances in their favor in the struggle for existence.
not know anything about math, the bees would be (through trial and error? Theory does not say) falling by chance on an optimal structure giving them an economy of wax decisive for their survival and multiplication in greater numbers. This explanation was so successful that the honeycombs have changed sides ideological argument divine they became the classic illustration of the dramatic effects of natural selection, which is found on most sites dealing with the question ( here or there example).
A purely mechanical explanation?
Yet there is one obvious problem with the Darwinian explanation: it is unprovable. Why imagine too convoluted stories, fulminated D'Arcy Thompson (which I mentioned in this post or it ) while the hexagonal honeycombs can be explained by the simple laws of physics? Enjoy the last beautiful days to watch the foam of your beer, you'll see that pressed against each other, the bubbles are also adopting a more or less hexagonal (right diagram, source here).
In two dimensions, the mechanism is simple to understand (left): Initially each bubble is circular and its neighbors in six key points. Under the effect of pressure, the six contact points are transformed into six straight lines and circles change into hexagons tight against each other
When one is in three dimensions, things are a little more complicated because you can only pave a curved surface with hexagons. This explains the somewhat quirky form bubbles in beer foam alternating hexagons and pentagons, like a football. We find these hexagonal shapes everywhere in nature, as soon as disks, spheres or cylinders are compressed against each other. Try for example with egg yolks in a dish:
When the cylinders are molten magma which press against each Other cooling, it gives these extraordinary formations of Giant's Causeway in Ireland:
In the area of living, networks of hexagons appear whenever a large number cells in round-gate crowd against each other under the effect of growth. Here, it's not a honeycomb cells but the eye of the American horsefly. The hexagons are irregular because the surface of the eye is spherical 
There are plenty of examples like this in all areas of chemistry and biology at all scales, from molecules to the hexagon Saturn .
How bee it take?
Could it be that the nests of bees owe their beauty or the genius of bees, or the hand of God, or even natural selection, but to simple mechanical effects on pockets of soft wax? The idea is not new: already in the seventeenth century, Erasmus Bartholin , a Danish mathematician doubted that research economy is behind those pretty patterns. He proposed that the hexagonal cells was simply the result of the effort of each bee to enlarge up the cell constructs, by analogy with the pressure in each bubble. Besides, the bees wax is initially very fluid, just as a soapy film. This hypothesis would explain why the hexagonal shape of cells is between the cells and not on the edge of the nest. I saw myself this summer on a nest of wasps in training (that would not we do for science!) That the outer walls are shaped arc, exactly as predicted by the Bartholin hypothesis:
Good, but can still be skeptical about the hypothesis "of bees that grow." How intermittent efforts as those of bees on the cells they can also lead to regular buildings? D'Arcy Thompson has a much simpler explanation (p133): "It seems much more probable it is in reality a question of power: the walls actually adopt their configuration when they are in a state semi-fluid, due to the presence of residual water in the pulp plant, or under the effect of softening the wax caused by the heat generated by all the bees at work in the hive. " D'Arcy too strong: in 2004 researchers seem to give reason for artificially recreating the hexagonal structure of a honeycomb without bees, by simply casting a liquid wax hot rolls around and tight against each other (which include bees in real life). It would therefore simply the bees to spread the liquid wax around them so that it will eventually take the form of hexagons almost perfect under the sole effect of physical laws:
And the optimum wax, that is he?
Opposite to these results, the hypothesis of natural selection of thrifty bees wax it still holds the road? The researchers of that study (( The amazing bee , P175) "if one were to include in computing the hem of wax that covers the edge of the cells, the 30% wax would negate any additional balance sheet optimization. D Moreover, as noted by D'Arcy Thompson (p 131), "the bee is not sparing of his work, it is not only the fineness or accuracy sufficient for it to take advantage of a any wax economy by building its socket according to these theoretical standards (...) When a bee built an isolated cell or a small group of cells destined to give rise to eggs queens, the building is of poor quality. The alveoli are the queens of small clusters of crude wax, where the cavity is barely sketched sharp blows of jaws, like a roughly hewn tree trunk that bore traces of a blunt tool. "
Natural selection therefore clearly has not selected specimens of bees wax more efficient. However, when one thinks of the energy they need to spend to produce all the wax, how to imagine that parsimony has played no role in the evolution of bees? I wonder if we should not reverse outright the argument of the classical natural selection. Is the need to make full backing of nests, which have developed in bees the instinct to save money? Or would it not rather the economy of wax that provides a mechanically collective nest would have favored the evolutionary success of social species?
Basically, all this is rather reassuring: the bees are not math geniuses, or the fierce economic and physical laws are probably sufficient to explain their prodigious constructions. Natural selection would play a role well, but not necessarily the one usually depicted. One thing is puzzling in this story: why, despite its inconsistencies and its unverifiable nature, the hypothesis of natural selection of thrifty bees wax it is still widespread, including in books or scientific sites? And conversely why do we find as little mechanical explanation for these structures, however, former explanation, consistent, backed by experience and in line with comparable phenomena in other areas? I suppose that such an explanation is difficult to popularize because it contradicts the representation of classical Darwinian evolution and because it is underpinned by any more comprehensive theory, mechanics this time of evolution. "You Can not Beat Something With Nothing", in science as elsewhere.
Sources:
This blog post General Knowledge
This article Science News on the conjecture of the honeycomb and the other the University of Montreal. This site
on diatoms and it on the hexagon in nature. And of course
Growth and Form by D'Arcy Thompson ...
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Ticket X rated power on the origin of spirals in nature
Maya against invaders : another post about the incredible ruses bees
PS. 
We also attempted to explain the shape of the bottom of the cells (right, source Wikipedia ) by arguments of economy of wax. The cells are stacked into each other in several layers and each is closed by three planar faces (called rhombs) that join, as a hexagonal pencil, the point would be cut by three strokes of the knife (left figure). This form allows cells to fit perfectly into each other. It is certainly more efficient than a flat-bottomed hexagonal wax, but in 1964 the Hungarian mathematician Fejes Toth proved that it is less a background consisting of two hexagons and two smaller diamonds (right diagram). 
This is played very little (the economy would be only 0.35%) but why natural selection would she despised that little optimization? researchers wanted to test an artificial if they could bring about a This form only by the laws of physics. In trapping bubbles between glass plates, they got two layers of hexagonal cells and observed how they fit into one another. Bingo! Depending on the amount of trapped liquid, the substance of these cells was sometimes that of three rhombs, sometimes described by Toth. Certainly, the physical mechanisms are far from clear and material saving is not the only variable. 
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