The Life of the Bee

by Maurice Maeterlinck

*It was as well, perhaps, that this standard was not
adopted. For although the diameter of the cells is admirably
regular, it is, like all things produced by a living
organism, not mathematically invariable in the same hive.
Further, as M. Maurice Girard has pointed out, the apothem
of the cell varies among different races of bees, so that
the standard would alter from hive to hive, according to the
species of bee that inhabited it.

Each of the cells is an hexagonal tube placed on a pyramidal base; and two layers of these tubes form the comb, their bases being opposed to each other in such fashion that each of the three rhombs or lozenges which on one side constitute the pyramidal base of one cell, composes at the same time the pyramidal base of three cells on the other. It is in these prismatic tubes that the honey is stored; and to prevent its escaping during the period of maturation,—which would infallibly happen if the tubes were as strictly horizontal as they appear to be,—the bees incline them slightly, to an angle of 4 deg or 5 deg.

"Besides the economy of wax," says Reaumur, when considering this marvellous construction in its entirety, "besides the economy of wax that results from the disposition of the cells, and the fact that this arrangement allows the bees to fill the comb without leaving a single spot vacant, there are other advantages also with respect to the solidity of the work. The angle at the base of each cell, the apex of the pyramidal cavity, is buttressed by the ridge formed by two faces of the hexagon of another cell. The two triangles, or extensions of the hexagon faces which fill one of the convergent angles of the cavity enclosed by the three rhombs, form by their junction a plane angle on the side they touch; each of these angles, concave within the cell, supports, on its convex side, one of the sheets employed to form the hexagon of another cell; the sheet, pressing on this angle, resists the force which is tending to push it outwards; and in this fashion the angles are strengthened. Every advantage that could be desired with regard to the solidity of each cell is procured by its own formation and its position with reference to the others."

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"There are only," says Dr. Reid, "three possible figures of the cells which can make them all equal and similar, without any useless interstices. These are the equilateral triangle, the square, and the regular hexagon. Mathematicians know that there is not a fourth way possible in which a plane shall be cut into little spaces that shall be equal, similar, and regular, without useless spaces. Of the three figures, the hexagon is the most proper for convenience and strength. Bees, as if they knew this, make their cells regular hexagons.

"Again, it has been demonstrated that, by making the bottoms of the cells to consist of three planes meeting in a point, there is a saving of material and labour in no way inconsiderable. The bees, as if acquainted with these principles of solid geometry, follow them most accurately. It is a curious mathematical problem at what precise angle the three planes which compose the bottom of a cell ought to meet, in order to make the greatest possible saving, or the least expense of material and labour.* This is one of the problems which belong to the higher parts of mathematics. It has accordingly been resolved by some mathematicians, particularly by the ingenious Maclaurin, by a fluctionary calculation which is to be found in the Transactions of the Royal Society of London. He has determined precisely the angle required, and he found, by the most exact mensuration the subject would admit, that it is the very angle in which the three planes at the bottom of the cell of a honey comb do actually meet."

*Reaumur suggested the following problem to the celebrated
mathematician Koenig: "Of all possible hexagonal cells with
pyramidal base composed of three equal and similar rhombs,
to find the one whose construction would need the least
material." Koenig's answer was, the cell that had for its
base three rhombs whose large angle was 109 deg 26', and the
small 70 deg 34'. Another savant, Maraldi, had measured as
exactly as possible the angles of the rhombs constructed by
the bees, and discovered the larger to be 109 deg 28', and
the other 70 deg 32'. Between the two solutions there was a
difference, therefore, of only 2'. It is probable that the
error, if error there be, should be attributed to Maraldi
rather than to the bees; for it is impossible for any
instrument to measure the angles of the cells, which are not
very clearly defined, with infallible precision.

The problem suggested to Koenig was put to another mathematician, Cramer, whose solution came even closer to that of the bees, viz., 109 deg 28 1/2' for the large angle, and 70 deg 31 1/2' for the small.

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I myself do not believe that the bees indulge in these abstruse calculations; but, on the other hand, it seems equally impossible to me that such astounding results can be due to chance alone, or to the mere force of circumstance. The wasps, for instance, also build combs with hexagonal cells, so that for them the problem was identical, and they have solved it in a far less ingenious fashion. Their combs have only one layer of cells, thus lacking the common base that serves the bees for their two opposite layers. The wasps' comb, therefore, is not only less regular, but also less substantial; and so wastefully constructed that, besides loss of material, they must sacrifice about a third of the available space and a quarter of the energy they put forth. Again, we find that the trigonae and meliponae, which are veritable and domesticated bees, though of less advanced civilisation, erect only one row of rearing-cells, and support their horizontal, superposed combs on shapeless and costly columns of wax. Their provision-cells are merely great pots, gathered together without any order; and, at the point between the spheres where these might have intersected and induced a profitable economy of space and material, the meliponae clumsily insert a section of cells with flat walls. Indeed, to compare one of their nests with the mathematical cities of our own honey-flies, is like imagining a hamlet composed of primitive huts side by side with a modern town; whose ruthless regularity is the logical, though perhaps somewhat charmless, result of the genius of man, that to-day, more fiercely than ever before, seeks to conquer space, matter, and time.