Insect Architecture

by James Rennie

The obstruction of which M. Huber complains only operated as a stimulus to his ingenuity in contriving how he might continue his interesting observations. From the time of Pappus to the present day, mathematicians have applied the principles of geometry to explain the construction of the cells of a bee-hive; but though their extraordinary regularity, and wonderfully-selected form, had so often been investigated by men of the greatest talent, and skilled in all the refinements of science, the process by which they are constructed, involving also the causes of their regularity of form, had not been traced till M. Huber devoted himself to the inquiry.

As the wax-workers secrete only a limited quantity of wax, it is indispensably requisite that as little as possible of it should be consumed, and that none of it should be wasted. Bees, therefore, as M. Réaumur well remarks,[AQ] have to solve this difficult geometrical problem:—a quantity of wax being given, to form of it similar and equal cells of a determinate capacity, but of the largest size in proportion to the quantity of matter employed, and disposed in such a manner as to occupy the least possible space in the hive. This problem is solved by bees in all its conditions. The cylindrical form would seem to be best adapted to the shape of the insect; but had the cells been cylindrical, they could not have been applied to each other without leaving a vacant and superfluous space between every three contiguous cells. Had the cells, on the other hand, been square or triangular, they might have been constructed without unnecessary vacancies; but these forms would have both required more material, and have been very unsuitable to the shape of a bee’s body. The six-sided form of the cells obviates every objection; and while it fulfils the conditions of the problem, it is equally adapted with a cylinder to the shape of the bee.

M. Réaumur further remarks, that the base of each cell, instead of forming a plane, is usually composed of three pieces in the shape of the diamonds on playing cards, and placed in such a manner as to form a hollow pyramid. This structure, it may be observed, imparts a greater degree of strength, and, still keeping the solution of the problem in view, gives a great capacity with the smallest expenditure of material. This has actually, indeed, been ascertained by mathematical measurement and calculation. Maraldi, the inventor of glass hives, determined, by minutely measuring these angles, that the greater were 109° 28’, and the smaller 70° 32’; and M. Réaumur, being desirous to know why these particular angles are selected, requested M. Kœnig, a skilful mathematician (without informing him of his design, or telling him of Maraldi’s researches), to determine by calculation what ought to be the angle of a six-sided cell, with a concave pyramidal base, formed of three similar and equal rhomboid plates, so that the least possible matter should enter into its construction. By employing what geometricians denominate the infinitesimal calculus, M. Kœnig found that the angles should be 109° 26’ for the greater, and 70° 34’ for the smaller, or about two-sixtieths of a degree, more or less, than the actual angles made choice of by bees. The equality of inclination in the angles has also been said to facilitate the construction of the cells.

M. Huber adds to these remarks, that the cells of the first row, by which the whole comb is attached to the roof of a hive, are not like the rest; for, instead of six sides, they have only five, of which the roof forms one. The base, also, is in these different, consisting of three pieces on the face of the comb, and on the other side of two: one of these only is diamond-shaped, while the other two are of an irregular four-sided figure. This arrangement, by bringing the greatest number of points in contact with the interior surface, insures the stability of the comb.

Arrangement of Cells.

It may, however, be said not to be quite certain, that Réaumur and others have not ascribed to bees the merit of ingenious mathematical contrivance and selection, when the construction of the cells may more probably originate in the form of their mandibles and the other instruments employed in their operations. In the case of other insects, we have, both in the preceding and subsequent pages of this volume, repeatedly noticed, that they use their bodies, or parts thereof, as the standards of measurement and modelling; and it is not impossible that bees may proceed on a similar principle. M. Huber replies to this objection, that bees are not provided with instruments corresponding to the angles of their cells; for there is no more resemblance between these and the form of their mandibles, than between the chisel of the sculptor and the work which he produces. The head, he thinks, does not furnish any better explanation. He admits that the antennæ are very flexible, so as to enable the insects to follow the outline of every object; but concludes that neither their structure, nor that of the limbs and mandibles, are adequate to explain the form of the cells, though all these are employed in the operations of building,—the effect, according to him, depending entirely on the object which the insect proposes.

We shall now follow M. Huber in the experiments which he contrived, in order to observe the operations of the bees subsequent to their laying a foundation for the first cell; and we shall again quote from his own narrative:—

“It appeared to me,” he says, "that the only method of isolating the architects, and bringing them individually into view, would be to induce them to change the direction of their operations and work upwards.

"I had a box made twelve inches square and nine deep, with a moveable glass lid. Combs, full of brood, honey, and pollen, were next selected from one of my leaf-hives, as containing what might interest the bees, and being cut into pieces a foot long, and four inches deep, they were arranged vertically at the bottom of the box, at the same intervals as the insects themselves usually leave between them. A small slip of wooden lath covered the upper edge of each. It was not probable that the bees would attempt to found new combs on the glass roof of the box, because its smoothness precluded the swarm from adhering to it; therefore, if disposed to build, they could do so over the slips resting on the combs, which left a vacuity five inches high above them. As we had foreseen, the swarm with which this box was peopled established itself among the combs below. We then observed the nurse-bees displaying their natural activity. They dispersed themselves throughout the hive, to feed the young grubs, to clear out their lodgment, and adapt it for their convenience. Certainly, the combs, which were roughly cut to fit the bottom of the box, and in some parts damaged, appeared to them shapeless and misplaced; for they speedily commenced their reparation. They beat down the old wax, kneaded it between their teeth, and thus formed binding materials to consolidate them. We were astonished beyond expression by such a multitude of workers employed at once in labours to which it did not appear they should have been called, at their coincidence, their zeal, and their prudence.