To lodge the mother’s inexhaustible supply of eggs, they have to build for her, within a limited space, the greatest possible number of cells, all of a capacity determined by the ultimate size of the larvæ. This condition exacts a strict economy of the available building-site. No empty gaps, therefore, which would take up unnecessary room and moreover compromise the general solidity of the structure.
Nor is this all. The business-man says to himself:
“Time is money.”
The Wasp, no less busy, says to herself:
“Time is paper; paper means a more spacious dwelling, holding a larger population. [[234]]Let us not waste our materials. Each partition must serve two neighbouring apartments.”
How will the insect set about solving its problem? To begin with, it abandons any circular form. The cylinder, the urn, the cup, the sphere, the gourd, the cupola and the other little structures of their customary art cannot be grouped together without leaving gaps; they supply no party-walls. Only flat surfaces, adjusted according to certain rules, can give the desired economy of space and material. The cells therefore will be prisms, of a length calculated by that of the larvæ.
It remains to decide what form of polygon will serve as the base of these prisms. First of all, it is evident that this polygon will be regular, because the capacity of the cells has to be constant. Once the condition obtains that the grouping must be effected without gaps, figures that were not regular would be subject to variation and would give different capacities in one cell and another. Now of the indefinite number of regular polygons only three can be constructed continuously, without leaving unoccupied spaces. These three are the equilateral triangle, the square, and the hexagon. Which are we to choose? [[235]]
The one that will approximate most closely to the circumference of a circle and hence be best adapted to the cylindrical form of the larva; the one that, with a containing wall of the same extent, will yield the greatest capacity, a condition essential to the free growth of the grubs. Of the three regular figures that can be assembled without vacant intervals, our geometry suggests the hexagon; and it is the hexagon and none other that is chosen by the geometry of the Wasps. The cells are hexagonal prisms.
Every high and harmonious achievement finds supersubtle minds that strive to degrade it. What has not been said on the subject of hexagonal cells, above all on the subject of the Bee’s, which are arranged in a double layer and united at the base? Reasons of economy of both wax and space demand that this base shall be a pyramid formed of three rhombs with angles of fixed value. Scientific calculations tell us the value of these angles in degrees, minutes and seconds. The goniometer subjects the work of the Bee to examination and finds that the value is precisely calculated to degrees, minutes and seconds. The insect’s work is in perfect agreement with the nicest speculations of our own geometry. [[236]]
There is no room for the glorious problem of the Bee-hive in these elementary essays. Let us confine ourselves to the Wasps. It has been said: