# Once a Week (magazine)/Series 1/Volume 2/A problem: algebra and the bees

The signature "E. J." represents Edward Jesse; the identity of the co-signer "A. A. M." is unknown.
A PROBLEM. ALGEBRA AND THE BEES.
“So work the honey bees;
Creatures that, by a rule of Nature, teach
The art of order to a perfect kingdom.”
Shakespeare.
Insect architecture has always appeared to me to be infinitely more curious than the structure of birds’ nests, most interesting as some of these latter most certainly are. For instance, we have the Tailor-bird’s nest, suspended at the end of a slender branch, out of the reach of monkeys, snakes, &c., and then the delicately-formed nests of various humming-birds, composed of cobwebs, thistledown, &c.; but all these and many others must give way to the architecture of bees. These insects, in the construction of their cells, have solved a problem, at whose solution the human mind could arrive only by the application of a high branch of analytical science.
It is well known that bees have chosen the hexagonal form for their cells, as being, mathematically, the most convenient and economical one. Any other form would indeed have either involved inconveniently-shaped corners, or have entailed an absolute waste of the material employed in their construction. The hexagonal form alone evades these disadvantages, and it, at the same time, includes the indispensable requirement that each wall shall serve as a common partition to the adjacent cells. Up to this point, then, we observe great judgment exercised in the choice of the form best adapted for convenience and economy. In selecting the particular kind of roof, however, a still greater difficulty had to be overcome, and in mastering this, something more than judgment would appear to have been exercised. As in the case of the sides, owing to the bees having only a limited supply of material for building, it is necessary that the roof of each cell shall serve also for the flooring of the cells in the upper story, the plan of building the cells in tiers over each other having been originally adopted to avoid extending the building over too much superficial space. The scheme accordingly devised for the roofage is the following. Each hexagonal case is covered by a roof, composed of three perfect rhombi, inclined to each other at a certain angle, and terminating in a common vertex G, as in the annexed figure. Thus the complete roof of one cell contributes to the flooring of three upper ones. The three angles at G are equal, and of the same magnitude in every hive, and by careful measurement have been ascertained to be invariably equal to 10${\frac {7}{13}}$ degrees. Now it is in the selection of the particular size of this angle that the masterpiece of calculation, or whatever it may be called, is exhibited. Had this angle been chosen larger or smaller, the amount of wax required to enclose the same must, in either case, have been greater. By means of the differential calculus—an algebraical process of a high order—it has been ascertained, in the course of a long and elaborate investigation, that in order to enclose a maximum of space with given material within a hexagonal ​cell thus roofed, the larger angle in each rhombus of the triple ceiling must exactly equal 109${\frac {7}{13}}$ degrees.
In the choice of this angle great latitude might have been allowed to these little architects, had the questions of arrangement and convenience been the only ones to be considered. But their supply of material being limited, it was absolutely necessary to ascertain even the fractional parts of a degree in the required angle, in order that the largest possible space might be enclosed with a given amount of wax.
I am not aware that this wonderful and minute accuracy in the construction of the cells of bees has been noticed by others. At all events, it solves a curious and interesting problem, and as such is submitted to the consideration of the mathematician.
A. A. M. & E. J.