The Lore of the Honey-Bee

by Tickner Edwardes

Turning the comb over, we see that the cells on this side also have pyramidal bottoms. If the depth of a cell on one side of the comb be taken, and added to the depth of a cell on the other side, and then the width of the whole comb be measured, it will be found that the combined depth of the two cells perceptibly exceeds the width of the whole comb. At first glance this seems like a case of the less including the greater, which is a manifest impossibility. But, holding the comb up to the light, a further discovery is made, and the seeming paradox is eliminated. The bottoms of the cells are so thin as to be almost transparent, and it is at once seen that the cells are not built end to end, in line, but that each cell-base on one side of the comb covers part of three cell-bases on the other. If the three diamonds, composing between them the triangular base of a single cell, be perforated with a needle, and the comb turned over, it will be found that the three perforations come each in a separate cell. Thus the saving in the total width of the comb is effected by allowing the pyramidal bases on each side to engage alternately like the teeth of a trap; instead of meeting point-blank, they overlap each other, and the faces of the pyramids are so contrived that each of them helps to close two cells.

There is another advantage in this arrangement which will be immediately obvious. The apex and three ribs of each pyramidal cell-base form foundation-lines for the cell-walls on the other side of the comb. This means that not only do all cell-walls abut on an arch, but that every cell-base is strengthened throughout by a triple girdering. The result is that the amount of wax required in the construction of the comb can be everywhere reduced to an absolute minimum. It becomes merely a question of what thickness of wax will retain the honey; and this experience proves to be no more than 1/180 part of an inch. The whole thing, indeed, might very well be taken as an ideal exemplar of the triumph of mind over matter.

The geometric principles brought into play in the construction of honey-comb have been a favourite study with mathematicians of all ages, and especially this rhombiform method adopted by the bee in flooring her cells. The rhomb is best described as a plane-figure whose four sides are equal, like those of a square, but whose angles are not right angles. In such a figure there are necessarily two greater angles and two smaller, facing each other in pairs. The three rhombs composing the base of the honey-cell lean together, as has been seen, in the form of a blunt pyramid; and—treating all angles as negligible factors—the bluntness of this pyramid is found to coincide very aptly with the shape of the full-grown larvæ. But this is not the only reason for the particular inclination given by the bee to the rhombs forming the base of each cell. Economy rules here, as in everything else she undertakes; and the truth that she has chosen the one and only form of cell-base which takes the least possible material to construct has received very striking confirmation.

The story is an old and famous one, but it will bear repeating. A great naturalist once put himself to an infinity of trouble in measuring the angles formed by the rhombs in a vast number of comb-cell bases, and he found that these showed remarkable uniformity. It will be clear that the hollow pyramid of the cell-bottom will be either deep or shallow, according to the shape of the three rhombs composing it. The apex of the pyramid is formed by the meeting of three equal angles, one from each rhomb; and it is plain that this apex will be sharp or blunt, according to whether the meeting angles are wide or narrow. It was, of course, impossible to ascertain the dimensions of these angles with absolutely microscopical nicety; but, dealing only with the most perfect comb, the naturalist found that the two greater angles in the rhombs measured very nearly 110°, and the two lesser angles 70°. He also found that the angles formed by the conjunction of the cell-sides with the bases had the same dimensions as those of the rhombs. Assuming therefore that, mathematically, the angles of the rhombs and cell-sides should be equal, he was able to calculate exactly the angles for which the bees were evidently striving in the construction of the rhombs—109° 28′ and 70° 32′.

Another bee-lover scientist, ruminating over these figures, was much impressed by them, and determined to find out the reason why the bee made such constant choice of this particular shape of rhomb. He therefore conceived the idea of submitting the bee’s judgment on this cell-base question to an independent authority. Without disclosing his object, he propounded the following problem to one of the greatest mathematicians of the day.

“Supposing,” said he, in effect, “you were required to close the end of an hexagonal vessel by three rhombs or diamond-shaped plates, what angles must be given to these rhombs so that the greatest amount of space would be enclosed by the least amount of material?”

It was a difficult problem, but the mathematician worked it out at last, and his answer was “109° 26′ and 70° 34.”

Now, the difference between the calculation of the man and the calculation of the bee was an exceedingly small one. No one thought of calling into question the work of the man, who was pre-eminent in his world of figures. It was therefore accepted as a fact that the bee had made a trifling mistake—so trifling, however, that, in the matter of comb-building, it was of no importance. Her reputation was unimpaired: to all intents and purposes the honey-cell was still a perfect example of utmost capacity secured by least material.

But another mathematician—a Scotsman this time—went over the whole business again, and he proved conclusively that the bee was right, while the first mathematician was wrong. He showed that the true answer to the problem of the angles was 109° 28′ and 70° 32′—identically the figures obtained by estimation of the honey-comb.