“And yet the innumerable family must be lodged, honey store-rooms must multiply to supply the wants of the community. Moreover, it is necessary that these store-rooms and nurseries take up as little room as possible, so as not to encumber the hive, and to permit free circulation to the twenty or thirty thousand inhabitants of the city. In fine, one of the hardest problems is presented to the bees: they must make the greatest possible number of cells in the least space and with the least wax possible. Well, friend Jules, do you think you could solve the bees’ problem?” “Alas! Uncle, I hardly understand the statement of it.” “To economize the wax, a very simple way suggests itself at the outset: it is to make the partitions of the cells very thin. You may be quite sure the bees are equal to this elementary requirement. They make the wax walls scarcely as thick as a sheet of paper. But that is not enough: it is necessary above all to take the form into consideration and to seek the most economical shape. Let us try. What shape shall we give the cells to satisfy the conditions of economy in space and wax? “First of all let us suppose them to be round. Let us trace on paper some circles of equal size and touching one another. Between three of these contiguous circles there will always be an unoccupied space. The round form will not do, then, for the cells, since there will always be a waste of space, or empty intervals. “Let us make them square. We will trace equal squares on the paper. In going about it properly we can arrange the squares side by side without leaving any empty spaces between them. Look at the inlaid floor of this room, composed of little square red bricks. These bricks leave no intervening spaces; they touch on every side. The square form, therefore, suits the first condition, namely: to utilize all the space. “But here is where another difficulty arises. Cells fashioned on the square model would not hold enough honey for the quantity of wax used in constructing them. In order to increase their capacity, you must increase as much as possible the number of their facets. I will not try to demonstrate to you this beautiful truth; it is beyond your intelligence. Geometry affirms it; let us consider it a fact. “Starting from that, the choice is soon made. Among all the regular figures that can be placed side by side without leaving an unoccupied space, you must choose that which has the greatest number of sides, for that is the one that will hold the most honey for the same quantity of wax used. “Geometry teaches that the only regular figures that can be arranged without waste of space are: the three-sided figure, or triangle; the four-sided, or square; and the six-sided, or hexagon. That is all: no other regular figures touch all around so as to leave no empty spaces between them.