Let the angle NVX, which is half the solid angle of the prism, =ã₤ö¡; the side of the hexagon, as AB, =ã₤a; and the height, as Aa, =ã₤h.
Then,
And VX =ã₤aã₤ãã₤sinã₤ö¡ (from inspection of the triangle LXB)
Therefore the area of the rhombus VAXC =ã₤(aÿ£¢2ã₤ãÿ£¢3)ã₤ãã₤(2 sinã₤ö¡).
And the area of AabX =ã₤(aã₤ãã₤2)(2hã₤ãã₤ô§VXã₤cosã₤ö¡)
Therefore the total area of the figure
Therefore
But this expression vanishes, that is to say, d(Area)ã₤ãã₤dö¡ =ã₤0, when cosã₤ö¡ =ã₤1ã₤ãã₤ãÿ£¢3, that is when ö¡ =ã₤54ô¯ã₤44ÿ£¢ãýã₤8ÿ£¢ã°
This then is the condition under which the total area of the figure has its minimal value.
That the beautiful regularity of the beeãs architecture is due to some automatic play of the physical forces, and that it were fantastic to assume (with Pappus and Rûˋaumur) that the bee intentionally seeks for a method of economising wax, is certain, but the precise manner of this automatic action is not so clear. When the hive-bee builds a solitary cell, or a small cluster of cells, as it does for those eggs which are to develop into queens, it makes but a rude production. The queen-cells are lumps of coarse wax hollowed out and roughly bitten into shape, bearing the marks of the beeãs jaws, like the marks of a blunt adze on a rough-hewn log. Omitting the simplest of all cases, when (as among some humble-bees) the old cocoons are used to hold honey, the cells built by the ãsolitaryã wasps and bees are of various kinds. They may be formed by partitioning off little chambers in a hollow stem; {332} they may be rounded or oval capsules, often very neatly constructed, out of mud, or vegetable fibre or little stones, agglutinated together with a salivary glue; but they shew, except for their rounded or tubular form, no mathôÙeôÙmatôÙiôÙcal symmetry. The social wasps and many bees build, usually out of vegetable matter chewed into a paste with saliva, very beautiful nests of ãcombsã; and the close-set papery cells which constitute these combs are just as regularly hexagonal as are the waxen cells of the hive-bee. But in these cases (or nearly all of them) the cells are in a single row; their sides are regularly hexagonal, but their ends, from the want of opponent forces, remain simply spherical. In Melipona domestica (of which Darwin epitomises Pierre Huberãs description) ãthe large waxen honey-cells are nearly spherical, nearly equal in size, and are aggregated into an irregular mass.ã But the spherical form is only seen on the outside of the mass; for inwardly each cell is flattened into ãtwo, three or more flat surfaces, according as the cell adjoins two, three or more other cells. When one cell rests on three other cells, which from the spheres being nearly of the same size is very frequently and necessarily the case, the three flat surfaces are united into a pyramid; and this pyramid, as Huber has remarked, is manifestly a gross imitation of the three-sided pyramidal base of the cell of the hive-bee31.ã The question is, to what particular force are we to ascribe the plane surfaces and definite angles which define the sides of the cell in all these cases, and the ends of the cell in cases where one row meets and opposes another. We have seen that Bartholin suggested, and it is still commonly believed, that this result is due to simple physical pressure, each bee enlarging as much as it can the cell which it is a-building, and nudging its wall outwards till it fills every intervening gap and presses hard against the similar efforts of its neighbour in the cell next door32. But it is very