Reflecting on this case, it occurred to me that if the Melipona had
made its spheres at some given distance from each other, and had made
them of equal sizes and had arranged them symmetrically in a double
layer, the resulting structure would probably have been as perfect as
the comb of the hive-bee. Accordingly I wrote to Professor Miller, of
Cambridge, and this geometer has kindly read over the following
statement, drawn up from his information, and tells me that it is
strictly correct:--
If a number of equal spheres be described with their centres placed in
two parallel layers; with the centre of each sphere at the distance of
radius x the square root of 2 or radius x 1.41421 (or at some lesser
distance), from the centres of the six surrounding spheres in the same
layer; and at the same distance from the centres of the adjoining
spheres in the other and parallel layer; then, if planes of
intersection between the several spheres in both layers be formed,
there will result a double layer of hexagonal prisms united together
by pyramidal bases formed of three rhombs; and the rhombs and the
sides of the hexagonal prisms will have every angle identically the
same with the best measurements which have been made of the cells of
the hive-bee.
Hence we may safely conclude that if we could slightly modify the
instincts already possessed by the Melipona, and in themselves not
very wonderful, this bee would make a structure as wonderfully perfect
as that of the hive-bee.
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