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 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
sqrt(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. But I hear from Professor Wyman, who has made numerous careful
measurements, that the accuracy of the workmanship of the bee has been
greatly exaggerated; so much so, that whatever the typical form of the cell
may be, it is rarely, if ever, realised.
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