IVAX AND COMB CONSTRUCTION, 
177 
The cells are hexagonal, and there is a very good 
reason for this form. Mathematicians have shown, as 
Dr. Reid (140) points out : 
‘There are only three possible figures of the cells, 
which can make them all equal and similar without any 
useless interstices. These are the equilateral triangle, the 
square, and the regular hexagon. It is well known to 
mathematicians that there is not a fourth way possible in 
which a plane may be cut into little spaces that shall be 
equal, similar, and regular, without leaving any interstices.’ 
The square and the triangle would be unsuitable, 
owing to their angles, for the round body of the 
chrysalis, which could only utilise the enclosed circle, 
but the hexagon is a nearer approach to this than 
either the square or triangle. The hexagon has also 
a smaller circumference than either of the other two 
figures, so that there would be an economy in material 
in constructing such cells. But there is also an 
economy of material in the bases, while best fitting 
them to the shape of the chrysalis. For if any other 
angle had been adopted, Lhuilier (97) pointed out 
much more wax would have been required, and if the 
midrib had been fiat he calculated that as much wax 
would be required to construct fifty cells as it takes to 
make fifty-one with pyramidal bases. It is known 
that Maraldi (106) had studied the form of the cells 
of the bee, and had described the angles of the 
rhomb, as well as Reaumur (139), who did so at a 
later date ; and it is part of the history of the subject 
that Koenig’s results, to whom the problem was sub- 
mitted for solution, differed from those of Maraldi by 
N 
