CHAPTER VIII 



THE FORMS OF TISSUES OR CELL-AGGREGATES {continued) 



The problems which we have been considering, and especially that 

 of the bee's cell, belong to a class of " isoperimetrical " problems, 

 which, deal with figures whose surface is a minimum for a definite 

 content or volume. Such problems soon become difficult*, but we 

 may find many easy examples which lead us towards the explanation 

 of biological phenomena; and the particular subject which we shall 

 find most easy of approach is that of the division, in definite pro- 

 portions, of some definite portion of space, by a partition- wall of 

 minimal area. The theoretical principles so arrived at we shall then 

 attempt to apply, after the manner of Berthold and Errera, to the 

 biological phenomena of cell-division. 



This investigation may be approached in two ways: by con- 

 sidering the partitioning off from some given space or area of one-half 

 (or some other fraction) of its content; or again, by dealing with the 

 partitions necessary for the breaking up of a given space into a 

 definite numbet of compartments. 



If We begin with the simple case of a cubical cell, it is obvious that, 

 to divide it into two halves, the smallest partition-wall is one which 

 runs parallel to, and midway between, two of its opposite sides. 

 If we call a the length of one of the edges of the cube, then a^ is the 

 area, ahke of one of its sides and of the partition which 'we have 

 interposed parallel thereto. But if we now consider the bisected 

 cube, and wish to divide the one-half of it again, it is obvious that 

 another partition parallel to the first, so far from being the smallest 

 possible, is twice the size of a cross-partition perpendicular to it; 

 for the area of this new partition is a x a/2. And again, for a 

 third bisection, our next partition must be perpendicular to the other 

 two, and is obviously a Httle scj^uare, with an area of (|a)2 = ^a^. 



* Minkowski, and others have shewn how hard it is, for instance, to prove the 

 seemingly obvious proposition that the sphere, of all figures, has the greatest volume 

 for a given surface; cf. (e.g.) T. Bonneson, Les problemes des isoperimetres et des 

 iseplphanes, Paris, 1929. For a historical account of this class of problems, see 

 G. Enestrom, in Bibl. Math. 1888. 



