VIIl] 



THE PARTITIONING OF SPACE 



383 



to angles, and to the number, not to the length of the intermediate 

 partitions ; it is to a great extent by variations in the length of these 

 that the magnitudes of the cells may be equalised, or otherwise 

 balanced, and the whole system brought into equilibrium. Lastly, 

 there is a curious point to consider, in regard to the number of 

 actual contacts, in the various cases, between cell and cell. If we 

 inspect the diagrams in Fig. 169 (which represent three out of our 

 thirteen possible arrangements of eight cells) we shall see that, in 

 the case of type b, two cells are each in contact with two others, 

 two cells with three others, and four cells each with four other cells. 

 In type a four cells are each in contact with two, two with four, 

 and two with five. In type/, two are in contact with two, four 

 with three, and one with no less than seven. In all cases the 



Fig. 169. 



number of contacts is twenty-six in all ; or, in other words, there 

 are thirteen internal partitions, besides the eight peripheral walls. 

 For it is easy to see that, in all cases of n cells with a common 

 external boundary, the number of internal partitions is 2n — 3 ; 

 or the number of what we call the internal or interfacial contacts 

 is 2 (2n — 3). But it would appear that the most stable arrange- 

 ments are those in which the total number of contacts is most 

 evenly divided, and the least stable are those in which some one 

 cell has, as in type /, a predominant number of contacts. In a 

 well-known series of experiments, Roux has shewn how, by means 

 of oil-drops, various arrangements, or aggregations, of cells can 

 be simulated ; and in Fig. 170 I shew a number of Roux's figures, 

 and have ascribed them to what seem to be their appropriate 

 "types" among those which we have just been considering; but 



