VIIl] 



THE PARTITIONING OF SPACE 



375 



internally, they must meet in a point ; furthermore, they tend to 

 do so at equal angles of 120°, and there is an end of the matter. 

 If we have four cells, then, as we have already seen, the conditions 

 are satisfied by interposing a little intermediate wall, the two 

 extremities of which constitute the meeting-points of three cells 

 each, and the upper edge of which marks the "polar furrow." 

 Similarly, in the case of five cells, we require two little intermediate 

 walls, and two polar furrows ; and we soon arrive at the rule that, 

 for n cells, we require w — 3 little longitudinal partitions (and 

 corresponding polar furrows), connecting the triple junctions of 



8 CeJ/s \Tvpe 



Fig. 158. Various possible arrangements of intermediate partitions, in 

 groups of 4, 5, 6, 7 or 8 cells. 



the cells ; and these little walls, like all the rest within the system, 

 must be inchned to one another at angles of 120°. Where we 

 have only one such wall (as in the case of four cells), or only two 

 (as in the case of five cells), there is no room for ambiguity. But 

 where we have three little connecting- walls, as in the case of six 

 cells, it is obvious that we can arrange them in three different 

 ways, as in the annexed Fig. 159. In the system of seven cells, 

 the four partitions can be arranged in four ways; and the five 

 partitions required in the case of eight cells can be arranged in no 

 less than thirteen different ways, of which Fig. 158 shews some 

 half-dozen only. It does not follow that, so to speak, these various 



