VIII] THE PARTITIONING OF SPACE 597 



rifts and cracks in a broken surface. The phenomeiion is in the 

 first instance mathematical, in the second physical ; and. the limited 

 number of possible arrangements appear and reappear in the most 

 diverse fields, and are capable of representation by the same 

 diagrams. We have seen in Fig. 196 how the cracks in drying mud 

 exhibit to perfection the polar furrow joining two three-way nodes, 

 which is the characteristic feature of the four-celled stage of a 

 segmenting egg. 



The possible arrangements of the intermediate partitions becomes 

 a question of permutations. Let us call the flexure between two 

 consecutive furrows a or 6, according to its direction, right or left; 

 and let a triple conjunction be called c. Then the three possible 

 arrangements in a system of six cells are aa, ah, c; the four in a 

 system of seven cells are aaa, aab, aba, ac; and the twelve possible 

 arrangements in a system of eight cells are as follows* : 



I cc 



i 



We may classify, and may denote or symbolise, these several 

 arrangements in various ways. In the following table we see: 

 A, the twelve arrangements of the five intermediate partitions 

 which are necessary to enable all the boundary walls of a plane 

 assemblage of eight cells (none being "insular") to meet in three- 

 way junctions; B, the literal permutations which symbolise the 

 same; C, the number of sides (other than the external boundary) 

 which in each case each cell possesses, i.e. the number of contacts 

 each makes with its neighbours.^ The total number of contacts 

 (as we shall see presently) is 26, and the mean number 3-25; if we 

 take the departures from the mean, and sum them irrespective of 

 sign, the sum is shewn under D. 



* I believe that Kirkman, in a paper of more than 80 years ago, said that the 

 number of S-sided convex, Eulerian polyhedra, with trihedral comers, was thirteen. 



