VIIl] 



THE PARTITIONING OF SPACE 



599 



serving (so to speak) as a base under all the rest. Then, calling 

 ^3, ^4, etc., the number of three-sided and four-sided facets, we re- 

 classify our twelve configurations as follows : 



Polyhedral arrangements of eight cells (none of them ''insular'*)^ 

 considered as part of a nine-faced polyhedron, whose ninth face 

 is octagonal. 



^3 ^4 



F, F, 



I 



ae 



cd 



b 



i 



h 



f 

 gk 



J 



2 — 



— 1 



2 — 



1 — 



It is of interest, and of more than mere mathematical interest, 

 to know, not only that these possible arrangements are few, but 

 that they are strictly defined as to the number and form of the 

 respective faces. For we know that we are limited to three-way 

 corners or nodes; and, that being so, the fpllowing simple rule holds 

 for the facets — a rule which we shall use later on in still more 

 curious circumstances, and which may be easily verified in any line 

 of the foregoing table : 



ZF^ + 2F, + F,± O.F, -F,- 2F, - etc. = 12. 



We may produce and illustrate all these configurations by blowing 

 bubbles in a dish and here (Fig. 247) is the complete series, up to 

 seven cells. They correspond precisely to the diagrams shewn on 

 p. 596, and their resemblance to embryological diagrams is only 

 cloaked a little by the circular outhne, the artificial boundary of 

 the system. Of the twelve eight-celled arrangements, four seem 

 unstable; these include the one case (i) where one cell of the eight 

 is in contact with all seven others, and the three cases (a, e, h) 

 where one is in contact with six others. The reason of this insta- 

 bility is, I imagine, that the internal angles cannot be angles of 

 120°, as equilibrium demands, unless the sides be curved, and 

 convex inwards; but this implies a combined pressure from without 

 on the large cell in the middle. While it adjusts its walls, then, 

 to the required angles, the large cell tends to close up, to lose hold 



