VIl] 



OF THE PARTITIONING OF SPACE 



337 



of intersection, shall balance ; and finally, that no more than three 

 such interfaces may meet in a line or edge, whence it follows that 

 the angle of intersection of the film-surfaces must be exactly 120°. 

 An assemblage of equal and similar rhombic dodecahedra goes far 

 to meet the case : it completely fills up space ; all its surfaces or 

 interfaces are planes, that is to say, surfaces of constant curvature 

 throughout ; and these surfaces all meet together at angles of 120°. 

 Nevertheless, the proof that our rhombic dodecahedron (such as 

 we find exemplified in the bee's cell) is a surface of minimal area, 

 is not a comprehensive proof; it is limited to certain conditions, 

 and practically amounts to no more than this, that of the regular 

 solids, with all sides plane and similar, this one has the least surface 

 for its solid content. 



The rhombic dodecahedron has six tetrahedral angles, and 

 eight trihedral angles ; and it is obvious, on consideration, that 

 at each of the former six dodecahedra meet in a point, and that, 

 where the four tetrahedral facets of each coalesce with their 

 neighbours, we have twelve plane films, or interfaces, meeting in 

 a point. In a precisely similar fashion, we may imagine twelve 

 plane films, drawn inwards from the twelve edges of a cube, to 

 meet at a point in the centre of the cube. But, as Plateau dis- 

 covered*, when we dip a cubical 

 wire skeleton into soap-solution and 

 take it out again, the twelve films 

 which are thus generated do not 

 meet in a point, but are grouped 

 around a small central, plane, quadri- 

 lateral film (Fig. 134). In other 

 words, twelve plane films, meeting in 

 a point, are essentially unstable. If 

 we blow upon our artificial film- 

 system, the little quadrilateral alters 

 its place, setting itself parallel now to one and now to another of 

 the paired faces of the cube ; but we never get rid of it. Moreover, 

 the size and shape of the quadrilateral, as of all the other films in the 

 system, are perfectly definite. Of the twelve films (which we had 



* Also discovered independently by Sir David Brewster, Trans. R.S.E. xxiv, 

 p. 505, 1867, XXV, p. 115, 1869. 



T. G. 22 



Fig. 134. 



