VII 



OF THE PARTITIONING OF SPACE 



549 



practically amounts to no more than this, that of the ordinary 

 space-filling soHds 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. At each of the latter three, and at each of the 

 former six, dodecahedra meet in a point in close packing; and foiir 

 edges meet in a point in the one case and eight in the other. This 

 is enough to shew that' the conditions for minimal area are not 

 rigorously metr In one of Plateau's most beautiful experiments*, 

 a wire cube is dipped in soap-solution. When hfted out, a film is 

 seen to pass inwards from each of the twelve edges of the cube, and 

 these twelve films meet, three by three, in eight edges, running 

 inwards from the eight corners of the 

 cube; but the twelve films and their 

 eight edges do not meet in a point, but 

 are grouped around a small central 

 quadrilateral fihn (Fig. 212). Two of 

 the eight edges run to each corner of the 

 little square, and, with the two sides of 

 the square itself, make up the four edges 

 meeting in a point which the theory of 

 area minima requires. We may sub- ^^' 



stitute (by a second dip) a little cube for the little square; now an 

 edge from each corner of the outer cube runs to the corresponding 

 corner of the inner one, and with the three adjacent edges of the 

 little cube itself the number four is still maintained. Twelve films, 

 and eight edges meeting in a point, w^ere essentially unstable; but 

 the introduction of the little square or cube meets most of the 

 conditions of stabihty which Plateau was the first to lay down. 

 One more condition has to be met, namely the equality of angles at 

 which the four edges meet in each conjunction. These co-equal 

 "Maraldi angles" at each corner of the square can only be con- 

 structed by help of a slight curvature of the sides, and the httle 

 square is seen to have its sides curved into circular arcs accordingly; 

 moreover its size and shape, as that of all the other films in the 

 system, are perfectly definite. It is all one, according to the 



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

 p. 505, 1867; xxv, p. 115, 1869. 



