548 THE FORMS OF TISSUES [ch. 



actually occurs in the rush is tantamount to this, but not absolutely 

 identical. It is not so much the pith-cells which tend to shrink 

 within a boundary of constant size, but rather the boundary wall 

 which continues to expand after the pith-cells which it encloses have 

 ceased to grow or to multiply. The points of attachment on the 

 surface of each little pith-cell are drawn asunder, but the content 

 of the cell does not correspondingly increase; and the remaining 

 portions of the surface shrink inwards, accordingly, and gradually 

 constitute the complicated . figure which Kepler called a star- 

 dodecahedron, which is still a symmetrical figure, and is still a 

 surface of minimal area under the new and altered conditions. 



The tetrakaidekahedron 



A few years after the pubhcation of Plateau's book, Lord Kelvin 

 shewed, in a short but very beautiful paper*, that we must not 

 hastily assume from such arguments as the foregoing that a close- 

 packed assemblage of rhombic dodecahedra will be the true and 

 general solution of the problem of dividing space with a minimum 

 partitional area, or will be present in a liquid ''foam," in which the 

 general problem is completely and automatically solved. The 

 general mathematical solution of the problem (as we have already 

 indicated) is, that every interface or partition-wall must have con- 

 stant mean curvature throughout ; that where these partitions meet 

 in an edge, they must intersect at angles such that equal forces, in 

 planes perpendicular to the line of intersection, shall balance ; that 

 no more than three such interfaces may meet in a line or edge, 

 whence it follows (for symmetry) that the angle of intersection of 

 all surfaces or facets must be 120°; and that neither more nor less 

 than four edges meet in a point or corner. An assemblage of rhombic 

 dodecahedra goes far to meet the case. It fills space; its surfaces 

 or interfaces are planes, and therefore surfaces of constant curvature 

 throughout; and they meet together at angles of 120°. Never^ 

 theless, the proof that the rhombic dodecahedron (which we find 

 exemplified in the bee's cell) is a figure of minimal area is not 

 a comprehensive proof; it is Hmited to certain conditions, and 



* Sir W. Thomson, On the division of space with minimum partitional acea, 

 Phil. Mag. (5), xxiv, pp. 503-514, Dec. 1887; cf. Baltimore Lectures, 1904, p. 615; 

 Molecular tactics of a crystal (Robert Boyle Lecture), 1894, pp. 21-25. 



