336 THE FORMS OF TISSUES [ch. 



result will obviously be that the intercellular spaces will increase ; 

 the six equatorial attachments of each cell (Fig. 133, a) (or its twelve 

 attachments in all, to adjacent cells) will remain fixed, and the 

 portions of cell-wall between these points of attachment will be 

 withdrawn in a symmetrical fashion (b) towards the centre. As 

 the final result (c) we shall have a " dodecahedral star" or star- 

 polygon, which appears in section as a six-rayed figure. It is 

 obviously necessary that the pith-cells should not only be attached 

 to one another, but that the outermost layer should be firmly 

 attached to a boundary wall, so as to preserve the symmetry of 

 the system. What actually occurs in the rush is tantamount to 

 this, but not absolutely identical. Here it is not so much the 

 pith-cells which tend to shrivel within a boundary of constant 

 size, but rather the boundary wall (that is, the peripheral ring of 

 woody and other tissues) which continues to expand after the 

 pith-cells which it encloses have ceased to grow or to multiply. 

 The twelve points of attachment on the spherical surface of each 

 little pith-cell are uniformly drawn asunder ; but the content, or 

 volume, of the cell does not increase correspondingly; and the 

 remaining portions of the surface, accordingly, shrink inwards and 

 gradually constitute the complicated surface of a twelve-pointed 

 star, which is still a symmetrical figure and is still also a surface 

 of minimal area under the new conditions. 



A few years after the publication 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 cellular liquid 

 "foam," in which it is manifest that the problem is actually and 

 automatically solved. The general mathematical solution of the 

 problem (as we have already indicated) is, that every interface or 

 partition- wall must have constant curvature throughout; that 

 where such partitions meet in an edge, they must intersect at 

 angles such that equal forces, in planes perpendicular to the line 



* Sir W. Thomson, On the Division of Space with Minimum Partitional Area, 

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



