THE SHAPE OF CELLS. 



539 



dodecahedra occur more often than the typical form. The upper and 

 basal surfaces of such cells are hexagonal instead of quadrilateral, and 

 the two quadrilaterals which previously extended from the base to 

 the top likewise become hexagons. Such a form, with four hexagonal 

 and eight quadrilateral surfaces, Kieser regarded as typical for the 

 parenchyma of bark and pith, being impressed with the way in which 

 it provided hexagonal sections when cut in several planes. The flat- 

 tened form (Figure 1, d) will be seen to be a most interesting approxi- 

 mation to the shape of cells actually found by methods not at Kieser's 

 disposal. In an excellent diagrammatic figure he showed how the 

 truncate dodecahedra would appear when combined to make a tissue, 

 and this perhaps led DeCandolle to warn his readers that "cells are 

 not as regular as the published figures might lead one to believe." 



Thus the matter rested until Plateau, in 1873, brought together his 

 masterly researches on the statics of liquids. By means of wire frames 

 dipped in glycerinated or soapy solutions he produced "liquid poly- 

 hedra," studying the production and arrangement of the films which 

 make their walls. In 1886 this work in physics was applied to the 

 problem of cell forms both by Berthold in his Studien uber Proto- 

 plasm amechanik and by Errera, Ueber Zellenformen und Seifenblasen. 

 Neither writer discloses the shape of parenchymal cells — Errera 

 refers to the endless diversity of cell forms — but the inclination and 

 curvature of cell walls, as seen in sections, are carefully studied, and 

 compared with inert liquid films and "artificial cell-tissues." It is 

 concluded that the phenomena of surface tension are of controlling 

 importance in the shaping of cells. 



In accordance with the mathematics of his time, Kieser had declared 

 that, of all bodies which may be combined to fill space without inter- 

 stices, "the rhombic dodecahedron encloses the greatest space with 

 the least extent of surface." But in 1887, Lord Kelvin, after experi- 

 menting with Plateau's cubic skeleton frame, demonstrated that for 

 the division of space with minimal partitional area the rhombic dode- 

 cahedron is rivaled by a more stable tetrakaidecahedron, or figure 

 having fourteen surfaces, certain of which are slightly curved. Neg- 

 lecting the curvature of such films, which he calculated with precision, 

 as being lost in greater curves and irregularities due to shrinkage in 

 preserved tissue, the form of this tetrakaidecahedron deserves careful 

 attention. It has six quadrilateral and eight hexagonal surfaces, 

 which form "thirty-six edges of intersection between faces, and 

 twenty-four corners, in each of which three faces intersect." If the 

 quadrilateral surfaces are equal squares, and the hexagons are equal, 

 equilateral and equiangular hexagons, the figure becomes what Lord 



