THE SHAPE OF CELLS. 547 



same letters. Every square has become a hexagon, and is above or 

 below a quadrilateral portion cut off from an original hexagon. Every 

 hexagon of the original cell has produced a quadrilateral above and 

 below, and its middle part remains hexagonal. 



Can any such geometrical process be assumed to occur in actual 

 cells? In Figure 2, at b and c, there are two cells which are shown 

 inverted in Figure 10. Except that one of these has three more sur- 

 faces than the other (16 and 13 respectively) their configuration is 

 suggestive of twins. On the side hidden in the figure each cell presents 

 four surfaces meeting at a salient point, such tetrahedral angles being 

 very exceptional and contrary to the arrangement of the experimental 

 soap-films. There is also a striking duplication of surfaces shown in 

 Figure 10, and they are so oriented as to indicate that this pair of cells 

 resulted from a transverse division. The surfaces a-e would then 

 correspond with those of the same letters in Figures 8 and 9. There 

 is an additional fifth contact of a at x: typically this surface should be 

 quadrilateral and its upper border horizontal. The surfaces b and c 

 are in contact with one and the same cell and are typical, as are all 

 four surfaces on the right. There a single cell is in contact with the 

 quadrilateral and the hexagon below it. On the left, the upper hexa- 

 gon is typical, but below it there are two surfaces instead of three, and 

 these are in contact with a single cell. Evidently the cell correspond- 

 ing with that adjoining g, h, and i in Figure 9 failed to divide with the 

 rest. Consequently the surface d has five instead of six sides, and 

 since the quadrilateral, i, is lacking, the entire cell has thirteen in- 

 stead of fourteen contacts. The general arrangement of these faces 

 is convincingly like the theoretical pattern. 



Vertical division of the type shown in Figure 12 should produce a 

 characteristic cell, having four pentagonal surfaces (two of which can 

 be seen in the diagram, Figure 13) and a hexagonal face extending 

 uninterruptedly from the top pentagon to the basal pentagon. Figure 

 14 represents an actual cell, which as seen from one side exhibits pre- 

 cisely these features. From other points of view it is not perfectly 

 typical; yet it has exactly eleven surfaces, and clearly represents a 

 cell which, after vertical division, failed to regain the tetrakaidecahe- 

 dral form. Vertical division occurring directly over a cell dividing 

 transversely would subdivide the hexagon beneath, producing on the 

 top of the underlying cell an additional surface like that marked x 

 in Figure 10. 



Cells more difficult to interpret are shown in Figure 16, which 

 represents a pair evidently derived from the transverse division of a 



