THE SHAPE OF CELLS. 545 



tetrakaidecahedral, with subsequent arrest of development. The 

 irregularities of the cells, due to their manner of growth, should not 

 be underestimated. Among the forty-two cells modeled, there is not 

 one with fourteen surfaces, eight of which are hexagons and six quadri- 

 laterals, arranged as in the tetrakaidecahedron of Figure 1, and yet 

 they seem to show convincingly that such is their typical shape, or the 

 form from which they may all have been derived. 



The typical shape is almost realized in a cell with fifteen contacts 

 shown near the base of Figure 2 and labeled c. If its lower lateral 

 wall on the right had taken the course of the dotted line in the figure, 

 its fifteenth contact (with the cell/) would have been eliminated. This 

 disturbing contact is shown as a small triangular area, 3, in the basal 

 view of the model, Figure 4 (Plate 1). It is there shown that the 

 central hexagon is not surrounded by alternating quadrilaterals and 

 hexagons, inasmuch as there are two adjacent pentagons on the left, 

 and a third pentagon, due to the intrusion of the triangular surface 3, 

 on the right. The same cell, in lateral view, looking directly at the 

 quadrilateral surface shown in Figure 4, is seen in Figure 5. Below 

 the quadrilateral of the upper tier is a hexagon, and below the adjacent 

 hexagon of the upper tier is a quadrilateral, which with the hexagon on 

 the under surface are reproducing exactly the complex pattern of the 

 type. Turning to a lateral view of this cell facing the adjacent pen- 

 tagons of the upper tier, it is seen, in Figure 6, that there are "also 

 pentagons below them. If this cell were absolutely typical, its lower 

 tier should present a succession of surfaces with the following numbers 

 of sides — 4, 6, 4, 6 — whereas it does show 4, 5, 5, 6; and above 

 them, in place of 6, 5, 5, there should be 6, 4, 6. But all these devia- 

 tions would be rectified if the boundary between the pentagons in the 

 upper tier were swung like a pendulum to the left, passing the bound- 

 ary between the pentagons in the lower tier. In other words, dif- 

 ferences in the relative volume of adjacent cells may convert two 

 quadrilaterals and two hexagons into four pentagons. Thus it is 

 seen that with the shifting of a single boundary and the elimination 

 of a small triangular fifteenth contact, the cell shown in Figure 4 

 would be typical throughout. 



It may be noted, however, that this cell is not at all an orthic tetra- 

 kaidecahedron, but, as if from flattening, the upper and lower tiers of 

 lateral surfaces tend to meet at an equatorial ridge. In the orthic 

 cells in the diagram, Figure 3, the squares make one third and the 

 hexagons two thirds, of the distance from the top of the cell to the 

 bottom; but in actual cells with an equatorial ridge, the squares almost 



