548 lewis. 



single predecessor. In one of their lateral aspects they present, as 

 seen in the figure, four superimposed irregular pentagons, with angles 

 jutting alternately to the left and right. The regularly superimposed 

 quadrilaterals and hexagons on either side of them can not, by any 

 change in the pentagons, be brought into harmonious relations with 

 each other. It happens, however, that the cell shown in Figure 15 

 may explain the situation. This cell presents an atypical feature 

 which is not very unusual, namely three, instead of two, superimposed 

 lateral surfaces. A hexagon has a quadrilateral both above and 

 below it. This would happen if the cell shown in Figure 9 failed to 

 divide at the time when division took place in an adjoining cell at the 

 plane indicated by the dotted line between a and b. This would 

 convert the quadrilaterals / and j into pentagons ; they would become 

 hexagons through the transverse division of certain cells on the hidden 

 side of the model. Thus the atypical features of the cell shown in 

 Figure 15 may be satisfactorily accounted for. Now if this cell should 

 divide transversely along the line of dashes, and adjacent cells against 

 the faces a and d should divide along the dotted lines, a pair of cells 

 almost identical with those in Figure 16 would result. The conditions 

 in twelve of the thirteen surfaces there shown would be exactly re- 

 produced. Division of a cell on the back of the model in Figure 15 

 would change the pentagonal surface c to a hexagon and make the 

 analogy perfect. 



There is sometimes evidence of an unequal cell division, as with the 

 pair of cells shown in Figure 17. The cell from which they came was 

 under the average size, and was pentagonal above and below, having 

 twelve surfaces altogether. It exhibited a pronounced equatorial 

 ridge. Transverse division took place below the ridge in a plane 

 passing through a quadrilateral surface and separating scarcely more 

 than one third of the cell from the rest. The volumes of the models 

 are 55 c.c. and 30 c.c. respectively. In the drawing a dotted line 

 indicates approximately the plane of normal division. As a result of 

 the unequal partition, the upper cell retained a portion of all the 

 original surfaces except that which formed the base, and with a new 

 pentagonal basal surface, it continued to have twelve facets. The 

 lower cell, however, has only eight facets, of which the three lateral 

 ones shown in the figure extend continuously from the top to the 

 bottom surfaces. The latter, as before mentioned, are pentagonal. 

 On the side away from the observer, similar conditions obtain except 

 that a small triangular fragment of one of the surfaces of the upper tier 

 has been included in the lower cell. 



