The Cellular Basis of Growth 53 



polygons should these faces be? At first it was believed that cells were 

 12-sided figures, since when spheres are stacked together like cannon 

 balls, each touches 12 others. Lord Kelvin (1887, 1894), approaching the 

 problem mathematically, showed that when space is divided into similar 

 units, each with a minimum area of partition and with stable angles, 

 each unit will be a 14-faced figure or a tetrakaidecahedron and that eight 

 of its faces will be hexagons and six will be squares. This, he thought, 

 was what the shape of a cell in pith or similar tissues theoretically 

 should be. F. T. Lewis ( 1923 and others ) found that the average number 

 of faces in such cells was, indeed, close to 14 but that only very rarely 

 did a cell with this number of faces show eight hexagons and six squares. 



This problem has been studied with particular care by Matzke and 

 his students and reported in a series of papers. The results have been 

 briefly reviewed by Matzke (1950), who cites the more important papers 

 from his laboratory. The general conclusion is that parenchyma cells do 

 tend to have 14 sides but that "ideal" ones, conforming to Lord Kelvin's 

 rule, occur very infrequently. Matzke points out that many factors other 

 than mathematically ideal space-filling are involved in determining cell 

 shape, among them pressure, surface forces, differences in cell size, direc- 

 tion of cell division, unequal growth, and genetic constitution. The prob- 

 lem is being attacked developmentally by an analysis of cell shapes at the 

 meristem (Matzke and Duffy, 1956). In dividing cells, the number of 

 faces here rises to about 17 and in daughter cells drops at first to be- 

 tween 12 and 13. The total cell population has an average number of 

 about 14 faces. 



In more specialized tissues there is a wide variety of cell shapes. 

 Palisade cells are elongated at right angles to the leaf surface. Most cells 

 of the vascular and conducting tissues are elongated parallel to the axis. 

 Hairs and glandular cells have many forms. Some cells expand equally 

 on all sides. Others, like root hairs, grow only at one point. Still others, 

 such as the more fantastic sclereids, have many growing regions ( Foster, 

 1955, and others ) . Galston, Baker, and King ( 1953 ) found that benzimida- 

 zole promotes the transverse as opposed to the longitudinal extension of 

 cortex cells in the pea epicotyl. Doubtless the polarity, or polarities, of the 

 cell and the plasticity, elasticity, and microstructure of its walls are in- 

 volved in shape differences. 



Tenopyr ( 1918 ) found that in leaves of different shapes the shapes and 

 sizes of cells were constant. Riidiger ( 1952 ) , however, observed that in 

 tetraploid plants the subepidermal cells of leaves, hypocotyls, and other 

 organs were not only absolutely but relatively wider than in diploids, a 

 fact which he relates to the greater relative width found in most tetra- 

 ploid organs as compared with diploid ones. 



Even in microorganisms where the cells are free from contact with 



