The Cellular Basis of Growth 



45 



Fig. 3-13. Hypothetical cube of film 

 with a film partition moving from left 

 to right across it. When a position is 

 reached 31.8 per cent of the distance 

 from the right-hand side, this partition 

 slips into the corner and becomes 

 curved, as shown. ( From D'Arcy 

 Thompson. ) 



will now be curved instead of straight as it was before (Fig. 3-13). The 

 point where this shift occurs is the point where the wall, now ( in section ) 

 a quarter of the circumference of a circle, has the same length that the 

 flat partition wall had, for the wall will have the least possible area that 

 will enclose the volume of the smaller "cell," the latter now being part 

 of a cylinder. If this smaller cell is then made still smaller, the wall that 

 separates it from the larger one will continue to be curved and to be less 

 than any other wall area that could bound the volume of the smaller cell. 

 Just where the point of shift from flat wall to curved will occur can be 

 calculated by determining the point at which (before the shift) the length 

 of a curved wall (a quarter of the circumference of a circle) across the 

 corner will be the same as that of the flat partition wall. Both will enclose 

 the same area. If it is assumed that each side of the cube, and thus the 

 length of the flat partition, equals 1, then 



2irr 



= 1 %rr = 4 and r = ^- = 0.637 



Ait 



The area of a quarter circle with this radius is tt( 0.637 ) 2 /4 = 0.318. This 

 also is the area of the smaller rectangular cell just before the shift. Thus 

 the distance from the partition wall to the side wall, when it shifts from 

 straight to curved, is 0.318 of the diameter of the cube. Experiments with 

 films essentially confirm this theoretical expectation. In cases of unequal 

 division of actual cells, such as the formation of companion cells in sieve 

 tubes, the new cell is usually cut out of a corner of the old one, as this 

 theory of least surfaces requires. 



Many dividing cells in plants and animals are spherical, and here the 



