516 THE FORMS OF TISSUES [ch. 



average angles of 120°, the polygonal areas must, on the average, 

 be hexagonal. This, then, is the simple geometrical explanation, 

 apart from any physical one, of the widespread appearance of the 

 pattern of hexagons. 



If the law of minimal areas holds good in a "cellular" structure, 

 as in a froth of soap-bubbles or in a vegetable parenchyma, then 

 not merely on the average, but actually at every node, three partition- 

 walls (in plane projection) meet together. Under perfect symmetry 

 they do so at co-equal angles of 120°, and the assemblage consists 



Fig. 196. Cracks in drying mud; a thread encircles and marks out a "'polar 

 furrow"; cf. p. 487. From R. H. Wodehouse. 



(in plane projection) of co-equal hexagons; but the angles may vary, 

 the cells be unequal, and the hexagons interspersed with other 

 polygonal figures. Nevertheless, so long as three partition- walls and 

 only three meet together, the cells are, ipso facto, on the average 

 hexagonal*. 



We may count the cells if we please. A section of Cycas-petiole 

 gave the following numbers: ^ 



Number of sides 3 4 5 6 7 8 9 



„ instances 8 97 207 96 9 

 Mean number of sides: 6-00. 



The fine emulsion of an Agfa plate shews a beautiful polygonal 

 pattern which obeys the law of the triple node; and a patch of a 



* A more elaborate proof is given by W. C. Goldstein, On the average number 

 of sides of polygons of a net, Ann. Math. (2), xxxii, pp. 149-153, 1931. 



