VII] OF HEXAGONAL SYMMETKY 323 



cells of the epidermis appear as a necessary consequence of the 

 simpler laws of growth which gave its simple form to the leaf as 

 a whole. In this last case, however, as in all the others, the rule 

 still holds that only three partitions (in surface view) meet in a 

 point; and at their point of meeting the walls are for a short 

 distance manifestly curved, so as to permit the junction to take 

 place at or nearly at the normal angle of 120°. 



Briefly speaking, wherever we have a system of cylinders or 

 spheres, associated together with sufficient mutual interaction to 

 bring them into complete surface contact, there, in section or in 

 surface view, we tend to get a pattern of hexagons. 



While the formation of an hexagonal pattern on the basis of ready-formed 

 and symmetrically arranged material units is a very common, and indeed the 

 general way, it does not follow that there are not others by which such a 

 pattern can be obtained. For instance, if we take a little triangular dish of 

 mercury and set it vibrating (either by help of a tuning-fork, or by simply 

 tapping on the sides) we shall have a series of little waves or ripples starting 

 inwards from each of the three faces ; and the intercrossing, or interference 

 of these three sets of waves procUices crests and hollows, and intermediate 

 points of no disturbance, whose loci are seen as a beautiful pattern of minute 

 hexagons. It is possible that the very minute and astonishingly regular 

 pattern of hexagons which we see, for instance, on the surface of many diatoms, 

 may be a phenomenon of this order*. The same maybe the case also inArcella, 

 where an apparently hexagonal pattern is found not to consist of simple 

 hexagons, but of "straight lines in three sets of parallels, the lines of each 

 set making an angle of sixty degrees with those of the other two sets f." We 

 must also bear in mind, in the case of the minuter forms, the large possibilities 

 of optical illusion. For instance, in one of Abbe's "diffraction-plates," a 

 pattern of dots, set at equal interspaces, is reproduced on a very minute scale 

 by photography ; but under certain conditions of microscopic illumination 

 and focussing, these isolated dots appear as a pattern of hexagons. 



A symmetrical arrangement of hexagons, such as we have just been 

 studying, suggests various simple geometrical corollaries, of which the following 

 may perhaps be a useful one. 



We may sometimes desire to estimate the number of hexagonal areas or 

 facets in some structure where these are numerous, such for instance as the 



* Cf. some of J. H. Vincent's photographs of ripples, in Phil. Mag. 1897-1899; 

 or those of F. R. Watson, in Phys. Review, 1897, 1901, 1916. The appearance will 

 depend on the rate of the wave, and in turn on the surface-tension; with a low 

 tension one would probably see only a moving "jabble." FitzGerald thought 

 diatom-patterns might be due to electromagnetic vibrations ( Wori.s, p. 503, 1902). 



t Cushman, J. A. and Henderson, W. P., Amer. Nat. xl, pp. 797-802, 1906. 



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