510. THE FORMS OF TISSUES [ch. 



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

 pattern of hexagons which we see on the surface of many diatoms (Fig. 189) 

 may be a phenomenon of this order*. The same may be the case also in Arcella, 

 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 setst-" 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 focusing, these isolated dots appear as a pattern of hexagons. 



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

 suggests various geometrical corollaries, of which the following may be a 

 useful one. We sometimes desire to estimate the number of hexagonal areas or 

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

 cornea of an insect's eye, or in the minute pattern of hexagons on many 

 diatoms. An approxipiate enumeration is easily made as follows. 



For the area of a hexagon (if we call 8 the short diameter, that namely 

 which bisects two of the opposite sides) is 8^ x v 3/2, the area of a circle 

 being ci^.7r/4. Then, if the diameter {d) of a circular area include n hexagons, 

 the area of that circle equals {n.Sfxn/i:. And, dividing this number by 

 the area of a single hexagon, we obtain for the number of areas in the circle, 

 each equal to a hexagonal facet, the expression w^ x7r/4 x 2/\/3 = 0'9077i^ or 

 9/lO.w^ nearly. 



This calculation deals, not only with the complete facets, but with the 

 areas of the broken hexagons at the periphery of the circle. If we neglect 

 these latter, and consider our whole field as consisting of successive rings of 

 hexagons about a central one, we obtain a simpler rule. For obviously, 

 around our central hexagon there stands a zone of six, and around these 



* Cf. some of J. H. Vincent's photographs of ripples, in Phil. Mag. 1897-99; 

 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." Cf. also Faraday, On 

 the crispations of fluids resting upon a vibrating support, Phil. Mag. 1831, p. 299; 

 and Rayleigh, Sound, u, p. 346, 1896. FitzGerald thought diatom -patterns might 

 be due to electromagnetic vibrations (Works, p. 503, 1902); with which of. W. D. 

 Dye, Vibration-patterns of quartz plates, Proc. R.S. (A), cxxxviii, p. 1, 1932. 

 Dye's Fig. 17, which he calls "one of the most beautiful types of minor vibration 

 met with in discs", is closely akin to the diatom Orthoneis splendida. In both cases 

 two nodal systems, conjugate to one another, are based on two foci near the ends of 

 an elliptical plate; . but bands in the experimental plate are further broken up into 

 rows of dots in the diatom. See also Max Schultze, Die Struktur der Diatomeenschale 

 verglichen mit gewi.ssen aus Fluorkiesel kiinstlich darstellbaren Kieselhauten, Verh. 

 nnturh. Ver. Bonn, xx, pp. 1-42, 1863; Trans. Microsc. Soc. (N.S.), xi, pp. 120-136, 

 1863; H. J. Slack, Monthly Microsc. Journ. 1870, p. 183. 



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



