324 THE FOErMS OF TISSUES [ch 



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

 An approximate 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 ^3/2, the area of a circle 

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

 the area of that circle equals {n . 8)^ x 7r/4. 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 n^ x 7r/4 x 2/s^B — 0-907»^, or 

 9/10 . 71^, 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 may obtain a still simpler rule*. For 

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

 these a zone of twelve, and around these a zone of eighteen, and so on. And 

 the total number, excluding the central hexagon, is accordingly: 



and so forth. If N be the number of zones, and if we add one to the above 

 numbers for the odd central hexagon, the rule evidently is, that the total 

 number, H, = 3N {N + 1) + 1. Thus, if in a preparation of a fly's cornea, 

 I can count twenty-five facets in a line from a central one, the total number 

 in the entire circular field is (3 x 25 x 26) + 1 = 1951t- 



The same principles which account for the development of 

 hexagonal symmetry hold true, as a matter of course, not only 

 of individual cells (in the biological sense), but of any close- 

 packed bodies of uniform size and originally circular outline; 

 and the hexagonal pattern is therefore of very common occurrence, 

 under widely different circumstances. The curious reader may 

 consult Sir Thomas Browne's quaint and beautiful account, in the 

 Garden of Cyrus, of hexagonal (and also of quincuncial) symmetry 

 in plants and animals, which "doth neatly declare how nature 

 Geometrizeth, and observeth order in all things." 



* This does not merely neglect the broken ones but all whose centres He between 

 this circle and a hexagon inscribed in it. 



f For more detailed calculations see a paper by "H.M." [? H. Munro], in 

 Q. J. M. S. VI, p. 83, 1858. 



