VII] OF HEXAGONAL SYMMETRY 519 



neighbourhood of each corner will the sides tend to curve in, so 

 forming a notch whose curved sides have tangents approximately 

 120° apart, again just as in our projection of the surface of a froth 

 (p. 494). Already these considerations lead us to a fair sketch of, 

 or first approximation to, the carapace of a tortoise; but we may 

 go on a httle further. The horny plates and the bony carapace 

 below must grow at such a rate as to keep pace, more or less exactly, 

 with one another; but it does not follow that they will keep time 

 precisely. If the horny. plates grow ever so httle faster than the 

 bones below, they will fail to fit, will overcrowd one another, and 

 will be forced to bulge or wrinkle. Both of these things they often, 

 and even characteristically, do; the wrinkles appear in orderly, 

 parallel folds, pointing to alternate periods, or spurts, of faster and 

 slower growth; and the characteristic patterns which. ensue are 

 the visible expression of these differential growth-rates. 



In all this we assume tHat the plates are lying in one and the 

 same plane or even surface, abutting against one another as they 

 grow, and so crowding and squeezing one another into the fornj of 

 straight-edged polygons. The result will be very different if they 

 overlap, after the manner of slates on a roof: the difference is what 

 we have seen to exist between the cones of Pinus and of Larix. 

 The overlapping edges will be* free to grow into natural, rounded 

 curves ; each plate, uncrowded and unconstrained, will stay smooth 

 and un wrinkled; the number and order of the plates will be the 

 same as before — but the shell will be no longer that of a tortoise, 

 but of the turtle from which " tortoise-shell" is obtained. 



A snow-crystal is a very beautiful example of hexagonal sym- 

 metry. It belongs to another order of things to those we have been 

 speaking of: for in substance it is a solid, and in form it is a crystal, 

 and its own intrinsic molecular forces build it up in its own way. 

 But (as we have mentioned once before) it is an exquisite illus- 

 tration of Nature's way of producing infinite variety from the 

 permutations and combinations of a single type. The snowflake is 

 a crystal formed by sublimation, that is to say by precipitation 

 from a vapour without passing through a liquid phase. It begins 

 as a tiny hexagon, the making of which tends to use up the vapour 

 near by; the angles of the hexagon jut out, so to speak, into regions 

 of greater, or less depleted, vapour-pressure, and at these corners 



