XIV] OR PHYLLOTAXIS \ 921 



a series of symmetrical patterns, whose nature will depend upon the 

 form of the entire surface. If the surface be that of a cylinder, we 

 shall have a system, or systems, of spiral helices: if it be a plane 

 with an infinitely distant focus, such as we obtain by "unwrapping" 

 our cylindrical surface, we shall have straight lines; if it be a plane 

 containing the focus within itself, or if it be any other symmetrical 

 surface containing the focus, then we shall have a system of loga- 

 rithmic spirals. The appearance of these spirals is sometimes spoken 

 of as a "subjective" phenomenon, but the description is inaccurate: 

 it is a purely mathematical phenomenon, an inseparable secondary 

 result of other arrangements which we, for the time being, regard 

 as primary. When the bricklayer builds a factory chimney, he lays 

 his bricks in a certain steady, orderly way, with no thought of the 

 spiral patterns to which this orderly sequence inevitably leads, and 

 which spiral patterns are by no means "subjective." The designer 

 of a wall-paper not only has no intention of producing a pattern of 

 criss-cross lines, but on the contrary he does his best to avoid them ; 

 nevertheless, so long as his design is a symmetrical one, the criss-cross 

 intersections inevitably come. And as the train carries us past an 

 orchard we see not one single symmetrical configuration, but a 

 multiplicity of collineations among the trees. 



Let us, however, lea'C^e this discussion, and return to the facts of 

 the case. 



Our second question, which relates to the numerical coincidences 

 so familiar to all students of phyllotaxis, is not to be set and 

 answered in a word. 



Let us, for simplicity's sake, avoid consideration of simultaneous 

 or whorled leaf origins, and consider only the more frequent cases 

 where a single "genetic spiral" can be traced throughout the 'entire 

 system. 



It is seldom that this primary, genetic spiral catches the eye, for 

 the leaves which immediately succeed one another in this genetic 

 order are usually far apart on the circumference of the stem, and it 

 is only in close-packed arrangements that the eye readily apprehends 

 the continuous series. Accordingly in such a case as a fir-cone, for 

 instance, it is certain of the secondary spirals or " parastichies " 

 which catch the eye ; and among fir-cones, we can easily count these, 



