170 



already pointed to in^GGl^JbjrJKe^pler in some of his botanical 

 speculations. 



But it must be clear from the above, that in the light of this theory 

 all supposed analogy with the arrangement of the molecules in 

 crystals, as suggested by Wulff, vanishes completely. Church 

 expressly points out that no A rchi median spirals ever play a role 

 in natural phyllotaxis l ), and, therefore, the development of such 

 a spiral in a plane does not give a system of points endowed with 

 the peculiarities of a Bravais' net-plane. 



In this case the result will rather be a system of logarithmic 

 curves, to which no reasonings as brought to the fore by Wulff 

 can be immediately applied. Only more complicated and elongated 

 relations exist between these logarithmic spirals 2 ) and the helices 

 on a cylindrical or conical surface. There is no question about such 

 simple connections between the internal structure of crystals and 

 the phenomena of phyllotaxis, as suspected by the Russian scientist. 

 The essential difference between the two cases remains this, that 

 all kinds of net-planes can, with a greater or smaller probability 

 of occurrence, have the function of crystal-facets, while of all theore- 

 tically possible regular distributions of the loci of leaf-attachment 

 in plants only such are realised by nature, whose divergencies 

 approach closely to those expressed by the fractions of the principal 

 series indicated in the above, or of series deduced from it in a 

 simple way. 



Finally, in connection with the subject dealt with in the last 

 paragraphs, attention may be drawn here also to the works of T. A 

 Cook and of S. Co 1m an, 3 ) who strongly emphasize the general 

 importance of s/^>0/-structures in art and nature. In the books 

 mentioned also a great number of excellent figures are reproduced, 

 which may be of value to all those who wish to study phenomena 

 of this kind more in detail, and certainly will contribute considerably 

 to awaken the interest "of the reader for the incomparable beauty 

 of the structure of the forms in living nature and of many products 

 of plastic and ornamental art. 



1 ) Cf.: G. van Iterson Jr., loco cit., p. 1, (1907). 



2 ) A. H. Church, loco cit. 



3 ) Th. A. Cook, The Curves of Life, London, (1914); S. Colman, Nature's 

 Harmonic Unity: A Treatise on its Relation to Proportional Form, New- York, (1912). 



