Symmetry 151 



any other aspect of the problem of organic symmetry. The external sym- 

 metry of an upright stem is typically radial and often very regularly so. In 

 the simplest cases, as in certain mosses such as Fontinalis, this is related to 

 the activity of a three-sided apical cell, the segments cut off from its three 

 sides giving rise to three rows of leaves. In higher plants, however, leaf 

 arrangement is not related to meristematic structure. 



In stems with opposite leaves, at successive nodes the leaf pairs rotate 

 through 90°. This decussate phyllotaxy thus shows four rows of leaves 

 along the stem. More frequently, phyllotaxy shows a spiral character. 

 Sometimes this is manifest, even in opposite-leaved types, by a twisting of 

 the whole axis so that members of successive pairs are a little more than 

 90° apart. Spirality more commonly expresses itself, however, in the ar- 

 rangement of so-called "alternate" leaves. These are rarely exactly alter- 

 nate but are so dispersed that a line connecting the points of attachment 

 of successive ones follows a regular spiral course around the stem. The 

 fact of this spiral and the various types in which it is manifest have for 

 many years attracted the interest of botanists and mathematicians. Many 

 of the discussions and speculations that have centered about the phyllo- 

 tactic spiral are of no great significance for morphogenesis. Some are 

 highly theoretical or even almost mystical. The developmental origin of 

 the various types of phyllotaxy, however, is an important morphogenetic 

 problem, and a knowledge of the factors involved may contribute to an 

 understanding of the origin of organic form. 



Goethe was greatly attracted by the spirality of leaf arrangement and 

 made it the basis of one of his theories. Charles Bonnet ( 1754 ) in the 

 middle of the eighteenth century discussed the spiral structure of the pine 

 cone. It was the work of Schimper (1836) and Braun (1831), however, 

 that established the study of phyllotaxy on its modern basis. Various ex- 

 planations of the origin and significance of spiral leaf arrangement have 

 been proposed, and the literature of the subject is extensive. No compre- 

 hensive review of it is available, though the earlier literature has been 

 surveyed by C. de Candolle ( 1881 ) . For the more important ideas the 

 reader is also referred to the works of the brothers Bravais (1837), Hof- 

 meister (1868), Wright (1873), Schwendener (1878), Schoute (1913, 

 1914), Church (1920), Hirmer (1922), Crow (1928), Goebel (1928), 

 Snow and Snow (1934), D'Arcy Thompson (1942), Plantefol (1948), 

 Wardlaw (1949a), and Richards (1950). 



Spiral phyllotaxy is not an example of symmetry in the strict sense since 

 planes of symmetry, in the crystallographic meaning of the term, are 

 absent. The leaves do have regular positions along the axis, however, with 

 reference to each other, and these, under proper analysis, can be ex- 

 pressed in terms of geometrical symmetry. The spiral formed by the 

 points of attachment of successive leaves— the genetic or developmental 



