A Model of Phyllotaxis

Phyllotaxis is the study of the symmetrical arrangements of plant organs, and most often associated with the Fibonacci series of numbers. The present work points out that the well known Helmholtz equation of mathematical physics correlates all of the well known patterns in one simple algorithm, involving two integerspandq:p>q>0 accounts for spiral patterns, including “jugate” patterns,p=qgives the alternating whorl patterns, whilep>0,q=0 gives superposed whorls. In spiral patterns, the integersp,qunderlie the larger and more usual integersm,n. The integer number of leavesNin a pattern is given in all cases by the expressionN=(p2±q2)/J, whereJis an integer giving the number of leaves on a single (e.g. stem) level. A biochemical origin of the algorithm is suggested.