Phyllotaxis and Farey sequences

This appendix is a self-contained presentation of the basic concepts of phyllotaxis in a more general setting which has led to the general form of the fundamental theorem of phyllotaxis. The theory presented here (Jean, 1988a) concerns lattices from a general point of view, but it can be particularized to give results known in phyllotaxis. The discussion below will allow the reader to better grasp the concepts of visible and of opposed parastichy pairs.

Figure A4.1 shows a regular point lattice similar to the ones obtained in the study of phyllotaxis. The lattice is made with a divergence d = 85°, or 85/360 = 17/72. This is the abscissa of point 1, a number always taken to be smaller than 1/2. The coordinates of point n = 1,2,3,…, are (nd–k, nr) where the riser is the ordinate of point 1, and k is any integer. We concentrate here on the points whose abscissae are between or equal to –0.5 and +0.5, included in an infinite vertical strip denoted by S (the past analyses, e.g., Coxeter, 1972; Adler, 1977a; Marzec & Kappraff, 1983, are performed in the region of the lattice of points having abscissae between 0 and 1). In this sector of the lattice the coordinates of point n are (nd–(nd), nr), where (nd) is the integer nearest to nd.