EIGENSHAPE ANALYSIS OF A CUT-GROW MAPPING FOR TRIANGLES, AND ITS APPLICATION TO PHYLLOTAXIS IN PLANTS

Phyllotaxis, the regular arrangement of leaves in plants, has been extensively studied geometrically and through theories of mechanism. Recent observations of the biophysics of cell growth and division at the plant apex during leaf formation lead to a "cut and grow" algorithm, in which a newly formed leaf is first cut off from the meristem and then grows rapidly, deforming the adjacent tissue to a new configuration. The properties of the sequence of shapes constructed by a cut-grow map of triangles are discussed. The cut-grow map excises a triangular segment of a triangle and dilates the newly formed edge. In shape space, the cut-grow map is equivalent to a reflection followed by inversion. Each map has a conjugate pair of "eigenshapes." In the limiting case when the eigenshapes are equal, the shapes converge with successive iterations at rate 1/n. However, an arbitrarily small perturbation from the eigenshape can initialize a divergent sequence of arbitrary length, before the shapes begin to converge. When the eigenshape are unequal, the shapes at successive iterations traverse a circle, with center offset from both eigenshapes, and with period pi/theta, where 2-theta is the angle between the conjugate pair of eigenshapes. The phyllotactic pattern generated by traversals of the circle is a flattened version of regular phyllotaxis. It is concluded that an apex "programmed" with a given cut and grow algorithm will generate any given pattern of spiral phyllotaxis.