Multidimensional generalization of phyllotaxis

Abstract

The regular spiral arrangement of various parts of biological objects (leaves, florets, etc.), known as phyllotaxis, could not find an explanation during several centuries. Some quantitative parameters of the phyllotaxis (the divergence angle being the principal one) show that the organization in question is, in a sense, the same in a large family of living objects, and the values of the divergence angle that are close to the golden number prevail. This was a mystery, and explanations of this phenomenon long remained “lyrical”. Later, similar patterns were discovered in inorganic objects. After a series of computer models, it was only in the XXI century that the rigorous explanation of the appearance of the golden number in a simple mathematical model has been given. The resulting pattern is related to stable fixed points of some operator and depends on a real parameter. The variation of this parameter leads to an interesting bifurcation diagram where the limiting object is the SL(2;Z)-orbit of the golden number on the segment [0,1]. We present a survey of the problem and introduce a multidimensional analog of phyllotaxis patterns. A conjecture about the object that plays the role of the golden number is given.