Phyllotaxis, or the properties of spiral lattices. - I. Shape invariance under compression

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Abstract
From the crystallographical point of view, phyllotaxis can be identified with the study of spiral lattices. In this paper, we devote our attention to plane lattices of points. Through a conformal transformation, one gets a lattice of points aligned along a logarithmic spiral. Centuries ago, one recognized the central role the Fibonacci sequence plays in phyllotaxis, as well as the golden ratio τ = 1/2 (1 + √5): the divergence angle (the angular distance between two consecutive points of the spiral) equals 2 π ι-1. We define some class of divergence angles more general than the « golden divergence » 2 π ι -1. This class insures a peculiar shape invariance of the lattice with respect to the change of the plastochrone ratio (or relative rate of growth of the logarithmic spiral)