Shape Limit in Triangular Spiral Tilings

Phyllotaxis is the study of arrangements of leafs and florets. The topology of triangular spiral (multiple) tilings with opposed parastichy pairs is intimately related to the phyllotaxis theory and continued fractions. It is shown that, if the divergence angle of the genetic spiral is given as a quadratic irrational and fixed, then the limit set of the shape parameters of triangular tiles, as the parastichy numbers tend to infinity, is a finite set. In particular, the limit is the golden section if the divergence angle is `ultimately golden'.