Animation of a Blooming Flower Using a Family of Complex Functions ${M_{\zeta ,\alpha }}\left( z \right) = \exp \left( { - \alpha \frac{{\zeta + z}}{{\zeta - z}}} \right)$

Abstract

Recently, Kim et. al.(1992) addressed the properties of a family of complex functions M _{ζ, α}(z) = exp$( - \alpha \frac{{\zeta + z}}{{\zeta - z}})$ where α > 0 and |ζ| = 1. When Newton’s method is applied to solve M _ζ,α (z) − 1 = 0, the basins of attraction for its roots show flower-like self-similar structures which vary according to the value of α. From an artistic point of view, we explore those self-similar strucures for an animated sequence of flower blooming by extending M _ζ,α (z) for α ≠ 0.