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2.7 Selfsimilarity : A Signature of Identical Iterative Growth Process
Selfsimilarity underlies all growth processes in nature. Jean (1994 References ) has emphasized the selfsimilar geometry of botanical elements. Selfsimilar structures are generated by iteration (repetition) of simple rules for growth processes on all scales of space and time. Such iterative processes are simulated mathematically by numerical computations such as Xn+1 = F(Xn ) where Xn+1 , the value of the variable at (n+1) th computational step is a function F of its earlier value Xn . Mathematical models of real world dynamical systems are basically such iterative computational schemes implemented on finite precision digital computers. Computer precision imposes a limit (finite precision) on the accuracy (number of decimals) for numerical representation of X. Since X is a real number (infinite number of decimals) finite precision introduces round-off error in iterative computations from the first stage of computation. The model iterative dynamical system therefore incorporates round-off error growth. Computed growth patterns exhibit selfsimilar fractal structure which incorporate the golden mean (Stewart, 1992aReferences). The new science of nonlinear dynamics and chaos seeks to understand the physics of such selfsimilar patterns in computed and real world dynamical systems.