2.7 Selfsimilarity : A Signature of Identical Iterative
Selfsimilarity underlies all growth processes in nature. Jean
) 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, 1992a References).
The new science of nonlinear dynamics and chaos seeks
to understand the physics of such selfsimilar patterns in
computed and real world dynamical systems.