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, 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.