More Fractal Geometry/Chaos Theory

[Juan Cires Martinez]

Having found five minites totally unexpectedly, I'll elaborate a tiny
bit about the difference between Fractal Geometry and Chaos Theory.

Chaos Theory is the study of dynamical systems that exhibit chaotic
behavior.  Dynamical systems are defined through equations that express
their evolution in time.  If these equations are non-linear, chaotic
behavior may occur.  This chaotic behavior is defined by extreme
sensitivity to initial conditions, that is, a small change in initial
conditions of the system (the flapping of wings of the proverbial 
butterfly in Manchuria) induces dramatic changes in behavior (the
proverbial storm).
 
Fractal Geometry studies geometric objects whose dimension is not
integral -- they are not exactly points, lines, surfaces ...  These
objects exhibit self-similarity, that is, they look the same at
different scales.  They are ideal mathematical objects defined in the 
same way as circles, lines, spheres and polygons.  And some of these 
objects arrise in the study of chaotic systems.  One can apply the
concept of fractal dimension to the study of natural objects such as
coastlines, with limitations, since they are not infinite objects.

Saludos, my five minutes are up, Juan.