For example:

            ART MATRIX  PO 880  Ithaca, NY  14851-0880  USA
                   (607) 277-0959, Fax (607) 277-8913
            'The Paths of Lovers Cross in the Line of Duty.'
                             THE WHY IS GOD
                 Copyright (C) 1990 Homer Wilson Smith
                          All Rights Reserved
     One of the great discoveries of the 19th century is that
everything in the physical universe (except maybe consciousness)
follows mathematical laws.  All of the relationships between objects
in space and time can be measured in terms of numbers, and all of the
causal influences between them can also be described in terms of
equations and mathematical relationships.
     Apparently God was a mathematician first, and a purveyor of
damnation second.
     Thus the study of math becomes of paramount importance in the
process of understanding and controlling the universe around us.
     It goes without saying that if the universe is controlled by
mathematical laws then the qualities and conditions found in those
mathematical laws will be manifest in the universe that is controlled
by them.
     Thus the study of pure mathematics is beneficial to us all, for
as we gain deeper insight into the nature of the math we use, we are
more able to find these same things manifested in the exterior
physical universe.  If it were not for the mathematician making the
discovery on the purely theoretical level, these manifestations would
go forever unnoticed or unexplained in the physical universe.
     It is quite apparent that the mathematics that governs the macro
scale world is based mostly on the simple force laws of Newton, Gauss,
Maxwell and Co. namely the 1/R*R laws that relate to the propagation
of force lines through out the surface of a surrounding sphere.  This
one idea alone brought science into the 20th century and with good
reason.  It is basically correct enough.
     We find this simple mathematical equation expressed not only in
the laws of gravity, but also in electromagnetism and electricity
without which our world would still be cold and dark.
     Thus it would seem that if the mathematicians are to stay ahead
of the game they should spend some time studying the ins and outs of
the 1/R*R laws and their related equations, such as derivatives and
     For the longest time, the missing part to this whole story was
the concept of iteration.  For example in the force law F = GMM/R*R we
see Force expressed as a function of mass and position, but we do not
see POSITION expressed as a function of mass and POSITION.  Thus the
use of the F=GMM/R*R law is strictly on an evaluation basis.  You find
the inputs and you get one output.  There is no place to stick the
output back into the input because the Force term is not in the input,
and the input terms are not in the output.
     Most of physics went this way for eons although it need not have.
People have long known of differential equations wherein the present
CHANGE in state is a function of the present state.  But really it
took some very bright people looking into living biological systems to
start coming up with equations that expressed the PRESENT STATE
directly in terms of the prior PRESENT STATE (just a moment before!)
     This of course gives the needed conditions for ITERATION, or
repeated evaluation of the equations involved.
     It is quite possible to then go back to all the old familiar
stuff, like planets moving around the sun, and REWRITE the equations
in terms of iteration, wherein the present position is a function of
past position.  In fact the rocket guys do it all the time, as simple
algebraic solutions to multi bodied planetary systems are either
impossible or difficult.  Actually the use of iterated differential
systems has been with us for a long time.
     So if we go back and look at the world in this new light, all of
a sudden we find we have a plethora of iterated expressions with
little to no understanding of what their MATHEMATICAL nature is.
     So once you start looking into the matter you find a wonderful
world of structure and 'nature' in the subject.  You find there are
fixed points, and period cycles, and chaotic cycles, there are
repulsive and attractive points with basins of attraction, and there
are open and closed Julia Sets which give rise to all sorts of
fantastic Mandelbrot Sets.  One finds that all these things depend on
the values of the parameters involved in the equations, and that all
non linear equations manifest these things in one form or another.
     One also finds that for many of them the various boundaries
surrounding basins of attraction or what have you, are quite FRACTAL,
an incidental but very important new thing to know about the matter.
     By fractal we mean infinitely convoluted, not a straight line in
sight, no matter how small.
     Not as long as the equations which manifest such things are key
causal descriptors in the physical universe in question.
     First you find out what is true MATHEMATICALLY, and THEN you find
out where those things are applied in the real universe.  I assure you
these things do not exist in a vacuum of vanity.  If they exist
mathematically, they ARE applied.
     Now the important equations of the physical universe have not
changed, Y = X*X + C is still Y = X*X + C, but the CONCEPT of what you
are going to DO with it has changed, from simple evaluation which
produces parabolas, to multiple iteration which produces Mandelbrot
Sets and Julia Sets.  And all of a sudden this entire new landscape of
structure and nature appears to unfold as if out of nowhere.
     It may seem like it is new, but you and I know it has been there
all along.
     The physical universe has been using these same equations to
march its wheels of time forever, and you can be sure that if these
equations show some inner structure or 'nature' then that same nature
will be manifested outwardly in the physical universe expression of
that equation.
     The physical universe is basically an iterated system, so
actually it is surprising we have made the progress we have, using
only simple evaluation.  The equations have been around forever.  The
physical universe has been USING them almost forever.  The equations
have as part of their nature things like fixed points, period cycles,
chaotic cycles, basins of attraction, etc., so you can be sure all
these things are manifested in the physical universe INCLUDING
     To say therefore that fractals have nothing to with anything and
have not explained or proven useful in our understanding of the
universe is more a statement about the people who are working with
fractals rather than a statement about the pertinence of fractals to
the world at large.
     Fractals are so pertinent to the universe no one can see it yet.
     Long time ago, they thought math did not pertain either.  The
'why' was God.
     The 'why' may still be God, but if it is, then clearly God is a
mathematician of significant merit, and no doubt a fractal enthusiast.