As water freezes, crystals send
out tips; tips grow, their boundaries become unstable and new
tips shoot out from the sides. This creates a highly nonlinear ,
unstable free boundary problem, the essence of chaos. Crystals
are products of imbalance in the flow of energy from one piece
of nature to another. Their visual outcome is dependent on the
flow of the air and the flow of the water around them.
THROUGH THE THORNS
The biological world fulfills a
design shaped by natural selection. A living organism has the
astonishing gift of concentrating a 'stream of order' on itself.
The beauty in biology lies in the patterns born amidst
formlessness. Life takes its order from a sea of disorder.
VIEW FROM THE AIR
One of the most exquisite forms in
nature is the meanderings of a river bed with its rich detail.
In the past, man saw nature as something wild and hostile that
needed to be tamed. They divided the planet in an orderly
fashion according to classical Euclidian geometry where scale is
a notion so obvious that it is of little or no importance.
Euclidian geometry can be very dull while irregularity is much
more natural and exciting. With fractal geometry, these
irregularities can be studied and measured.
FROM FRACTAL TO GEOMETRIC
Euclidian geometry has controlled
the shape of our man-made world and is still the only geometry
that most people ever learn. For 2,000 years we have measured
things for their regularity, not their irregularity. With
fractal geometry, all facets of life are being re-examined and
irregularities have value as never seen before. Perhaps in the
future, our man-made world will be in better harmony with
The fungal feature commonly called
toadstool is really only a minuscule tip of an enormous
underground web of living threads called hyphae. Fungi, with
their threads penetrating the food of the forest but excised by
environmental contingency tomorrow are truly fractal organisms.
AMIDST THE MOSSY CRAGS
The broken forms of the craggy
hillside show the element of repetition so common in nature and
essential to beauty. As you look closer, the rocks become more
complicated. The mica fragments that I have used to represent
the rocks can stand alone as being fractal. Euclidian geometry
set aside such forms in nature as being formless. Fractal
geometry embraces such forms and allows us "to cope with
the range of changing dimensions of the earth. It gives
mathematics and geometry tools to describe and to make
predictions." (Christopher Scholz)
When the potato is left in
conditions conducive to growth, the eyes quickly grow into
sprouts that seek the vital elements necessary to sustain life.
If planted, these same shoots would become the roots of the
plant and other shoots would emerge to become the plant. What
drives the bifurcations is the need to survive. If the plant's
needs are not satisfied, the outcome is death and the potato
quickly becomes a nutrient for another living system.
ABOVE THE MULCH
Plants grow in forms to meet their
needs...to capture sunlight, resist wind, catch moisture and
seek nutrients from the earth. They do this with fractal
branches, fractal leaves and fractal root systems. Fractal
scaling is universal in morphogenises. Branching behaves
consistently at large scales as well as small.
When you look at the osage orange, you would immediately notice the bumps on its surface.
Looking closer, you would see the irregularities of the surface.
If you were to cut it in half, the interior shows even more
irregularities. Neither words nor concepts of Euclidean geometry
serve to describe such an object. In 1979, Benoit Mandelbrot
discovered that he could create one image in a complex plane
that would serve as a catalogue of what is known as "Julia
sets". With the help of a computer, fractal shapes can be
formed by the iteration of complicated processes, equations with
square roots, sines and cosines.
THE BUTTERFLY EFFECT
The notion that a butterfly
stirring the air today in Peking can transform storm systems
next month in New York is the exaggeration of the phenomena
technically named: sensitive dependence on initial conditions.
In science as in life, it is well known that a chain of events
can have a point of crises that could magnify small changes.
Edward Lorenze called this phenomena, "The Butterfly
Rust, as seen on this corn sheller
is iron oxide, which is the ash caused by the burning of iron
and oxygen. The reaction releases energy and creates disorder in
the surroundings. Overall there is an increase in the chaos in
the world when iron and oxygen react according to the
"Butterfly Effect". In chemistry as in physics, the
driving force in natural change is the chaotic, purposeless,
undirected dispersal of energy.
Fire is high temperature,
self-sustaining oxidation. It is also a "strange
attractor". One of the difficulties in studying flames is
the variety and the relativity of the fuels. It was hard to give
a full mathematical description of fire because we have no
control over the mixing of fuel vapor and air. The movement of
gases and the reactions which occur depend on the temperature,
which itself depends on heat evolved by the reaction, on the
loss of useful energy by movement of gases and outward radiation
, and on the retention of heat by radiation back to the reactive
As a wave rushes to the
shore, then slowly retreats back into the ocean, one can see
undulations in the sand around the remaining objects such as the
shells. The shells can be identified as "strange
attractors". The surprising, erratic behavior of the water
flow comes from a nonlinear twist in the flow of energy around
the shell. This unpredictability has been a fascination for both
physicists and mathematicians, who discovered that a chaotic
system could be stable if its particular brand of irregularity
persisted in the face of small disturbances.
SWEPT AWAY BY WIND AND
The channel below a rock in a
stream becomes a whirling vortex that grows, splits off and
spins downstream. This can be seen if you drop a feather in the
stream. This same turbulence is seen when you drop a feather in
the wind. Neither the water, the air nor the feather move
MOUNTAINS AND CLOUDS
Seen from twenty miles away, the
mountains' outline is very recognizable, yet at the same time,
it is very irregular. The closer you get, the more detail
becomes obvious. The complex systems of nature seem to preserve
their irregularities when they are seen at any distance. These
same mountains and clouds could have also been rendered on a
computer using mathematical formulas from fractal geometry.
Fractal geometry is the correct geometry for natural phenomena
such as this.
BIFURCATIONS ABOVE AND
Both the lightning and the
drainage trenches are samples of bifurcations. A bifurcation in
a system is a vital instant when the system is undergoing a flux
and being offered a "choice of orders". The internal
feedback of some of the choices are so complex that there is a
virtual infinity of the degrees of freedom and eventually it
turns to chaos.
The artist David Hockney said,
"With a fractal, you look in and in and in and it always
goes on being fractal. It's a way toward a greater awareness of
unity." Looking through these pipes is like looking into
space. They are colored according to the wave length of the
light spectrum. In astronomy, stars , nebulae and galaxies are
perceived by the light they emit and analyzing the spectrum of
light is the main way in which astronomers draw conclusions
about the nature of celestial objects.
BIFURCATIONS TO THE MOON
Each split in a bifurcation
represents a choice. Mankind has been given choices since the
beginning of his existence. Some of the choices have been
beneficial to our planet and some have had disastrous effects.
Looking toward the future, the noted biologist, Rene Dubos
wrote; "The behavior of our societies toward the earth must
be based on a new kind of ethics, embracing the land as well as
animals and plants. In principle, we all have the same rights,
but in practice, we can enjoy these rights only to the
extent hat the earth is maintained in a healthy
In terms of aesthetic values, the
new math of fractal geometry brought hard science in tune with
the peculiarly modern feeling for untamed, uncivilized,
undomesticated nature. We find that the application of human
reasoning in it most refined and formalized sense to mathematics is full of paradox and uncertainty. However, there
will always be truth that lies beyond rational scientific
inquiry and logical reasoning. It is called mysticism. Some of
the greatest revelations in science have come in the way of a
mystical experience. Could the cosmos be controlled by a
super intelligence who guides its evolution through quantum
processes? How we have become linked into this cosmic dimension
is a mystery, but the linkage cannot be denied.