24 Square Feet of Fractals

TWENTY-FOUR SQUARE FEET OF FRACTALS

An Exploration of the Science of Chaos through the eyes of an artist

What is a fractal? A fractal, in mathematics, is a geometric shape that is complex and detailed in structure at any level of magnification. Fractals are an outcropping of the mathematics of chaos which relates to random behavior occurring in a deterministic system such as the Laws of Nature.

Albert Einstein spoke in reference to this new way of thinking," You believe in a God that plays dice, and I in complete law and order." The science of chaos is forcing scientists to rethink even the most fundamental ideas about the way in which the universe behaves. It may not be the end all solution but it will cast a shadow of doubt on the numerous systems already in place and lead scientists to new answers.

The images in this exhibit portray some of the natural phenomena that is being measured by fractal geometry. These images also express some of the ideas being explored in this new science of chaos, such as predicting weather, ecological questions, the origins of life, the origins of the universe and mysticism. Immanuel Kant wrote that "God has put a secret art into the forces of Nature so as to enable it to fashion itself out of chaos into a perfect world system.

THE PAINTINGS

All of the paintings in this exhibit are watercolors and all are copyrighted

THE BIRTH OF A FRACTAL

Life seems to have originated in whatever were the primordial ancestors of modern bacteria. Chemical systems that became biological systems, these first beings would have metabolized and incorporated energy, nutrients, water and salts into their developing selves.

THE CLASSICAL NAUTILUS

The spiral shows up in motifs all over the world. Carl Jung said that such images are archetypes or universal structures in the collective unconscious of mankind. Could such a collective wisdom be expressing its intuitions of the wholeness within nature, the order and simplicity, chance and predictability that lie in the interlocking and unfolding of things?

THE PRIMAL SOURCE

Life resembles a fractal. The fractals of life are cells, many celled organisms, communities of organisms and ecosystems of communities. The motivating force behind living fractals is the reproduction of cells.

IN OUR GARDEN

This garden could be in our back yard or it could be growing on a reef in the ocean. It is the garden of the mind. In the cerebellum, these spectacular cells act as the control and coordination center for our movement. At the base of each of these cells is the axon which transmits the main signals from the cerebellum. The cerebellum carries within it "programs" of previously learned movements which are activated by the cerebral cortex.

FERN MENAGERIE

Much of nature has a hidden order. There is only so much information in the spore that encodes one fern. Using a computer, Michael Barnesly, a mathematician, played what he called the "Chaos Game". By placing random dots on the computer screen over and over, they unfolded into the shape of a fern and other natural arrangements with greater sharpness as the game proceeded. All the necessary information was encoded in a few simple rules. It was then determined that other natural patterns could be generated and measured with fractal processes on the computer.

EXPANSIVE BEAUTY

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 nature.

MUSHROOM CLUSTER

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

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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)

POTATO JUNGLE

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.

OSAGE ORANGE

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 Effect".

RUSTED ANTIQUITY

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

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 zone.

STRANGE ATTRACTORS

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 WATER

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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 independently.

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 BELOW

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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.

FRACTAL DIMENSIONS

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 state."

MYSTERIOUS FRONTIERS

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.

Alice Webb