Brian Sullivan


by Sally Morem, helium.com
Process rules our universe. The Big Bang brings forth energy, matter, space and time. Quarks form subatomic particles, which in turn form atoms. Atoms bind themselves together into molecules. Molecules assemble themselves into crystals and living matter, which form complex relationships with one another.


French mathematician and astronomer Laplace believed that our jangling, booming, buzzing confusion of a universe was, at least in theory, fully comprehensible. If we could imagine a consciousness great enough to know the exact locations and velocities of all the objects existing in the universe at the present instant, as well as the exact nature of the basic forces of nature, then there could be, in principle, no secrets from that consciousness. It could calculate anything about the past or the future according to strict laws of cause and effect.


Laplace was profoundly wrong. In order to carry out those calculations, that consciousness would have to know the exact state of the entire universe at that instant. But, given the immense size of the universe and the inherent fuzziness of the subatomic realm implied by quantum physics, that is precisely what no one will ever be able to do.


A powerful intelligence could only specify approximations of approximations. But, in a universe where so many systems are poised on the edge of chaos, approximations just won't do. The error rate would multiply quickly as events follow events after the predictions were made. As a result, the universe would quickly veer off its predicted path. No matter how many errors were accounted for, new ones would pop up and stymie any accurate prognostication.


The universe does follow strict cause-effect rules. It is deterministic. But it is also inherently unpredictable.


Stuart Kauffman, a leading researcher in complexity theory, explained one aspect of the universe's self-organized criticality with a simple illustration: The central image here is of a sand pile on a table to which grains of sand are added at a slow, constant rate. Eventually, as the sand piles up, avalanches begin. What one finds are lots of small avalanches and few large ones.


Kauffman emphasized the central fact of this system by pointing out what happens at criticality. One additional grain of sand can tip the system in one of three directions: toward a small avalanche, a large avalanche, or no avalanche. Additionally, an observer can never tell before the instant the grain of sand is dropped which way the system will go. This is called bifurcation. A system can apparently "jump" in one of several directions without being constrained by a simple linear cause-effect relationship. Bifurcation is ubiquitous in systems that ride close to the edge of chaos.


How can we comprehend such bumpy, branching, flowing, milling, irregular systems? In comparison, the neatness of Euclidean geometry and Newtonian physics seems the essence of logic, and yet they're wholly unable to model these real life examples of complex systems teetering on the edge.


Belgian mathematician Benoit Mandelbrot asked the question a different way: Why is geometry often described as barren and dry? He concluded that ordinary geometry couldn't describe the shape of clouds, mountain ranges, a coastline, or a tree, the kinds of things humans find aesthetically pleasing. Thus, ordinary geometry left most humans cold. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth. Nor do lightning or river systems travel in straight lines.


Nature demonstrates not simply a higher degree of complexity, but an altogether different kind of complexity. The number of distinct scales of length and pattern is for all practical purposes infinite.


Mandelbrot furnished us with a much more robust model of nature, a branch of mathematics he named fractal geometry. "Fractal" from the Latin "fractus," meaning "broken."


One of Mandelbrot's mathematical forebears, Gaston Julia, defined many such irregular shapes during his work during World War I. They are now referred to as Julia sets. One of his contemporaries described these shapes as incredibly varied: "Some are of fatty clouds, others are of skinny, bushy brambles. Some look like the sparks which float in the air after fireworks have gone off. One has the shape of a rabbit. Lots of them have seahorse tails."


Check out some of these amazing shapes here:


Mandelbrot took an important step toward developing the logic underlying fractal geometry when he discovered that the same mathematical patterns could be detected in many time frames and scales in such strikingly different systems as the stock market, population growth and decline, weather fronts, and coastlines.


For example, when he studied coastlines closely, he realized that no one knew or could describe a means by which the true length of any coastline could be measured. Why? Because at each level of magnification, they retain the same complex assortment of bays and peninsulas. Which scale is best - miles, yards, feet, inches? A surveyor would be in a quandary. All measurements would be equally true, yet each would lead to a vastly different result.


Also, this scalar property of coastlines would make it particularly difficult to pinpoint the true nature of a specific portion of them without an outside reference. For instance, you can't tell if you're looking at a photo of the White Cliffs of Dover or an eroded outcropping of chalk a few feet high without, say, a human being standing next to it.


Mandelbrot's fractals consist of just these kinds of nested, repeating patterns. When you see a fractal built up from tinier and tinier triangles, its self-referential nature is quite unmistakable. The overall structure of triangles within triangles is repeated with some variations (or none at all) all the way to infinity.


We see fractal aspects of repetitive branchings and clumpings in a wide variety of natural objects and phenomena such as trees, lightning, and river watersheds. We see other aspects of fractal self-reference when water crystallizes in the winter sky, forming a growing tip, becoming unstable, and then sending branches out to form a snowflake.


Similar patterns can be seen in a clear glass of super-saturated sugar water. Add a string and watch sugar crystals grow. Window panes are painted with a forest of fractal frost on a cold winter's day. Create fractals on a stovetop. Boil water and vegetable oil in a saucepan. Let it cool. Note how oil puddles into fractal-like circles on the surface of the water. Sometimes they look like Julia sets as small archipelagoes of circles float next to larger circles.


When a chemist heats dyed viscous fluids that mix as poorly as oil and water, they begin to get wavy, tendrils form, which draw on themselves over and over again. Loops and marble-like whorls pinwheel, creating images of stunning beauty.


Jupiter's Great Red Spot is the most spectacular example of non-mixing fluids at work. This vast oval near the equator is actually a giant hurricane bigger than the Earth. It has lasted for centuries, never moves with respect to the other bands of dense hydrogen and helium of the planet, and despite the jostling, retains its overall characteristic shape.


The Spot was a mystery for years after it was first imaged. What could that apparently enormous structure be? Wouldn't such a thing collapse in on itself? Using fluid equations, scientists modeled the Spot on computers and found that it was indeed a self-organizing system composed of chaotic atmospheric flow. If you look closely at movies constructed from successive photos taken by flyby missions past Jupiter, you will note its fractal nature. Eddies within eddies constantly break off and other eddies join it.





Scientists model many other aspects of nature by testing simple equations on computers and plotting the results on Cartesian graphs. One of the most famous examples of this is the building of a fractal fern. The equations are loaded and the basic fern structure takes shape on the monitor. Leaves are added as the stem grows. Note the tinier leaves growing on tinier stems within the larger structures, and then yet even tinier structures filling in as fern leaves grow their own little leaves. The fern becomes astonishingly lifelike in form surprisingly quickly.


Watch a fractal fern grow here:


This reminds me of a poem that I though was written by Ogden Nash, but which was actually penned by Augustus De Morgan. It's a refinement of an even earlier poem by Jonathan Swift:


Great fleas have little fleas

Upon their backs to bite 'em

And little fleas have lesser fleas

And so ad infinitum


Scalar self-referential systems were familiar to scientists and intellectuals for centuries. But as computer graphics became ubiquitous in the Seventies and Eighties, scientists have had much more success modeling them.


Some fractals are built up over time on the computer's Cartesian graph program in an apparently haphazard way. As the computer runs the equation, solutions are posted on the graph in the form of dots. As the dots grow in number, a pattern emerges. The dots cluster around one, two, or more areas of the graph, areas mathematicians call Strange Attractors. Out of very simple equations, complexity grows.


Here are some startlingly beautiful examples of Strange Attractors:


Consider the most striking form of fractal: the Mandelbrot Set. Mandelbrot generated a collection of points on a graph by taking a series of complex numbers, squaring each of them, adding the original number to each, and squaring them again and again. If the number remains finite after many such iterations, it remains in the Set and is plotted on the graph.


The resulting shape is composed of successful solutions to what is truly an extraordinarily simple equation. Paradoxically or not, it's the most complex and beautiful object in mathematics. When software displays the Set in false colors, a viewer can be excused for concluding that the Set is a Set of infinities.


Software displays segments of the Set along its rich, complex edges, under greater and greater degrees of magnification. As each tiny portion is magnified, more detail emerges. Mathematicians insist that the Set holds the entire set of Julia sets in infinitely many places in its infinite numbers of levels of organization. An eternity would not be long enough to explore the Set's many hidden splendors.


Watch as "...its disks grow spikes of prickly thorns, spirals and filaments curl outward and around, bearing bulbous molecules that hang infinitely variegated like grapes on God's personal vine."


This lovely description of the Mandelbrot Set is taken from James Gleick's book, "Chaos," which I commend to anyone interested in a more in-depth exploration of fractals and chaos theory. Explore the Set in detail at this site:


Fractals aren't limited to two dimensions. Fractal geometry explores the dimensions between the traditional geometric concepts of the zero-dimensional point, the one-dimensional line, the two-dimensional plane, and the three-dimensional solid.


The dimensionality of fractal shapes can be calculated: a 2.636-dimension sponge, for example, is quite possible. A fractal sponge is generated by dividing a cube into nine cubes and removing the middle one, then subdividing each cube further, and removing the middle one at each turn. The process continues to infinity.


Fractals also include the nearly dimensionless particles known as fractal dust. They are generated by taking a line, dividing it into thirds, removing the middle third, dividing each remaining segment into thirds, removing in turn their middle thirds, and so on, ad infinitum. What remains is a ghostly faint line of points, virtually there or not there.


And then there is the famous Koch Curve or "fractal snowflake." Take an equilateral triangle, divide each side so that they extend outward, creating a hexagon. Then, divide each side again, so that the form begins to look like a snowflake. Continue dividing the sides of the snowflake forever, creating a boundary that encompasses a finite area but is itself infinite in length. The iterative process of generating a Koch Curve will look very much like the "fractal movie" posted on this site:


Just as Koch Curves can pack infinite length into a comparatively small area, fractal-like biological systems can pack an enormous amount of area into relatively small volumes.


Consider the circulatory system with its miles of veins and arteries, or the nervous system with its enormous arrays of nerve cells in the body and brain. Acres of bronchial tissue layer themselves in the lungs. Muscles cells, too, show signs of fractal organization.


With the enormous numbers of fractal patterns found in nature and in the human body, it's apparent that this is one of the most efficient patterns in which matter may arrange itself.


But, fractal organization isn't limited to nature. Software based on fractal geometry can be used to model complex patterns and events in the human world as well.


Mandelbrot realized that when he discovered that the same equations he used to derive the scalar organization of changes in stock market prices could also be used to model weather patterns, he knew he had stumbled onto an important insight. Similar equations can be used to analyze election returns, traffic patterns, population densities and distribution patterns, migration patterns, rates of information production and distribution, and the patterns in which job specializations and sub-specializations fall.


The clumpiness of buildings in a modern city's downtown skyline strongly resembles fractal formations such as fractal sponges. So do the networks within networks within networks that build upon one another to form computer chips, computer motherboards, computer networks, and the Internet.


Perhaps all of evolution - physical, biological, and cultural - is in some sense a fractal. If so, fractal geometry would have much to say about the spontaneous order of the universe and how it expresses itself in countless ways in our lives.


Stuart Kauffman reminds us: "As the scale of our activities in space and time has increased, we are being driven to understand the limited scope of our understanding and even our potential understanding."


As we learn how truly complex apparently simple things really are and as we learn what "complexity" really means, we are beginning to learn to accept our own human limits. We will never achieve the consciousness described earlier that can see all things and know all things in the universe. And now we know why.


Puncturing mortal hubris is not the least of the good insights we've received by studying fractals. We are all part of the process of fractal formation, created by it, creating it. We participate. We don't command. Perhaps this is as good a definition of evolutionary processes as any.


In the beginning was the Word - the Law. Fractal. The rest follows and we participate.



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