Overview of Mathematics in the 20^th Century

 

For almost a quarter of a century now mathematics and the natural sciences have been riding a wave, which, in its power, creativity and expanse, has become an interdisciplinary experience of the first order.

 

This wave has also been touching distant shores far beyond the sciences for some time now.  Never before have mathematical insights, usually seen as a boring subject, found such swift acceptance and created so much excitement in the public mind. 

 

Fractals and chaos have pretty much captured the attention, enthusiasm and interest of people worldwide.  To the casual observer, the color of their fundamental structures and their beauty and geometric form mesmerize the visual senses as not much else in mathematics has done. 

 

The student will bring mathematics out of the realm of ancient history into the twenty first century.  While to the scientist, fractals and chaos offer a rich environment for exploring and modeling the complexity of nature.

 

Mathematics in the twentieth century has reached the point of being recognized as such a phenomena.  The twentieth century proved to have spawned many great mathematicians.

 

In previous years as well, humans have noted that small causes could have large effects and that it was hard to predict anything for sure. In some systems small changes of initial conditions could lead to predictions so different that prediction itself becomes useless is what had caused a stir among scientists.

 

Toward the later part of the 19th century a French mathematician Jacques Hadamard proved a theorem.  It was a theorem on the sensitive dependence on initial conditions about the frictionless motion of a point on a surface.

 

That theorem was proved through an experiment that was about three billiard balls and why you can't predict what three of them will do when they careened off each other on the table. A French physicist Pierre Duhem knew and understood the significance of Hadamard’s theorem.

 

In 1906, he published a paper that made it clear that prediction was "forever unusable" because of the necessarily present uncertain initial conditions in Hadamard's theorem. The man who was recognized as the Father of the Chaos theory, Henri Poincere (1854 – 1912) didn’t note or even notice Hadamard’s theorem.

 

Poincare published in1908 a paper called “Science Et Methode.  It contained one sentence concerning the idea of chance being the determining factor in dynamic systems because of some factor in the beginning that we didn't know about.

 

These ideas went unnoted because quantum mechanics had disrupted the whole physics world of ideas.  This was because there were no tools such as ergodic theorems about the mathematics of measure.  Also because back in those days there were no computers to simulate what these theorems prove.

 

As mention previously, the one that is responsible for the study of Chaotic dynamics is the French mathematical physicist Poincaré.  In the late 1800's, Poincare tried to solve the celestial three-body problem (sun, planet and moon) experiencing mutual gravitational pull.

He was attempting to solve the age-old problem or question of whether the solar system was stable forever or if some planets would eventually fly off.  Poincaré started to look at the problem from a different point of view to try to answer this question.

Rather than following all the trajectories of every orbit, he instead worked out a geometric approach to investigate the problem. It was from this approach he was able to show that the three-body problem has complicated orbital dynamics, which we now call chaos.

Pierre Fatou

 

Poincare wasn’t the only great mathematician to come out of the twentieth century.  Pierre Fatou entered the Ecole Normanle Supérieure in Paris in 1898 to study mathematics and graduated in 1901.  After that he then decided that the chance of obtaining a mathematics post was so low and he applied for a position in the Paris Observatory.

Fatou continued to work on mathematics for his thesis since he had been appointed to the astronomy post. His thesis was submitted in 1906, which was on integration theory and complex function theory.  Fatou showed that if a function is Lebesgue integrable, then radial limits for the related Poisson integral exist almost everywhere.

Fatou’s result led to generalizations by Privalov, Plessner, and Marcel Riesz. His incomplete solution also made a big contribution to finding a solution to the related question of whether conformal mapping of Jordan regions onto the open disc can be extended continuously to the boundary. Fatou received his doctorate for this important work in 1907.

Fatou’s book gives a beautiful historical account of the global theory of iteration of complex analytic functions. Fatou enters this history in a rather complicated way.  The book does a wonderful job in telling an interesting episode in the history of mathematics.

In 1915, the Académie des Sciences in Paris provided the topic for its 1918 Grand Prix. The topic was for a study of iteration from a global point of view and there was a prize involved.

Mathematicians such as Appell, Emile Picard, and Koenigs had put forth the idea to the Académie des Sciences.  This was because they were hoping for developments of Montel’s concept of normal families.

In 1917, Fatou wrote a long memoir, which did indeed use Montel's idea of normal families to develop the fundamental theory of iteration.  Although it is not known for certain that he was intending to enter for the Grand Prix.  It seems almost certain that he did the work with that in mind.

It is not surprising that since there was a prize involved that another mathematician would also work on the topic.  Indeed another great mathematician named Gaston Julia came up with a long memoir developing the theory in a similar way to Fatou. Yet, the two chose different ways to go about it.

Toward mid-part of 1917 Julia enclosed his results in sealed envelopes and deposited them with the Académie des Sciences.  Fatou, on the other hand, published a statement of his results in a note in the December 1917 part of Comptes Rendus. Even though they both tried a different route, it later became evident that they had discovered very similar results.

Julia wrote a letter to Comptes Rendus concerning priority, which was published on 31 December 1917. Julia had asked the Académie des Sciences to inspect his sealed envelopes and Georges Humbert had been asked to take care of the task.

Not to far after Julia wrote that letter, the 31 December 1917 part of Comptes Rendus Georges Humbert has a letter reporting on Julia’s papers. It was pretty much certain as a result of these letters Fatou did not enter for the Grand Prix that the award went to Julia.

But it wasn’t a total loss for Fatou for he did not lose out completely. Even though he had not entered for the prize, the Académie des Sciences gave him an award for his outstanding paper on the topic.

The title “astronomer” was given to Fatou in 1928 and, as an astronomer; he also made contributions to that topic. Using existence theorems for the solutions to differential equations, Fatou was able to prove thoroughly certain results on planetary orbits, which Gauss had suggested by only verified with an intuitive argument.

Fatou also studied the motion of a planet in a resistant medium with the intention of explaining how twin stars would form with the capture of one moving in the atmosphere of the other.

Work on some Fatou’s important work has been mentioned, but it should be shown that he had other great works, for example, his work on Taylor series where he examined the convergence and the analytic extension of the series. Perhaps Fatou's most famous result is that a harmonic function u > 0 in a ball has a nontangential limit almost everywhere on the boundary.

Benoit Mandelbrot

 

Perhaps another great mathematician of the twentieth century was Benoit Mandelbrot, who was largely responsible for the present interest in fractal geometry. He showed how fractals could occur in many different places in both mathematics and elsewhere in nature.

Mandelbrot was born in Poland in 1924 into a family with an extremely academic tradition. His mother was a doctor, but his father, however, made his living buying and selling clothes.

Mandelbrot was introduced to mathematics by his two uncles as a young boy.  Then in 1936, Mandelbrot's family emigrated to France and his uncle Szolem Mandelbrot, who was Professor of Mathematics at the Collège de France and the successor of Hadamard’s in this post, took responsibility for his education.

Indeed the influence of Szolem Mandelbrot was both positive and negative since he was a great admirer of Hardy and Hardy's philosophy of mathematics.

This brought a reaction from Mandelbrot that set him against pure mathematics.   However, Mandelbrot himself says, he now understands how Hardy's deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war.

Up to the start of World War II, when his family moved to Tulle in central France, Mandelbrot attended the Lycée Rolin in Paris.  The war was a time of extraordinary difficulty for Mandelbrot because he feared for his life on many occasions.

Mandelbrot unconventional education is what he attributed much of his success to.  It allowed him to think in ways that might be difficult for someone who, through a conventional education, is strongly encouraged to think in typical ways.

 

It also permitted him to develop a highly geometrical approach to mathematics, and his outstanding geometric intuition and vision began to give him exceptional insights into mathematical problems.

Mandelbrot entered the Ecole Normale in Paris after studying at Lyon. It was one of the shortest lengths of time that anyone would study there, because Mandelbrot left there after just one day.

In 1944, after a very successful performance in the entrance examinations of the Ecole Polytechnique, Mandelbrot began his studies there.  It was at this institute where he studied under the direction of Paul Levy, who was another to strongly influence Mandelbrot.

Mandelbrot went to the United States after completing his studies at the Ecole Polytechnique; Mandelbrot went to the United States where he visited the California Institute of Technology. From there he went to the Institute for Advanced Study in Princeton where John von Neumann sponsored him.

In 1955, Mandelbrot returned to France.  There he worked at the Centre National de la Recherche Scientific and during this time back to France he married Ailette Kagan.  He did not stay there too long before returning to the United States.

IBM presented Mandelbrot with an environment, which let him to discover a wide variety of different ideas. He has told of how this freedom at IBM to choose the directions that he wanted to take in his research presented him with an opportunity, which no university post could have given him.

In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming that it was a work of art.  His uncle also said that it was a potential source of interesting problems, but Mandelbrot did not like it.

Indeed he didn’t react that well against the suggestions posed by his uncle, since he felt that his uncle’s attitude to mathematics was so different from that of his own.

As an alternative Mandelbrot chose his own very different course which, on the other hand brought him back to Julia’s paper in the 1970s.  This was after a path through many different sciences which some characterize as extremely individualistic or nomadic.

Actually, the choice by Mandelbrot at a young age to give aid to many different branches of science was a very purposeful one.  It is outstanding on how he was able to satisfy this ambition with such extraordinary success in so many areas.

Mandelbrot, who then worked at IBM's Watson Research Center, was able to show how Julia’s work, with the aid of computer graphics, is a source of some of the most beautiful fractals known today.

To accomplish this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics. !!!

His work was first put detailed in his book Les objets fractals, forn, hasard et dimension (1975) and more completely in The fractal geometry of nature in 1982.

Mandelbrot received his Honorary Degree of Doctor of Science from the University of St Andrews on 23 June 1999.  Mandelbrot was Professor of the Practice of Mathematics at Harvard University as well as IBM Fellow at the Watson Research Center.

He also was a Professor of Engineering at Yale, a Professor of Mathematics at the Ecole Polytechnique, a Professor of Economics at Harvard, and a Professor of Physiology at the Einstein College of Medicine.

 

Mandelbrot's interest and exploration into so many different branches of the sciences was, as mentioned above, no accident but a very deliberate decision on his part.

 

Mandelbrot has received many honors and prizes in recognition of his outstanding accomplishments.  For instance, in 1985 Mandelbrot was awarded the 'Barnard Medal for Meritorious Service to Science'.

 

The next year he received the Franklin Medal and in 1987 he was honored with the Alexander von Humboldt Prize.  In the following year he received the Steinmetz Medal.  He received many more awards including the Nevada Medal in 1991 and the Wolf prize for physics in 1993.

Edward Lorenz

The arrival of digital computers has accelerated the pace of development in the field of chaos. Computers permitted one to experiment with the nonlinear differential equations that were unfeasible before.

It is considered that Edward Lorenz a meteorologist was the first true experimenter in chaos. In 1960 he was working on the dilemma of weather prediction.  He had a computer set up with a set of twelve equations to model the weather.

Even though it didn't predict the weather itself, this computer program did theoretically predict what the weather might be.

Then on a particular day in 1961, he wanted to see a particular sequence again. Lorenz wanted to save time, so instead of starting at the beginning of the sequence, he started in the middle of the sequence. He left the computer running after he entered the number off his printout.

After an hour, he had come back to check on the status and the sequence had evolved differently. Instead of the same model as before, it diverged from the model, ending up wildly different from the original.

Lorenz eventually figured out what happened. The computer stored the numbers to six decimal places in its memory. He only had it print out three decimal places to save paper. In the original sequence, the number was .506127, and he had only typed the first three digits, .506.

With all the conventional ideas of the time, it should have worked. Therefore, he should have gotten a sequence close to the original sequence. When a scientist can get measurements with accuracy to three decimal places, he considers himself to be lucky.

Anyone would think that the fourth and fifth decimal points wouldn’t be that important.  Surely it would be impossible to measure using reasonable methods and can't have a huge effect on the outcome of the experiment.  Lorenz proved the idea wrong.

The butterfly effect is what this effect came to be known as.  The amount of difference in the starting points of the two curves is so small that it looks like and is comparable to a butterfly flapping its wings.

A single butterfly flapping its wings today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does wanders away from what it would have done.

In other words, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen or maybe one that wasn't going to happen does.

This phenomenon is also known as sensitive dependence on initial conditions and is common to the chaos theory.  Only a small change in the initial conditions can drastically change the long-term behavior of a system.

Experimental noise, background noise, or an inaccuracy of the equipment are such a small amount of difference in a measurement, but can have a drastic change in the behavior of a system.  Although, these things are impossible to avoid in even the most secluded and isolated lab.

With a starting number of 3, the final result can be entirely different from the similar system with a starting value of 3.000001.  Just try and measure something to the nearest millionth of an inch and see it is simply impossible to achieve this level of accuracy.

From that idea Lorenz stated that it is impossible to predict the weather accurately. Even though he couldn’t predict the weather, his discovery led Lorenz on to other aspects of what eventually came to be known as chaos theory.

Lorenz began to look for a simpler system that had sensitive dependence on initial conditions. His first discovery had twelve equations, and he wanted a much more straightforward version that still had this characteristic.

He obtained the equations for convection and made them unrealistically simple by stripping them down.  The system had only three equations this time and it no longer had anything to do with the convection, but it did have sensitive dependence on its initial conditions. 

Later, it was discovered that his equations accurately described a water wheel.

At the top, water drips steadily into containers hanging on the wheel's rim. Each container drips steadily from a small hole. If the stream of water is slow, the top containers never fill fast enough to overcome friction, but if the stream is faster, the weight starts to turn the wheel.

The rotation might become continuous. Or if the stream is so fast that the heavy containers swing all the way around the bottom and up the other side, the wheel might then slow, stop, and reverse its rotation, turning first one way and then the other.

The equations for this system also appear to give rise to entirely random behavior. Yet, a surprising thing happened when he graphed it, the output always stayed on a curve, a double spiral.

Previously, there were only two kinds of order.  One was a steady state, in which the variables never change.  The second one was periodic behavior, which the system goes into a loop, repeating itself indefinitely. Lorenz's equations are definitely controlled because they always followed a spiral.

They never settled down to a particular point, but since they never repeated the same thing, they weren't periodic either.  The image he got when he graphed the equations is called the Lorenz attractor.

Lorenz published a paper in 1963 describing what he had discovered. He discussed the types of equations that caused this type of behavior and included the unpredictability of the weather.

Although a meteorological journal was the journal he was able to publish in, because he was considered a meteorologist, not a mathematician or a physicist.

Consequently, Lorenz's discoveries weren't acknowledged until years later, when others rediscovered them. Lorenz had discovered something groundbreaking; now he had to wait for someone to discover him.

In addition, if he started the simulation over again, his solution would obviously diverge from the previous simulation and became entirely different after a period of time. This established that the system is inherently unpredictable and as a result any slight error in the initial conditions would be amplified rapidly.

In 1971 Ruelle and Takens came up with a new theory, based on the abstract concept of a strange attractor, for the onset of turbulence in fluids. Several years later May found an instance of chaos in the iterative map (logistic map) that was used to study population dynamics in biology.

 

Feigenbaum discovered that there are certain universal routes, which systems will take in transitioning from regular to irregular motion. This discovery provided the link between chaos and its transitioning phase.

 

In the late 1970's there were two additional important developments in the field. Mandelbrot discovered fractals and showed that they can be applied to other subjects.  And in the field of mathematical biology, Winfree used the geometric idea to study biological oscillations such as circadian rhythms and heart rhythms.

 

By 1980 the widespread interest in chaos, fractals, oscillators and their applications had taken a firm root for this emerging field of dynamics.

 

This resulted in the rapid growth of developing theories and other mathematical tools to deal with the systems. One of these would lead to the development of the control of chaos in the early 1990's.

 

**Chaos Theory

 

 

"It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible."
----Henri Poincare

 

The study of Chaos is part of a bigger program of study of a supposed “strongly” nonlinear system (Springer-Verlag 1). A nonlinear system is a system that is not in the first degree.  Multi-dimensional systems and systems that are not in a straight line are prime examples of a nonlinear system (1).

 

An example in the world of physics of such a system would be a fluid in turbulent motion (1). 

 

If this were not exactly the study of chaos, then the image of the turbulent motion would serve as a powerful symbol to remind a physicist of the sorts of problems he would like to understand (1).

 

Fluid turbulence certainly presents us with extremely erratic and only a somewhat predictable phenomena (1).  Since the historic times of La Place say, physical scientists have turned to the statistical methods when presented with problems that concern the joint behaviors of innumerably large number of pieces (1).

 

For the majority, chaos theory already belongs to the greatest achievements in the natural sciences in the twentieth century (9).

 

Indeed, it can be said that very few developments in natural science have gotten so much interest from the public (9).  Sometimes, people even hear of changing images of reality or of a revolution in the natural sciences (9).

 

Opponents of chaos theory have been inquiring about whether this popularity could have something to do with the witty choice of catchy terms (9).  Other critics say that an excuse could be the very human need for a theoretical explanation of chaos (9).

 

Chaos theory is once and awhile at risk of being overtaxed by being associated with everything that can be even superficially related to the concept of chaos (9).  Yet, what is it that makes chaos theory so fascinating?  

 

In particular cases one inferred that chaos was an unwanted anarchic quality.  In the past people have incorporated this idea of disorder into chaos. 

 

Dictionaries defined chaos as turmoil, turbulence, and primordial abyss.  Biblical references to Tohu and Bohu had the same referential character of undesired randomness.

 

Technically, Chaos implied the existence of the unwanted randomness, but the self-organization concept at the edge of chaos denoted the order we get out of chaos.

 

The American novelist and historian Henry Adams (1858-1918) wrote his scientific meaning of "chaos" succinctly: "Chaos often breeds life, when order breeds habit".

 

In a manner of speaking chaos theory came in the back door of the researcher's world.  It was not a law like thermodynamics or quantum physics, but it did allow the researcher to analyze events or areas with many problematic intricacies.

Several explanations for Chaos theory called for the words synthesis, cross-discipline, edge of chaos, dynamical, cellular automata, or neural networks, but all take with them the concept of complex systems.

The implications of Chaos are profound, for who could know the absolute conditions of any system for a complete prediction to be made of the behavior of that system?

To ancient humans, Chaos represented the unknown, nightmarish visions that reflected man’s fear of the irrational and the need to give shape and form to his apprehensions (Pickover 29).

 

Today Chaos usually involves the study of a range of phenomena exhibiting a sensitive dependence on initial conditions (29).  Which means that if you very slightly change a parameter in an equation or system, very different behaviors can result (29).

 

From chaotic toys with randomly blinking lights to wisps and eddies of cigarette smoke, chaotic behavior is generally irregular and disorderly (29).

 

Other examples include weather patterns, some neurological and cardiac activity, the stock market, and certain electrical networks of computers (29).

 

Our daily encounter with Chaos is seen in traffic flow, weather changes, population dynamics, organizational behavior, shifts in public opinion, urban development and decay, cardiological arrhythmias, epidemics.

 

It might be found in the operation of the communications and computer technologies on which we rely, the combustion processes in our automobiles, cell differentiation, immunology, decision-making, the fracture structures, and turbulence.

 

Chaos, fractals and dynamics, these are topics that are pretty much in the news today.  What do these teams mean though?  These are terms that come from the area of mathematics known as dynamical systems.

 

Dynamics is a natural topic to expose students from high school and college to and is a fascinating topic in mathematics. 

 

One reason it is so fascinating is that you will see is the computer graphic images that come out of the field of dynamics are so beautiful.  There’s something in these computer graphics that strikes the human eyes as quite alluring. 

 

Another reason for the fascination for this field of mathematics is the fact that is so exciting, even for people who despise mathematics.  Much of what you see in this field of study today is less than a quarter of a century old.  This is new mathematics.

 

One might think that this area of mathematics might be somewhat esoteric, since it’s so new, but in fact much of this mathematics is very accessible. 

 

We’re talking about quadratic functions, such as sin and cos, the kind of things that students in high school know quite a bit about.   

 

This area of mathematics involves in an essential way, the use of a computer.  Computers, computer graphics, and computer experiments are essential to understanding and working with dynamical systems. 

 

This area of mathematics promises to be quite applicable.  Anyone who has read the book by James Glick called Chaos: The Making of New Science will see the tremendous power of this mathematics in terms of applications in all areas of science.

 

One of the complicated objects in mathematics is the Julia set.  The Julia set is considered to be one of the most beautiful of these dynamical systems.  A process called iteration produces the images.

Julia Sets

Closely associated to the Mandelbrot set is the collection of fractals known as Julia sets. These sets were first examined by Gaston Julia (1893-1978) and Pierre Fatou (1878-1929).

Mathematicians originally characterized Julia sets as being "pathologically complicated," but it is their complexity that gives Julia sets their astonishing beauty.

Like the Mandelbrot set, Julia sets are the creation of dynamical systems based on the function f(x) = x² + c, where x and c are complex numbers. But for Julia sets each spot in the complex plane becomes an initial value of x, and the same fixed value for c is used for a complete set.

When the function is iterated, individual points in the plane will have open areas that either converge to some particular point or diverge to infinity. Those points in convergent regions form a Fatou set. The boundary of this set is a Julia set.

For each initial value x₀, the recursion x_n+1 = f(x_n) describes a sequence of points called the forward orbit of x₀. A set of points whose forward orbits move toward the same limit point is called a basin of attraction, and this limit point is known as the set's attractor.

A Julia set is a set of extraordinary points that separate different basins of attraction. It can be thought of as the repeller set of the iterative function, or as the attractor of the opposite relation. 

For the quadratic function f(x) = x² + c, the Julia set is the limit between the set of points that iterate to infinity and those that do not.

Assigning a value of 0 to c creates the function f(x) = z², which provides a helpful example. When this function is iterated, every point whose space from the origin is less than 1 converges to the origin.

Every point bigger than 1 unit from the origin iterates to infinity. Thus the Julia set of f(x) = x² is the unit circle centered at the origin. For nonzero values of c the matching Julia sets are fractals.

Two essentially different algorithms for producing Julia sets exist. One plots the boundary between basins of attraction; the other graphs the attractor of the inverse relation.

They construct somewhat different results, but neither is really superior to the other. The first method is more successful than the second for definite values of c, but less satisfactory for others. And both algorithms verify to be somewhat insufficient for a large number of Julia sets.

The first method is capable of generating color images by representing different convergence rates with different colors. The approach is very alike to that used to generate Mandelbrot set graphics.

For each pixel in the chosen domain, the function is iterated for the equivalent value of x. The iteration ceases when the size of x reaches 2 or the number of iterations arrives at a predetermined maximum. The last iteration count determines the color assigned to the pixel.

Infinity can be considered as an attractor for the iterative process defined by x_k+1 = x_k² + c. The pixels in its domain of attraction can be colored according to the how rapidly their corresponding points escape to infinity.

Most Julia sets are controlled within the square centered at the origin and having sides 3 units long, so this is the region usually selected for examination. A value for the complex parameter c must also be selected.

The maximum number of iterations to be permitted is assigned to the integer constant K.

Those points that do not converge in the direction of infinity after K steps will be shaded with one solid color. If the number of available colors is limited, they may be used in a periodic manner.

Furthermore, the computation time can be sliced in half by taking advantage of the Julia set's symmetry about the origin; the iteration counts for x = a + bi and -x = -a - bi will always be the same.

The value of c completely determines the figure of the associated Julia set. For values of c with a relatively small complete value, the Julia set is a simple closed curve.

As can be observed, the fractal curve is self-similar.  The location of c with reverence to the Mandelbrot set determines the general form of the corresponding Julia set. When c belongs to the Mandelbrot set, its Julia set is related.

But if c lies exterior the Mandelbrot set, its Julia set is "broken into infinitely many pieces."

When c lies within the huge atom of the Mandelbrot set's continental molecule, the related Julia set is a fractally distorted circle. If c is contained inside one of the continental molecule's smaller atoms, the Julia set consists of an infinite number of fractal loops, each surrounding a different basin of attraction.

If c is selected from the interior of one of the island molecules, the Julia set contains infinitely many copies of the Julia set that would be determined by the corresponding position in the continental molecule, all connected by a dendrite.

A value for c that lies on one of the filaments creates a similarly shaped Julia set, because infinity is the only attractor.  And if c lies outside the Mandelbrot set, the Julia set falls apart into a cloud of points called Fatou dust.  These points become sparser as c becomes more isolated from the Mandelbrot set.

What makes chaos theory so fascinating?  What do the supposed changes in the image of reality consist of?  To these subjects the philosophy of nature might help in the attempt to answer them and also impose some more questions.

 

And to do so, the topic of dynamical systems will be discussed.

 

Dynamical Systems

 

A dynamical system is any process that moves or changes in time.  Dynamical systems occur in every branch of science.  For example:  the motion of the planets, the weather, the stock market, and finally chemical reactions. 

 

The motions of the planets in celestial mechanics are a good example of a process of something that evolving in time.  The weather is another system that changes dramatically over time. 

 

Similarly, the Stock Market, economic systems are good examples of very chaotic at times, dynamical systems.  Finally, in chemistry, simple chemical reactions are examples of processes that evolve in time.

 

Can you predict what will happen?

 

When a scientist confronts a dynamical system, the question that she or he ask is can I predict what will happen in the future, Can I predict how this motion will evolve in time?  If you look at some of the examples giving of dynamical systems, it is clear that some of them are predictable.

 

The motion of the planets for example; you know that in the morning when you wake up the sun will rise.  Similarly, chemical reactions, you know that tomorrow morning when you put crème in your coffee, the resulting chemical reaction will not be an explosion. 

 

On the other hand, the weather or the stock market, those are examples of dynamical systems that seem to be unpredictable.  The question now, is why are they so unpredictable? 

 

A person might say I know why the weather and stock market are so unpredictable.  Those are dynamical systems that seem to depend on so many variables, that it would be impossible for anyone to know all of the variables at any one time so to make a prediction.

 

For example to predict the weather, you would have to know all elements of the weather around the globe instantaneously. You would have to know the barometric pressure, the wind speed and direction everywhere in the globe in order to predict what the weather will be like a week, hence.

 

The main maxim of science is its ability to relate to cause and effect.  On the basis of the laws of gravitation, for example, astronomical events such as eclipses and the appearances of comets can be predicted thousands of years in advance.

 

Other natural phenomena, however, appear to be much more difficult to predict.  Although the movements of the atmosphere, for example, obey the laws of physics just as much s the movements of the planets do, weather prediction is still rather problematic. 

 

We speak of the unpredictable aspects of weather just as if we were talking about rolling dice or letting an air balloon loose to observe its erratic path as the air is ejected.  Since there is no clear relation between cause and effect, such phenomena are said to have random elements.

 

Yet there was little reason to doubt that precise predictability could, in principle, be achieved.  It was assumed that it was only necessary to gather and process greater quantities of more precise information (e.g., through the use of denser networks of weather stations and more powerful computers dedicated solely to weather analysis).

 

Some of the first conclusions of chaos theory, however, have recently altered this viewpoint.  Simple deterministic systems with only a few elements can generate random behavior, and that randomness is fundamental; gathering more information does not make it disappear. This fundamental randomness has come to be called chaos.

 

Similarly, the stock market, to predict where the stock market will go, theoretically at least, you have to know the behavior of all elements of the economy, of all the consumers in the economy.  Clearly these are systems that depend on just too many variables to make a prediction. 

 

Well, it is certainly true that the weather and the stock market depend on too many variables, but on the other hand, that’s not necessarily the reason that makes these dynamical systems unpredictable. 

 

It’s one of the most remarkable discoveries of mathematicians in the last thirty years that very simple dynamical systems, systems that depend on only one variable, not billions of variables like meteorology or the economy,

 

systems that depend on a single variable can behave just as unpredictably, just as turbulently as the weather or the stock market.  That’s what will be discussed more, is how the simple dynamical systems can react or behave in a very strange and chaotic way.

 

An evident inconsistency is that chaos is deterministic, generated by fixed rules, which do not themselves involve any elements of change (Springer – Verlag 11).  People even talk about deterministic chaos (11). 

 

In principle, the future is completely determined by the past; but in small doubts, much like minute errors of measurement, which enter into calculations, is amplified, with the effect that even though the behavior is predictable in the short term, it is unpredictable over the long run (11).

 

A landmark achievement of tremendous, accelerating effect was made about three hundred years ago with the development of calculus by Sir Isaac Newton (1643 – 1727) and Gottfried Wilhelm Freiherr von Leibniz (1646 – 1716) (11). 

 

Through the universal mathematical ideas of calculus, the basis was given with which to they say that successfully model the laws of the movements of planets with as much aspect as that in the development of populations,

 

the spread of sound through gases, the conduction of heat in media, the interface of magnetism and electricity, or even the path of weather events (11).

 

Also growing during that time was the secret belief that the terms determinism and predictability were equivalent (11- 12).  Present, past, and the future are joined together by casual relationships;

 

and along with the views of determinists, the problem of an exact prognosis is only a matter of the difficulty of documenting all the relevant data (12). 

 

In addition, chaos and order, specifically the causality principle, can be observed in coincidence within the same system (12).  There may be a linear progression of errors characterizing a deterministic system, which is controlled by the causality principle (12).

 

While, in the same system, there can also be an exponential chain of errors, for example the butterfly effect, indicating that the causality principle breaks down (12).

 

In other words, one of the lessons coming out of chaos theory is that the soundness of the causality principle is narrowed by the uncertainty principle from one end as well as by the inherent properties of fundamental natural laws from the other end (12). 

 

SOLAR SYSTEM CHAOS

Chaos theory isn't new to astronomers. Most have long known that the solar system does not "run with the precision of a Swiss watch." Astronomers have uncovered certain kinds of instabilities that occur throughout the solar system in the motions of Saturn's moon Hyperion, in gaps in the asteroid belt between Mars and Jupiter, and in the orbits of the system's planets themselves.

As used by astronomers, the word chaos denotes an abrupt change in some property of an object's orbit. An object behaving in a chaotic manner may, for example, have an orbital eccentricity that varies cyclically within certain limits for thousands or even millions of years, and then abruptly its pattern of variation changes.

The result is a sharp break in the object's history -- its past behavior no longer says anything about its long-term future behavior. For centuries astronomers have tried to compare the solar system to a gigantic clock around the sun.

But they found that their equations never actually predicted the real planets' movement. This problem arises from two points, one theoretical, and the other, practical.

The theoretical difficulty was summed up by Henri Poincare around the turn of this century. He demonstrated that while astronomers can easily predict how any two bodies -- Earth and the Moon, for example -- will travel around their common center of gravity, introducing a third gravitating body (such as another planet or the Sun) prevents a definitive analytical solution to the equations of motion.

This makes the long-term evolution of the system impossible, in principle, to predict. The practical difficulties are the limits of computer power. Even with the help of calculators and desktop computers, the long-term calculations were too lengthy.

The conclusion from all this is that while new real-life chaos discoveries are being made, current computing technology cannot keep up with the pace.

CHAOS AND THE STOCK MARKET

 

According to respected authorities, stock markets are non-linear, dynamic systems. Chaos theory is the mathematics of studying such non-linear, dynamic systems. Chaos analysis has determined that market prices are highly random, but with a trend.

 

The amount of the trend varies from market to market and from time frame to time frame. A concept involved in chaotic systems is fractals. Fractals are objects that are "self-similar" in the sense that the individual parts are related to the whole.

 

A popular example of this is a tree. While the branches get smaller and smaller each is similar in structure to the larger branches and the tree as a whole. Similarly, in market price action, as you look at monthly, weekly, daily, and intra day bar charts, the structure has a similar appearance.

 

Just as with natural objects, as you move in closer and closer, you see more and more detail. Another characteristic of chaotic markets is called "sensitive dependence on initial conditions." This is what makes dynamic market systems so difficult to predict.

 

Because we cannot accurately describe the current situation band because errors in the description are hard to find due to the system's overall complexity, accurate predictions become impossible.

 

Even if we could predict tomorrow's stock market change exactly (which we can't), we would still have zero accuracy trying to predict only twenty days ahead.

A number of thoughtful traders and experts have suggested that those trading with intra day data such as five-minute bar charts are trading random noise and thus wasting their time.

Over time, they are doomed to failure by the costs of trading. At the same time these experts say that longer-term price action is not random. Traders can succeed trading from daily or weekly charts if they follow trends.

The question naturally arises how can short-term data be random and longer-term data not be in the same market? If short-term (random) data accumulates to form long-term data, wouldn't that also have to be random?

As it turns out, such a paradox can exist. A system can be random in the short-term and deterministic in the long term.

 

Mathematical Dynamical Systems

 

To simplify the situation, let’s begin by discussing mathematical dynamical systems, a very simple abstraction of the kinds of dynamical systems that arise in nature.

 

What’s a mathematical dynamical system?  Well, among the simplest mathematical dynamical systems are the so-called Iterated “functions.”

 

Start with any mathematical expression, for example the square root function and start with any number, say x.   How does one create a mathematical dynamical system?  Well, through the process of iterating this mathematical system.

 

That is accomplished by taking the initial number x and computing its square root, you get a new number.  Then take that number and compute its square root, you’ll get another new number and so forth.

 

This is the process of iteration.  It’s a dynamical system.  The numbers are changing in time.  The question to the mathematician is, just as in the case of the scientist, “Can you predict what will happen?”

 

“Can you predict what will happen when you iterate this function over and over again?”

 

Now one can easily see that this process is well suited for the use of a computer.  There’s nothing a computer can do better than iterate functions over and over again (VIDEO).

 

An iterator comes with a bunch of numbers that you can input together with a bunch of functions that you can iterate.  What functions the iterator has is up to whom ever programs the computer or calculator. 

 

 How does one do the process of iteration?  Well, for the simple minded you can use a calculator for the first few examples.  Well, with the square root example, start by imputing your favorite number into the calculator.

 

For example, you might put in the number 256.  You then iterate the square root function by pressing the square root button, then computing the square root of 256.  The answer is 16. 

 

To iterate, you would just do it again.  The square root of 16 is 4 and the square of 4 is 2.  The square root of 2 is 1,41…  Then you may ask, “What happens when we do this over and over again?”

 

Iterate the square root of 1.41…  Keep hitting the square root button and eventually you’ll see that no matter what number you started with, you’ll always end up with the number 1.

 

That is an excellent example of an iterated process that is completely predictable.  No matter what number you start with on the square root function, you always end up with the number 1.

 

Here’s another example:

 

Let’s take the function x^2.  Start with any number.  Let’s start with the number 2.  When you iterate the squaring function you first get 4.  Square 4, you get 16, square 16 and get 256. 

 

You can see what happens.  Square 256 and you get 256^2, a rather large number.  You see that upon iteration, repeated squaring when you start with any number greater than 1, it tends to infinity.

 

Once again, that’s an example of an iterated function whose behavior is completely predictable.

 

Here’s another example:

 

Take the sin function.  What happens when you iterate the sin function?  Well, start with any number, say 123 and iterating the sin of 123, you get -.45…  sin of that is -.43…  Iterate the sin again and you get -.41…

 

And you see what happens.  Iterating the sin function over and over again, eventually yields after 300 or more iterations the number 0.  So if you iterate the sin function, no matter what number you start with, you always end up with the number 0. 

 

Again, a perfectly predictable iterated process.

 

Another example: 

 

Instead of using the sin function, use the cos function.  What happens when you iterate the cos function?  Well let’s see.  Start with any number, say 123 and what do you think happens when you iterate the cos function? 

 

What happens when you iterate cos over and over again?  It turns out that with no matter what number you start with, when you iterate cos in radians, you always end up with the number .73908…

 

Where did that number come from?  That will be discussed later on, but for now notice that you know how to iterate or what the result of the iteration of cos will be.  No matter what number you start with, with cos you always end up with that strange number.

 

There are many dynamical systems that can produce chaos.  However, the focus will now be on only one particular transformation.  It is the quadratic transformation, which comes in different forms, for example, x -> ax(1 – x).

 

How about iterating the quadratic function 4x(1 – x).  What happens when you iterate this simple quadratic expression?  Well let’s start with any number.  Say the number .4 and what happens when you compute this quadratic expression? 

 

Now if you plug in .4 into the quadratic expression, you get .96.  Now iterate again and you get .154.  Iterate again and you get .521 and iterate again and you get .998.  Iterate yet another time and you get .008.  Iterate once more and the result is .032.  See the pattern?

 

Try some more iterating.  Iterate again and the result you get will be .123.  Iterate yet another time and you get .431.  Do it again and get .980, once more and the answer is .078.  See the pattern?  Probably not. Iterate again and you get .288, once more and you get .823. 

 

There is no pattern whatsoever when you iterate the quadratic expression, because this expression is Chaotic, totally unpredictable.  For all intense and purpose, iterating 4x(1 – x) is a random number generator.

 

Now most iterators or calculators don’t come with a 4x(1 – x) button, but that’s no problem.  On most computers you can easily program it to iterate a quadratic expression.

 

 

Fractals

Nature is full of shapes that are alike to themselves on different scales. A boulder looks like the mountain to which it was once attached. The structure of a twig is a lot like that of the tree from which it has fallen.  A coastline has the same irregular shape when viewed from various altitudes. 

The surfaces of certain cheeses and the random distribution of the stars in the sky display the property known as statistical self-similarity.  These phenomena and many others, such as the scattering of nuclear particles, are examples of fractals that happen in nature.

Many of nature's irregular and fragmented patterns exhibit a much greater level of complexity than can easily be explained with standard Euclidean geometry.  Such features have escaped the application of classical mathematics for a long time.

But now, due primarily to the work of Benoit Mandelbrot, this is quickly changing Mandelbrot is an IBM Fellow at the Thomas J. Watson Research Center.  His essay The Fractal Geometry of Nature is commonly received as the definitive work on the subject of fractals.

This Polish-born French mathematician has developed a new geometry and demonstrated its effectiveness as a model in a number of diverse fields.

Mandelbrot derived the word fractal from the Latin fractus, which means "fragmented" or "irregular." Frangere, the corresponding Latin verb, means "to break" or "to create irregular fragments." Fractals come in a broad assortment of visually fascinating patterns, many of which have practical scientific applications.

Some are referred to as "dragon curves," while others look precisely like mountain ranges. Fractals can mimic the ups and downs of the stock market, the erratic wanderings of molecular particles, or the growth of plants.

They have found applications in fields as different as physics, biology, sociology, and motion-picture simulation.  Mandelbrot has even used fractal geometry in the study of auditory noise transmission and galactic clustering.

Fractal geometry is, without a doubt, "one of the chief developments of twentieth century mathematics." While fractal geometry is relatively new, dating from about 1975, it builds upon the geometric measure theory for sets of integral and fractional dimension that was developed by pure mathematicians early in this century.

Number theory and the study of non-linear differential equations also give examples of fractal sets.  In addition, infinitely many fractal curves can be discovered in the complex plane. Julia sets, named after the mathematician Gaston Julia (1893-1978), and the Mandelbrot set are the chief examples.

Many fractals, especially those that copy natural phenomena, are generated with random numbers. The peaks and valleys of fractal mountain ranges are determined erratically, for instance; but uncertainty alone does not qualify a surface or curve as a fractal.

Many fractals, such as the Koch snowflake or the Harter-Heighway dragon, are not random at all. The necessary and sufficient property that distinguishes fractals is their fractional dimensionality. Fractals also exhibit self-similarity in one way or another, the smaller parts resembling the bigger, but this is not a mathematical requirement.

Mandelbrot has found order in places where others before him saw only chaos. In 1961 he established the importance of fractal geometry to economics. Next Mandelbrot recognized the central role that fractals play in many areas of physical science. He then discovered that the Hausdorff-Besicovitch dimension of certain sets has an essential application to fractals.

Mandelbrot suggested that the irregularity of a coastline could be measured by its Hausdorff dimension. In 1967 he posed the question, "How long is the coast of Britain?" The correct answer is, in his words, "It all depends." More specifically, it depends on the size of the instrument that a person is measuring with.

As the measurement becomes more and more precise, the measured length approaches infinity. But some coastline lengths are more infinite than others. Application of the Hausdorff dimension resolves this dilemma.  Euclidean geometry is sadly insufficient for the task.

Fractal graphics are almost impossible to generate without the aid of computers. The formulas that produce fractals are often fairly simple, but they must be calculated repeatedly, each iteration using the result of the previous one. Precise results are also best achieved by way of computers.

Computer graphics also facilitate comparisons between natural shapes and their computer imitations. Theories can be eliminated solely on the basis of the improper shapes they generate.

Depicting natural things such as clouds and mountain ranges has been a challenge for computer graphics systems based on everyday geometry. Modeling mountains with hyperboloids and clouds with ellipsoids is not very effective.

But the complexity of natural surfaces can be effectively modeled with fractal geometry methods. Fractal curves generated with probabilistic functions are particularly well suited for modeling nature's geological features.

The inclusion of the right amount of randomness in the generating algorithm can yield very realistic results.  Mandelbrot proposes Brownian motion as the basis for random fractals. He has demonstrated that this model provides for both the self-similarity and irregularity that fractals exhibit.

Fractal geometry provides scientists with a mathematical model that embraces the irregularities found in nature. Natural phenomena such as fluid turbulence can be described with the concepts of fractal geometry.

Consequently, fractals have become increasingly important. What began as a purely mathematical concept has now found many applications in the sciences.

The ability of fractals to mimic nature has led to the widespread acceptance of fractal techniques. Chemists, biologists, physicists, and statisticians have used fractals to model a wide variety of phenomena.

The vast number of fractals occurring in nature is enough to justify the study of fractals. Recognition of an object as a fractal can improve one's understanding of its behavior.  The growing interest in fractal graphics has also been affected by the proliferation of more powerful microcomputers.

Numerous articles on fractals have appeared in microcomputer magazines recently. Part of this interest stems from the unpredictable nature of certain fractals; one can spend hour after hour exploring the variety of shapes that a single program can create.

In some sense, fractal geometry is first and foremost a new ‘language’ used to describe, model and analyze the complex forms found in nature. 

 

But while the elements of the ‘traditional language’, the familiar Euclidean geometry, are basic visible forms such as lines, circles and spheres, those of the new language do not lend themselves to direct observation.

 

They are, namely, algorithms, which can be transformed into shapes and structures only with the help of computers.  In addition, the supply of these algorithmic elements is inexhaustibly large; and they are capable of providing us with a powerful descriptive tool.

 

Once this new language has been mastered, we can describe the form of a cloud as easily and precisely as an architect can describe a house using the language of traditional geometry.

 

When people think about fractals as images, forms or structures people usually perceive them as static objects.  This is a legitimate initial standpoint in many cases.

 

But this point of view tells people little about the evolution or generation of a given structure.  Often, as for example in botany, people like to discuss more than just the complexity of a ripe plant.  In fact, any geometric model of a plant, which does not also incorporate its dynamic growth plan for the plant, will not lead very far.

 

The same is actually true for mountains, whose geometry is a result of past tectonic activity as well as erosion processes which still and will forever shape what we see as a mountain.  We can also say the same for the deposit of zinc in an electrolytic experiment.

 

In other words, to talk about fractals while ignoring the dynamic processes which created them would be inadequate.  But in accepting this point of view we seem to enter very difficult waters.

 

What are these processes and what is the common mathematical thread in them?  Aren’t we proposing that the complexity of forms, which we see in nature, is a result of equally complicated processes?  This is true in many cases, but at the same time the long-standing paradigm is far from being true in general.

 

Rather, it seems – and this is one of the major surprising impacts of fractal geometry and chaos theory – that in the presence of complex pattern there is a good chance that a very simple process is responsible for it.

 

In other words, the simplicity of a process should not mislead us into concluding that it will be easy to understand its consequences.

 

As mentioned, Mandelbrot is often characterized as the father of fractal geometry.  Some people, however, remark that many of the fractal and their descriptions go back to classical mathematics and mathematicians of the past like George Cantor (1872), Giuseppe Peano (1890), David Hilbert (1891), Helge von Koch (1904), Waclaw Sierpinski (1916), Gaston Julia (1918), or Felix Hausdorff (1919), to just name a few.

 

Yes, indeed, it is true that the creations of these mathematicians played a key role in Mandelbrot’s conceptual steps towards a new perception or new geometry of nature.

Mandelbrot Set

The correlation of chaos and geometry is anything but coincidental.  Rather it is a witness to their deep kinship.  This kinship can best be seen in the Mandelbrot set.  It has been discovered by some scientist as the most complex, and possibly the most beautiful, object ever seen in mathematics. 

Its most fascinating characteristic, however, has only just recently been discovered: namely, that it can be interpreted as am illustrated encyclopedia of an infinite number of algorithms.

It is fantastically efficiently organized storehouse of images, and as such it is the example par excellence of order in chaos.  Here is an image of the magnificent Mandelbrot set.

The Mandelbrot set is probably the most widely recognized fractal. Mandelbrot's discovery resulted from his research in the area of iteration theory, also known as complex analytic dynamics.

This field dates back to the investigations of P. Fatou and G. Julia in the early part of this century.  A one-to-one correspondence exists between the complex numbers and the points in the complex plane.

Repeated application of a simple function causes some of these points to flee toward infinity, while others never wander far from the origin. The latter points form the Mandelbrot set, seen here.

Mandelbrot Set and Miniature

The boundary of this set is an infinitely complex and strangely beautiful fractal.  Although, the iterative function that produces the Mandelbrot set is quite simple, the complexity of the set itself is mind-boggling.

To fully appreciate this, one must explore the region of the complex plane near the boundary of the Mandelbrot set.  Examining any portion of the boundary in greater detail reveals new complexities. This property of the Mandelbrot set makes it an endless source of fascinating computer art.

Iterating the function x_n+1 = f(x_n) produces a dynamical system.  There are several possible outcomes for the sequence of points produced by such a system.

The sequence may diverge to infinity, converge to a finite limit, or repeat a cycle of points. In the case of the Mandelbrot set, the resulting sequence is determined by the initial value of z.

To decide whether a given point lies within the Mandelbrot set, the sequence f(z), f(f(z)), f(f(f(z))), ... must be evaluated as many as 1000 times or more, testing each new result for membership in the set.

The function f(z) = z² + c is the dynamic that generates the Mandelbrot set. z and c both represent complex numbers, where z is allowed to vary and c is kept constant.  Initializing z to 0 yields f(z) = c.

This result is then substituted for z in the next iteration. The iterative process continues in this fashion, the output of each step becoming the input for the next.

The Mandelbrot set is defined as the set of values of c for which z = 0 fails to iterate to infinity under f.  The boundary surrounding this set of points is jagged and nondifferentiable.

The computer is an invaluable tool for studying complex dynamical systems. One uses the computer as a sort of microscope to examine the Mandelbrot set's boundary.

Zooming in for closer looks at higher levels of magnification reveals the amazing similarities and differences that exist within the set.  The magnification that is attainable depends on the machine representation used for floating point values.

If the algorithm is to be programmed in a language that does not directly support complex numbers in the way that FORTRAN does, one must recall that i² equals -1, and therefore (a + bi)² equals a² + 2abi - b².

In order to write a feasible algorithm based upon the Mandelbrot set's definition, the circle of radius 2 can be used as a suitable neighborhood of infinity.

Once the iterative process yields a result whose size exceeds 2, the sequence will always iterate to infinity.  The size, or norm, of a complex number is simply its distance from the origin in the complex plane

The real and imaginary parts of c are plotted on the x and y axes, respectively. The algorithm assigns a value to c for each pixel and then counts the number of iterations required before the norm of z exceeds 2.

When color graphics are available, the iteration count can be used to determine each pixel's color.  Smaller pixel sizes will improve the resolution, but at the cost of increased computation times.

The maximum number of iterations to be allowed, typically anywhere from 100 to 1,000, must also be determined.  Higher values will yield more accurate results if longer computation times are acceptable.

A region of the complex plane must be specified.

The Mandelbrot set includes filaments that reach out in all directions and even miniature versions of the set itself.  Yet none of these miniatures are exactly like the parent set.

Some of these smaller Mandelbrots appear to float freely in the complex plane. But A. Douady and J. Hubbard have proven that the Mandelbrot set is connected. Thus these miniatures are actually attached to the rest of the set by fine filaments.

Hubbard has called the Mandelbrot set "the most complicated object in mathematics."  Mandelbrot has invented a descriptive terminology for discussing the set that bears his name.

The main "continental molecule" is surrounded by infinitely many "island molecules." Each molecule is comprised of infinitely many "atoms," any two of which may share a common point known as a "bond." The entire set forms a branched "polymer" that is without closed loops.

Mandelbrot has conjectured that the boundary of the set, which he modestly refers to as the M-set, is a curve whose fractal dimension is D = 2.

This figure examines a small portion of the Mandelbrot set at increasing magnifications. The second view is very similar to the first, but the strips of white outside the set are narrower. Sets whose fractal dimensions are known to be D = 2 exhibit this same characteristic.

Mandelbrot Set at Increasing Magnifications

 

Rather, what we know so well as the Cantor set, the Koch curve, the Peano curve, the Hilbert curve and the Sierpinski gasket, were regarded as exceptional objects, as counter examples

 

The most important example of a simple process with a very complicated behavior is the process determined by quadratic expressions, like x^2 + c, where c is considered to be a fixed constant, or p + rp(1 – p), where r is a constant.

 

Now, we're not done yet. The work shown above represents one iteration. We continue to run each new set of coordinates through the function until we can

 

prove that the point will a) leave the graph (example: on a ten by ten graph, the

 new coordinates are (-234, 97)) or b) never leave the graph (the rule is after 200

 

iterations, if the point is still on the graph, it will never leave.) This is how a color

is selected. If the point leaves after one iteration, it is assigned a color. Every

 

point after, that leaves the graph after one iteration, is that same color. All points

that leave after two iterations will be assigned a different color, and so on. Every

 

point that never leaves the screen is assigned one color, usually black. After

doing this process for each and every point of the graph, the result could look something like this Julian set.

 

 

As you can see, in many cases, 200 iterations are needed to assign only one point. On most PC's, a common number of points for a fractal is 303,200. This is why computers are needed to calculate the huge amount of iterations and to be precise.

Fractals do have a real-life purpose. Computers can take a normal shape, and run it through many iterations giving it a surrealistic look. A fractal equation can be made to make the seemingly randomness of clouds. Many movies use fractal landscapes to use as backdrops.

Here are some more images of fractals.

 

 

THE LORENZIAN WATERWHEEL

 

Most casual armchair scientists have no access to uniformly smooth boxes and elemental gases, much less instruments to measure the speed of the moving gases.

A metaphor for the gas chamber is found in the Lorenzian waterwheel. This is a thought experiment. Imagine a waterwheel, with a set number of buckets, usually more than seven, spaced equally around its rim.

The buckets are mounted on swivels, much like Ferris-wheel seats, so that the buckets will always open upwards. At the bottom of each bucket is a small hole. The entire waterwheel system is than mounted under a waterspout.

Begin pouring water from the waterspout. At low speeds, the water will trickle into the top bucket, and immediately trickle out through the hole in the bottom. Nothing happens.

Increase the flow a bit, however, and the waterwheel will begin to revolve as the buckets fill up faster than they can empty. The heavier buckets containing more water let water out as they descend, and when the water is gone, the now-light buckets ascend on the other side, ultimately, to be refilled.

The system is in a steady state; the wheel will, like a waterwheel mounted on a stream and hooked to grindstone, continue to spin at a fairly constant rate. But even this simple system, sans boxes or heated gases, exhibits chaotic motion. Increase the flow of water, and strange things will happen.

The waterwheel will revolve in one direction as before, and then suddenly jerk about and revolve in the other direction. The conditions of the buckets filling and emptying will no longer be so synchronous as to facilitate just simple rotation; chaos has taken over.

The explanation for the irregular movement of the gas lies at the molecular level. While the box sides may seem smooth and thus the flow of the should always be regular, at molecular levels the sides of the box are quite irregular due to the motion of atoms and molecules.

After all, in any solid not at absolute zero, total entropy is positive and there must be some irregularity in the molecular structure of the sides of the box. Molecular interactions are tiny, however.

How would such tiny things like slightly misplaced molecules affect the flow of the gas in such a profound way as to cause seemingly random motion? The theory behind how small deviations can lead to large deviations lies at the heart of chaos theory.

The explanation is simple, and in retrospect, obvious explanation commonly known as sensitive dependence on initial conditions.

Fractal Surfaces

The concept of fractal dimension can be extended to surfaces.  Most current methods for representing three-dimensional shapes are based on Plato's ideal forms, such as spheres, cylinders and cubes.

These methods are excellent for depicting man-made objects, but imitating complex natural surfaces presents a problem. Natural objects, such as mountains and bushes, contain too much variety and detail to be easily described by conventional means.

But since the roughness of a surface corresponds very well with its fractal dimension, fractal models are capable of describing such surfaces qualitatively.

Fractals are being used more and more in applications that require realistic simulation of natural phenomena.  Fractal models can describe a continuous range of surface textures, from perfectly smooth to extremely rough.

Simple random motion does not imitate nature accurately. But an image generation process that incorporates a scaling factor can create images that mimic nature quite well.

This technique has been especially successful in the generation of artificial landscape images.  Mountains, clouds, water, and plants have all been realistically portrayed using fractal techniques.

Fractal functions can accurately model natural surfaces because many physical processes produce fractal shapes.  Most forms that occur in nature are fractals. Any physical process that randomly modifies the shape of some surface through local action will usually create a fractal surface.

Different physical processes act over different ranges. Thus, the fractal dimension of a natural surface will depend on the dominant process at any particular scale.

Real surfaces cannot be true mathematical fractals. The size of a surface's basic particles prevents the infinite regression of detail that true fractals exhibit. But a surface can be called fractal if its fractal dimension is consistent over a wide range of scales.  This property is known as scale invariance.

Generating a fractal surface begins with the assignment of a real value between 0 and 1 to the fractal ratio r. This ratio determines the fractal dimension D of the surface; D = D_T + r, where D_T is the topological dimension of the surface (D_T = 2 for a plane).

Next n² large bumps are randomly placed on the plane, for some positive integer n. The altitudes of these bumps should have a Gaussian (normal) distribution with variance s². Then 4n² bumps are added, these being half the size of the first and having altitude variance s²r².

This is followed by 16n² bumps of one fourth the original size and variance s²r⁴, then 64n² bumps one eighth size with variance s²r⁶, and so on. The end result is a true Brownian fractal surface.

It is also possible to specify the location of the bumps down to a certain size, and then continue the process in a random fashion, specifying only the fractal ratio to determine the roughness of the surface.

In this way specific mountains or clouds, for example, may be described. The transition from a quantitative model to a qualitative one is quite effortless.  This technique imitates the method our brains seem to use when we store visual information mentally.

Creating landscapes in exact accordance with the above algorithm is computationally expensive.  Shortcuts have been devised by companies such as Pixar, which was formerly the Lucasfilm Computer Graphics Laboratory.

The most well known approach is the midpoint displacement algorithm. This method starts with a square and then increases or decreases the altitude of each side's midpoint by some random amount in proportion to the length of the side.

The point at the center of the square is also displaced vertically. The square is then divided into four smaller squares, and the process is repeated for each of the new squares until the desired resolution is achieved.

Using a triangle as the initial figure makes the construction process even easier. After displacing the midpoints of the three sides, they can be connected to form four new triangles.

Regardless of which initial configuration is used, this simple algorithm can produce a variety of complicated polygonal surfaces.

While the fractal ratio determines the dimension of the generated surface, the seed for the random number generator actually determines location of the peaks and valleys.

And a simple change of sign inverts the landscape, turning mountains into sinkholes, and vice versa.  The algorithm checks each segment's endpoints against the water level and makes the appropriate adjustments.

The actual display is created by plotting cross sections of the surface.  The fact that this technique cannot represent small details causes the landscape's apparent dimension to be somewhat smaller than D_T + r.

This approximation to the mathematically pure algorithm produces fairly realistic results, but it does not generate truly self-similar fractal shapes. Mandelbrot argues that the full algorithm, which is based on Brownian motion, must be used in order to obtain the most accurate imitations of natural landscapes.

Koch Snowflake

In 1904 the Swedish mathematician Helge von Koch proposed a method for constructing a "snowflake" curve.  Like the Sierpinski curve, the Koch curve is a closed limit curve of infinite length that bounds a region of finite area.  But unlike the Hilbert and Sierpinski curves, the Koch snowflake is a fractal curve that is not plane filling.

The method of construction that Koch proposed begins with an equilateral triangle with sides of unit length. Each side is then trisected, and each middle segment is replaced by a smaller equilateral triangle whose sides measure 1/3. The middle segments are deleted, resulting in a Star of David.

This process of trisecting the resulting sides and replacing them with smaller triangles is repeated ad infinitum.   Cesaro devised an alternate method for constructing the Koch snowflake in 1905.

His construction begins with a larger regular hexagon and proceeds by displacing the midpoints of each side inward. The limit snowflake curve lies between these outer approximations and the inner approximations of the original method.

Each iteration of Koch's algorithm increases the length of the curve by a factor of 4/3. Thus it is easy to see that the curve's length approaches infinity as the order of the curve increases without bound. The nth approximation, C_n, has four times as many sides as C_n-1.

Therefore the number of triangles to be added at each stage quadruples, while the triangles added to C_n have 1/9 the area of those added to C_n-1. Thus the areas enclosed by successive approximations to the Koch snowflake form an infinite geometric series with common ratio 4/9.

As n approaches infinity, the area of C_n approaches 8/5 that of the original triangle.  Thus the Koch snowflake is an infinitely long curve that encloses a finite area.

The Koch snowflake clearly illustrates the concept of fractal dimension. Each side is split into four new sides; N = 4. The length of each of these sides is 1/3 the length of the replaced side; r = 1/3.

Thus D = log 4/log 3, which is about 1.26181.  In addition, the curve is everywhere continuous but nowhere differentiable.  The absence of tangents and the curve's infinite length suggest that the Koch snowflake is comprised of "infinitely small deviations which one could not dream of tracing."

The Koch curve can be generalized in a number of ways. Any regular polygon can be used initially, and many methods for sectioning the sides exist in addition to trisection. The basic algorithm has also been applied in dimensions other than two.  This type of recursive procedure can generate an enormous variety of non-intersecting fractal curves and surfaces.

Applications

Fractals have been used to describe many aspects of nature. Mathematicians have used fractals to simulate the effect of shoreline decay on fisheries. Biochemists have investigated the influence of irregular protein surfaces on molecular interactions with fractals.

Climate and other apparently chaotic phenomena can be modeled and even predicted with fractal methods.  Studies of topics as diverse as fluid turbulence and bone structure have benefited from the use of fractal structures.

Fractal geometry has also provided the computer graphics artist with an exciting new palette of intriguing shapes and surfaces.

In 1974 Jerry P. Gollub and Harry L. Swinney performed an experiment in fluid turbulence, examining the flow patterns in a fluid contained between two concentric rotating cylinders.

Their findings supported the theory that chaotic attractors cause fluid turbulence. Chaotic attractors were also discovered by Robert S. Shaw in an experimental study that measured the time intervals between the drips of an ordinary faucet.

Another test, conducted by Mitchell J. Feigenbaum, Mogens H. Jensen and Itamar Procaccia, simultaneously applied heat and electricity to a chamber of mercury. The fluctuating temperatures that resulted revealed a fractal pattern.

Fractals have also been used to describe the irregularity of protein surfaces. Certain protein surfaces have a fractal dimension D of approximately 2.4, but this value varies considerably. Experiments suggest that variations in surface texture may affect molecular interactions.

The diffusion-limited aggregation model of Witten and Sander has generated a great deal of research activity during the past few years.  In 1984 Brady and Ball used this model to measure the electrode position of solid copper limited by the diffusion of Cu²⁺ ions.

In a similar study, Matsushita, Sano, Hayakawa, Honjo and Sawada measured the dimension of metallic zinc deposits to be 1.7, which agreed with the dimension of the computer-generated fractal.

Another phenomenon that exhibits fractal qualities was first observed by Henry S. Hele-Shaw in the 19th-century. When a less viscous fluid is injected into glycerin or oil, a finger-like pattern results.

Nittman, Daccord and Stanley realized the detrimental effect that viscous fingering might have on enhanced oil recovery, where water is pumped into porous rock to displace oil.

The patterns formed by viscous fingering strongly resemble those generated by the DLA method. Lincoln Paterson has explained the similarities between the principles underlying both processes.

Electrical discharges also produce forked, lightning-like patterns that bear a resemblance to the DLA model. These patterns are called Lichtenberg figures in honor of Georg Christoph Lichtenberg,

the 18th-century German physicist.  Niemeyer, Pietronero and Wiesmann have verified that the DLA model is appropriate when the ionized region surrounding a central electrode is at equipotential.

The sharp tips on the structure have large electric fields, which stimulate the growth rate at these points.  These diverse examples demonstrate the many forms that are governed by the DLA process.

Fractal geometry has even been applied to computer-aided music composition. Choosing a "generating motif" for the slowest moving line, adding faster repetitions of the motif, and then repeating the process results in a time-filling musical equivalent of a plane-filling curve.

Altering the motif changes the nature of the resulting music. The musical fractal sounds less mechanical when a certain degree of randomness is incorporated into the algorithm.

Dodge and Bahn suggest the addition of random Brownian offsets to the pitch levels of the higher layers. These offsets should be chosen from a range of six semitones above or below the original pitch. This technique is closely related to the method used to make fractal mountain ranges.

An unusual application for fractals has been found in acoustical research. Converting the Weierstrass function into audible sound has led to an interesting paradox.

Manfred R. Schroeder has devised a Weierstrass function that, when reproduced at twice the speed, creates a chord that is one semitone lower rather than an octave higher. This effect is in exact agreement with modern theories of pitch perception.

Alan Norton, a computer scientist at the Watson Research Center, has been using parallel-processing super-computers to investigate three- and even four-dimensional fractals, called quaternions.

Rather than evaluating every point in a large three-dimensional matrix, Norton has introduced a boundary-tracking algorithm that reduces both time and memory space requirements.

A line segment that crosses from the interior to the exterior of a given shape must contain at least one boundary point. This set of points then serves as a starting list for the determination of the surface.

Generating Mandelbrot set images is becoming a recognized benchmark for parallel-processing computer systems. An array of floating point "transputers" can generate a 512 x 512 pixel Mandelbrot image in a matter of seconds.  But the fractals that lie between one and two dimensions can be explored with even modest computer graphics hardware.

The opportunity to discover objects of abstract geometry that have never been seen before is very inviting. Computer graphics have made it possible for the non-mathematician to see the beauty of mathematics.

Mandelbrot suggests that fractals should be introduced to students when the idea of the derivative is first presented. The knowledge that continuous functions are not necessarily differentiable would strengthen the student's grasp of calculus.

Mandelbrot also supports the inclusion of fractals in the physics and geophysics curricula, where fractals play an important role. In the field of computer science, fractals should be taught in courses on computer graphics. Fractals might also be used as examples when teaching rational iteration theory.  According to physicist John A. Wheeler,

no one is considered scientifically literate today who does not know what a Gaussian distribution is, or the meaning and scope of the concept of entropy. It is possible to believe that no one will be considered scientifically literate tomorrow who is not equally familiar with fractals.

More Fractal Info

The combination of fractal geometry and computer graphics gives scientists a powerful tool for modeling the fragmentary aspects of nature. Many natural phenomena exhibit fractal dimension, the quality that all fractals have in common.

The fractal dimensions of many curves can be determined from the curves' definitions with the equation (1/r)^D = N. Each step in the production of the Hilbert and Sierpinski curves, for example, replaces one segment with four half-sized segments.

Therefore, the similarity ratio r = 1/2 and N = 4. Solving for the fractal dimension D gives D = 2. For the Koch snowflake, r = 1/3 and N = 4, which results in the fractal dimension D = log 4/log 3. In the case of the Harter-Heighway dragon, the fractal ratio is half the square root of two and N is two, so its fractal dimension is also two.

Because these fractals are determined by generator shapes, their fractal dimensions can also be derived from the recursive algorithms that generate them. For each increment in the order of the Hilbert and Sierpinski curves, the SET_UP function divides the length of each segment by two, giving r = 1/2. The PLOT functions that draw each of these two curves make four self-recursive calls, which means that N = 4.

In the Koch snowflake program, the SET_UP function divides the segment length by three every time the order is incremented, and the PLOT function calls itself recursively four times; therefore r = 1/3 and N = 4. The PLOT function in the dragon program calls itself only twice; thus N = 2.

Since the segments in this curve are alternately on- or off-axis, depending on the order of the curve being drawn, the similarity ratio is not as apparent as in the other examples. But the SET_UP and MOVE functions work together to produce the correct value.

In addition to having fractal dimension, fractals usually exhibit some form of self-similarity. This property is readily observed in the Mandelbrot set, as well as in other fractals. The next two figures demonstrate the reoccurrence of similar, but not identical, shapes in the Mandelbrot set. These four images show consecutively closer looks at the same portion of the complex plane.

Similar Mandelbrot Set Shapes

Similar Mandelbrot Set Shapes

Many shortcuts are often taken when fractal landscape scenes are generated. The midpoint displacement algorithm is itself an approximation. This algorithm is often further simplified by fixing the fractal dimension at D = 2.5 and using a random number generator with a uniform distribution.

The figure on top contains landscapes generated by this simplified algorithm. A qualitative comparison can be made between these surfaces.  The average displacement can be varied in either case.

The effect of increasing this parameter is seen in the4 figure above. While the fractal ratio determines the roughness of the surface, the average displacement controls the heights of the mountains. The midpoint displacement algorithm can generate landscapes that range from the realistic to the surrealistic.

Computation times for the different types of fractals vary a great deal. For fractal curves that are based on a generator shape, such as the Koch snowflake, computation times are proportional to N^m, where N is the number of parts in the generator and m is the order of curve being drawn.

The time required to generate a Julia set or some portion of the Mandelbrot set depends on a number of factors, such as the resolution of the output device, the maximum number of iterations permitted, and the portion of the complex plane being graphed.

The value chosen for the complex number c will also have an effect when generating a Julia set. The inverse iteration method for creating Julia sets is an infinite loop that may be terminated by the user at any time.

The original algorithm takes anywhere from one to five hours to generate a Julia set image on the equipment used here; with the inverse iteration method, little improvement in the image is observed after twenty or thirty minutes.

The computation time for the bifurcation diagram is proportional to the desired output resolution. It will also depend on the complexity of the dynamic that defines the system.

For the midpoint displacement algorithm, the number of elements in the surface matrix will determine the computation time. This matrix will always contain (2ⁿ + 1)² elements, for some positive integer n.

The diffusion-limited aggregation model is very expensive in terms of computer time. The maximum distance from the cluster that a particle may travel before being destroyed, the manner in which the step length is varied, and the desired cluster size all affect the computation time. Even with the large step sizes used here, this program took over nineteen hours to execute.

The number of possible applications for fractal computer graphics seems almost endless because so many natural phenomena have fractal characteristics. This is precisely the reason that the study of fractals is important.

Besides the many applications described, fractal landscapes could possibly be used in flight simulators. Pilots training in conventional simulators often memorize the terrain after a number of sessions. The creation of different and realistic fractal landscapes for different training sessions would eliminate this problem.

There are many ways in which this research could be expanded. The number of fractal curves based on different generator shapes is limitless. A program that would graph the fractals resulting from different user-defined generators would be an excellent tool for experimentation in this area.

Designing Mandelbrot and Julia set programs for direct output to the printer would improve the image resolution. A program that applied solid modeling and appropriate shading techniques to fractal landscapes would be a rewarding project for a computer graphics student. Such a program might allow the user to input the location of the observer and the direction of the sunlight.

The abundance of fractals in nature makes the study of fractals as important as it is fascinating. Fractal computer graphics continue to prove their value as a scientific modeling tool. Students of mathematics, natural science and computer science should all be introduced to this new field of study.

Conclusion

Fractals and modern chaos theory are also linked by the fact that many of the contemporary pace-setting discoveries in their fields were only possible using computers.

 

From the perspective of our inherited understanding of mathematics, this is a challenge, which is felt by some to be a powerful renewal and liberation and by others to be a degeneration.

 

However this dispute over the ‘right’ mathematics is decided, it is already clear that the history of the sciences has been enriched by an indispensable chapter.  Only superficially is the issue one of beautiful pictures or of perils of deterministic laws.

 

In essence, chaos theory and fractal geometry radically question our understanding of equilibria, and therefore of harmony and order, in nature as well as in other contexts.

 

They offer new holistic and integral model, which can encompass a part of the true complexity of nature for the first time.  It is highly probable that the new methods and terminologies will allow us, for example, a much more adequate understanding of ecology and climate developments, and thus they could contribute to our more effectively tackling our gigantic global problems.