FractaLog

Originally Posted by Sean Houlihan

So yes! This semester is finally at a close. Looking back, I feel as though I truly enjoyed this class. It was not only new and interesting, but it is fun and exciting. I learned many new things, or at least tried to. I may not have the whole concept of what chaos and fractals are all about, but at least I know some. I can have an educated conversation with someone about it at least. Also, the final project is something i can show off to may family and friends to blow their minds, which is cool. I would like to thank everyone in calss for a good semester, and especially RAD for allowing me to explore something i never would have had the chance to elsewhere.

Merry X-mas & Happy New Year!

Ben Franklin's Map: The Gulf Stream, Weather, and Climate

Click on map to enlarge. Read more about Franklin and the Gulf Stream at the NOAA library.

The special series that began in Sunday's Inquirer (12/18/2005) is an in-depth look at the gulf stream - from its earliest mapping (by Ben Franklin!) to the far-ranging effects the stream has on weather and climate.

The main articles are written by Anthony Wood, who has covered weather and climate issues for several years at the Inquirer. A number of other staffers have put together other goodies at the web site, including some animated graphics that illustrate the interaction between sun, wind, and earth's spin.

Check out the series by clicking here

In some ways the Gulf Stream reminds me of the Red Spot of Jupiter discussion in the Gleick book - a bit of order within the turbulent seas around it.

The articles do not have any references to chaos, non-linear dynamics, Edward Lorenz, and the theoretical problems with forecasting weather and climate. I believe that the public should be aware of the theoretical limitations faced by weather modelers - especially when modeling and predicting the weather and climate-altering events due to phenomena as large as the Gulfstream. Given its highly non-linear nature, predicting the on-going stability and variations of the Gulfstream may be as difficult as predicting the stability of the Solar System.

Is Fractal Art Really Art?

Originally Posted by Jeremiah Noll

"grassyknoll"
"orangefla"
"hazyire" Fractal art is a controversial matter, but what in the art world isn't controversial? Contrary the opinions of some dissenters fractals are not easy to make, there are infinite possibilities, it takes a great deal of talent, and it is in no way cheating. Fractal art is an innovative art form and it will likely be the next art fad. Some of the greatest art was at some time controversial.

The fractal software on the web that I recommend is Sterling Fractal Generator, Chaos Pro, and Terragen.

These are also free and can be downloaded with little trouble. You can also check out the fractal portion of my website.

By the way they are both unreal.

Reflections of Chaos and Fractals

Originally Posted by Jeremiah Noll

At the beginning of this course I was much more certain of things! I have learned quite a bit, but with this learning came new mysteries. Studying chaos has expanded my mind and taught me things I never would have dreamed. When you sit and think about the concept and the principles which founded chaos it just makes sense. Butterflies can start El Niño's and stock market is predictable to a degree. But sensitive dependence on initial conditions is not the only principle of chaos, there is also the fact that extremely simple rules can produce infinitely complex results. Who would have thought a couple lines of computer code could produce the Serpinsky Triangle or model the population growth of a small nation.

But the coolest things I have ever seen are fractals. Through this course I have developed a great understanding of the image, mapping, and product of fractal code. Simple functions and an elementary Cartesian plane can model infinity in the most accurate and beautiful way. I enjoyed fractals so much I decided to make them my semester project.

The value of this course is a tremendous amount of insight into how the world works and how things thought to be random can really be very simple. I feel as if I have been shown a piece of how God put this universe together and how he made it so complex in only six days. Like deciphering the human genome code chaos makes us more able to predict, heal, and understand the way things work and why events happen.

Chaos and Nature

Originally Posted by Joe Cosella

In my project, I described how chaos is involved with different natural disasters. The examples I looked at were turbulence within the Earth's crust that causes earthquakes, flooding patterns in Southern China, and hurricane statistics.

I also introduced a sandpile game which can be found here.

M.C. Escher - Art, Mathematics, Recursion, Infinity

M.C. Escher's name also came up in the fractal/art discussion. Escher (1898-1972) was a Dutch graphic artist whose most well-known work was highly mathematical, incredibly complex, and, unbelievably, hand-made. Making many of his works with woodcuts, his technical expertise was staggering - this is not someone using a computer program, but instead pencil, paper, wood, knives, and ink.

Many Escher prints suggest infinity , but in a somewhat different way from a basic fractal such as the Sierpinski triangle. The image at the top of this post - titled "Circle Limit III" - implies an infinity of interlocking fish, with the surface of the sphere as the "inhabitable" universe. The visualization of infinity is suggested by the diminishing views in perspective at the sphere boundary. While the creatures near the edge are self-similar to the middle-sphere images, the Sierpinski fractal's self-similarity is found by zooming Into a section, rather than moving to the boundaries. Thus the Escher image is more akin to the Mandelbrot bug, appearing at different sizes and orientations along tendrils emanating from the main bug. (For the mathematics behind the graphic, see The Trigonometry of Escher's Woodcut "Circle Limit III", by H. S. M. Coxeter.)


Ascending and DescendingEscher also encapsulated an infinity of time in some of his prints - most famously displayed in "Ascending and Descending," a print that was one of his set of "impossible structures." Are the figures in the picture walking up or down the stairs? And when do they reach the top (or bottom)?

Note that the Escher web site contains a free interactive puzzle in which your task is to build an impossible structure.

Read about more of the mathematics behind another of Escher's famous works - The Print Gallery - as well as the connection with Dutch chocolate at the web site Escher and the Droste Effect. (Be sure to check out the animations at this site.) Here you'll find ample mention of scaling - a necessary feature of fractals.

Fractals And Art

Image by Jarosław Wierny. Click to enlarge.

Originally Posted by Pat Rafferty

The project Sean and I worked on was concerned with fractals and art. There were two parts to the project: fractals in art, and fractals as art.

Fractals in art showed how various works of art were created using fractals. For thousands of years, dating back to ancient times, people have used fractals to create fantastic art pieces. Some examples of this include religious mandalas, Dali's Visage of War, and gothic architechure. These different art forms each portray a fractal nature, but are limited by the physical boundaries of reality.

Fractals as art considered whether or not fractal created images should fall under the same label as other artforms. Fractal landscapes and other fractal generated images were given as examples of some pieces that might be considered art. Our discussion concluded that art is a personal preference and that there is not right or wrong answer.

Chaos, Consciousness, & Dreaming

Originally Posted by Rachel Hensey and John Sehi

In a dream, Paul McCartney (1965) created the melody to the popular song, "Yesterday."  Likewise, Tse Wen (2003) was inspired in a dream in which he developed a breakthrough drug that greatly reduces the risks for people who suffer from peanut allergies. These two men are examples of scientists, musicians, athletes, mathematicians, writers, and artists who have reported accounts of moments of inspiration or breakthroughs during dreaming time.

This construction of nonlinear meaning that occurs during dreaming is an example of chaos theory at work in the brain. The brain is a complex, chaotic system, and small shifts in its input during any state can dramatically alter how it operates. Inputs of order and chaos cause a tension in the brain that is mandatory for proper growth. Without the input from order, the brain would dream too much and thus fall into irrationality. Without the input from chaos, the brain would no longer dream and thus would function like a robot/automaton, without any real creativity. The output (our thoughts, feelings, and behaviors) depends upon a proper blend of chaos and order. For additional information on chaos in the brain, click here.

Modeling the Arms Race

Nuclear Proliferation game box-top. Click to enlarge.Following up on Tom and Meredyth's presentation and post: the Saperstein article referenced dealt with modeling interactions among nuclear states. A recent article by William C. Potter, director of the Center for Nonproliferation Studies at the Monterey Institute of International Studies, entitled The Second Last Chance: American Power and Nuclear Nonproliferation points out the need for non-proliferation modeles and theorists to consider the effects of regimes and terrorist agents on our notions of non-proliferation.

Potter's article is an extensive review of the book Nuclear Terrorism: The Ultimate Preventable Catastrophe by Graham Allison (Harvard Kennedy School of Government). Because of its call for the U.S. to change its current theories and practices of approaching non-proliferation, the connection with Saperstein's model is an important one. I don't know whether Saperstein, or any think-tanker worrying about such modifications has the ear of the government. At the very least, the chaotic behavior of nuclear-arms possessing states as predicted by the Saperstein model will certainly be more prevalent as the highly non-linear interactions of terrorist agents and rogue states are factored into the model.

(Note: the image at the top of the post is from the card game Nuclear Proliferation by the FlyingBuffalo company. From the blurb on the box: "It's a sarcastic, humorous look at the futility of Atomic Warfare in the post-cold war 1990s.")

Chaos: Good For More Than Absolutely Nothing

Originally Posted by Meridyth Mascio and Tom Plick

Chaos, War, and Disarmament

Debate has raged over whether wars are due mainly to long-term causes (oppression, economic depression, nationalistic zealotry) or whether they only find their genesis in freak events (the murder of a ruler, the sinking of a fleet). Many believe the former; among these people was Lewis Richardson, who, in the 1930s, developed a simple model of an arms race between two countries. He related the size of a nation's stockpile to its willingness to go to war, for two reasons: Firstly, arms could be easily qualified, whereas the "feelings" of a nation were not easily transformed into numbers. Secondly, the larger a nation's stockpile, the greater its chance of winning a war, and thus, the greater its chance of starting a war. (When each nation's stockpile is sufficiently large, the system becomes crisis-unstable: each nation wants to attack the other(s) first, lest the other(s) make the first blow.)

Richardson's model was the first attempt at modeling an arms race mathematically. Many analysts point out the inapplicability of Richardson's model to modern conflicts - the model mishandles many situations, because of its simplicity and the assumptions made to secure that simplicity.

Others after him, including Alvin Saperstein, took Richardson's idea and elaborated on it, in an attempt to model reality more accurately. Saperstein modeled an arms race between three nations under two systems - in one, the nations were allowed to ally as they saw fit, so that a superpower might find the other two nations allied against it. In the other system, the nations were not permitted to ally, and thus were at odds with both of the other nations. Saperstein's model was complex, and non-linear; recall that non-linearity breeds chaos.

Saperstein showed the system permitting alliances to be stable - in time, he says, each country's stockpile will dwindle to zero. Saperstein uses this observation in support of encouraging cooperation between countries instead of competition.

More interesting than the alliance system is the system that disallows alliances.
In the independent-nations scenario, several outcomes are possible, depending on the initial stockpiles of the nations and the "fear and loathing" values between the nations, plus many other parameters defined by Saperstein.

1. Strong stability: Each nation's stockpile decreases to zero (that is, in the limit, not necessarily in any finite amount of time).
   
2. Weak stability: Each nation's stockpile decreases to some non-zero amount.
   
3. Weak chaos: The system has a strange attractor, but its basin of attraction is small, and so the system is still largely crisis-stable.
   
4. Strong chaos: The system has a strange attractor, whose basin encompasses the entire system.
   

According to Saperstein, the strange attractor signals crisis instability, since once on a strange attractor, what happens in the long-term is anyone's guess. The way to maintain peace, he says, is to avoid the appearance of the strange attractor. To him, knowing chaos is useful for avoiding it in world affairs.

You can see our Powerpoint presentation again here.