Press "Enter" to skip to content

Sudden Climate Change


Butterfly/Lorenz Attractor from the Chaos and Culture class website of Prof. Tom LoFaro, Gustavus Adolphus College
In the final article of the 3-part Inquirer Series on the Gulf Stream that appeared in December 2005, the connection with Chaos was at least implied. (See my earlier post on the surprising lack of chaos in the first two parts of the series.)

The article chronicles the discovery by Richard Alley in 1992 of extremely rapid temperature shifts of up to 15 degrees within a 30 year time span. Such a rapid temperature change today would lead to world catastrophe (regardless of whether the earth was cooling or warming).

Alley was investigating core ice samples in Greenland. Remarkably, based on the depth of the samples, he measured this rapid temperature shift as occurring approx. 12,000 years ago . The culprit? You guessed it - Ben Franklin’s Gulf Stream.

The connection with chaos is obvious to anyone who understands the separate wings of Lorenz’s butterfly. As the trajectory that describes the state of the weather moves around one wing, it will suddenly flip over to the other wing - a sudden reversal that "just happens."

As the Inquirer article points out, "until the 1950’s, climate was viewed as essentially a stable system."

Lorenz, Alley, and many others have certainly shown otherwise.

Categories Chaos Weather & Climate

SWARM: Art Through Multiplication


Fractal art abounds…consider the show now appeariing at the Fabric Workshop in Philadelphia, where all sorts of mathematical chicanery is used to produce stunning images and sculpture.The show is titled “Swarm.” From the Fabric Workshop website:

Swarm theory is an idea animating contemporary art, science, design, digital media, and social theory. “Swarm logic” is seen in works that use vast numbers of small parts to create systems whose final behavior or effect cannot be wholly predicted. Artists working with computers and new media construct rules that draw together data and generate behaviors that evolve over time. Sculptors and painters create structures and patterns based on the interrelationships and inherent properties of individual elements. Swarm connects the social life of bees, birds, crowds, and cities to contemporary aesthetics, as seen in the fascination of artists and designers with how simple, discrete units accumulate into complex systems.
Sounds like the ultimate Chaos Game to me!In his review article By Multiplication, Products of Art Inquirer art critic Edward Sozanski describes his ambivalence to mathematically-generated art:
I have long been ambivalent about art made by clumping. At a glance itcan appear simple-minded - take one “hand grenade” and multiply to infinity.There’s no denying, though, that in art as well as in nature simple multiplication can produce aesthetically powerful and beautiful results, especially when the individual building blocks aren’t very prepossessing.“Swarm” isn’t entirely about creating elegant structure; process and how it’s controlled are presented as equally important, especially in systems that exhibit communal decision-making.

Read the entire Sozanski piece - he did not like everything he saw. Nevertheless, isn’t the idea of SWARM further fuel against those who discredit fractal art?

It’s hard to argue with the fact that “simple multiplication can produce aesthetically powerful and beautiful results. “

To all fractal artists: keep on multiplying…to infinity and beyond…

“Swarm” continues at the Fabric Workshop and Museum, 1315 Cherry St., through March 18.

Categories Art Mathematics

Causality, Murder Novels, and Determinism


File this under books that might appear on the reading list of a future course in Chaos and Fractals…

A fascinating idea for a book: Stephen Kern’s A Cultural History of Causality: Science, Murder Novels, and Systems of Thought" looks at changing views of causality through time - as played out in murder novels and scientific theories.

Kern is attempting to follow changes in societal explanations for the causes of aberrant human behavior - in this case murder. He juxtaposes this with the increasing refinement in scientific models of causation.

Given our umbrella for this course of "modeling/prediction/understanding", Kern’s study touches on determinism, anti-determinism, and free will - perfect material to muck up our already chaotic understanding of the limits of predicting human behavior.

You can read the source of Kern’s ideas for the study in his own words by clicking here.

Categories Determinism Literature & Poetry


The fractal image appearing in the site header was produced using Ultra Fractal 3.05. The original image was designed by Damien M. Jones in 1999, and published to the Ultra Fractal Formula Database


Click to enlargeThe image is titled Melt-O-Waves. I varied the colors, zoom factor, layer blending method, and starting values for the underlying Mandlebrot set.

Please visit fractalus - this is an amazing site of fractal art that is managed by Damien Jones.

Categories Fractals Software

Fractal Image Generation, Part 1


The fractal images generated for this blog have been generated using UltraFractal 3.05 - an incredibly robust package that is effectively the PhotoShop of the fractal kingdom.In addition to all possible types of mathematical settings, UltraFractal allows layering (as in PhotoShop), with different blending styles adjustable (e.g. addition, difference, hue, etc.) The interface allows the designer to create or edit fractal formulas, set up color palettes, and add mathematical image transformations. There are many pre-loaded fractal types loaded - even with these you will have days of exploration as you vary coloring schemes and learn layering with fractals.There is also a very active user group that uploads new fractal parameter files and color templates - all available for download. You can find appropriate links, and much more at Janet Parke’s Ultra Fractal Resource Page.The result is a degree of image control that is state of the art.UltraFractal also contains a rendering engine with anti-aliasing for producing high-quality images for web and print applications.You can try UltraFractal free for a brief period.Note that ChaosPro is almost as powerful as UltraFractal and is freeware! Check it out!For a good source of chaos and fractal links check out by Juan Luis. This is a very nice site, and Juan is clearly interested in the fractal aesthetic:

Have in mind that this site,, focuses on the artistic side of fractals, not in their mathematical aspects, so these applications are primarily intended for picture or music creation, not for studying fractals as mathematical models or for in-depth experimentation (what most of them can do in any event).
Note that some of the software generates fractal music.

Categories Fractals Software

Global Warming & Prediction of Water Shortages


Click on glacier to enlargeMost Global Warming (GW ) scenarios predict melting of polar ice caps, leading to massive seaboard flooding as sea levels increase. A fact often overlooked (at least in the popular press) is that GW will most likely produce water shortages for many, even while other areas are inundated with too-much water.A recent account was picked up by the major newspapers: Tim Barnett of the Scripps Institution of Oceanography (La Jolla, CA) has developed a mathematical model predicting that up to 1/6th of the world population could face water shortages because of GW.So, if there is too much water, why the water shortage? The answer is that the water shortage is a fresh-water shortage. Fresh water will be lost due to GW in several ways:

  • Evaporation from fresh-water sources, such as rivers, may occur at a faster rate than any higher precipitated caused by GW.
  • GW will cause glaciers to melt. Glaciers are estimated to contain up to 70 percent of the world’s fresh water reserves. (And they are the natural feeders of rivers)
Island countries will be the hardest hit by GW because of the flooding and the fresh-water loss. There are already numerous calls around the world for action. The United Nations Environment Programme has recently released an urgent call for world attention to Water Shortages and Global Warming Risks for Indian Ocean Islands, where the potential for future tsunami is very real.We are at least at a point where most agree that GW is occurring, even though there is still much debate (ideological/political) about whether GW is caused by the effects of mankind. Usually these debates center on the mathematical model used. Mathematical modeling always involves compromise. Decisions are made to simplify the model by keeping the number of variables (and equations) manageable, and the interaction of these variables (e.g. solar energy and ocean condition) is usually done heuristically. With the models necessarily non-linear, the potential for chaotic behavior , especially SDIC, is very real. Because of this erratic-output, it then becomes very easy for some (and some governments) to blame the model, or the modeler for the bad news of GW, losing sight of the occuring phonemona of GW.GW still remains very real, however, whether or not a model can convince all of what what causes GW. But the models for water shortage due to GW are much simpler because of the basic correlation that more heat = more ice melting, and should be trusted by all.Predictions of severe water shortages due to GW are not new - they just don’t make it into MSM - until now. Is anyone listening?

Categories Modeling Understanding & Prediction Weather & Climate

The Why vs. When of Weather


It has been a relatively mild January in the Philadelphia area, with high temperatures significantly above the normal temperature range for this time of year. This type of occurrence is often thought to be a sign of global warming. This is not obvious - in fact there have been very large changes in January temperatures, both high and low, throughout the past century. (And January of 2005 was hotter than the current month!) This means that we cannot look at at our temperature record alone as a signature of global warming. This is true even if there is a string of higher-than-average-temperature days, months, or even years. Repeatability in weather patterns is certainly nothing one can count on.Edward Lorenz did show that a simplistic weather model based on three coupled non-linear differential equations produces weather patterns that never repeat. He used this fact, coupled with the sensitive dependence on initial conditions (SDIC) to make strong claims against the impossibility of accurate weather prediction. His model also predicted sudden swings in weather patterns - similar to what we are now experiencing.Jim Laver, the director of the Climate Prediction Center is quoted in today’s Inquirer, explaining why the weather has been warmer - it is due to easterly winds from the Pacific as opposed to cold Canadian air dominating our area this month. But why the shift in winds? There is this follow-up quote from Jim Poirer of the National Weather Service in Mount Holly, NJ: “If we knew why, we’d make money.“What’s wrong with this statement?According to Lorenz, we do know why - weather is a non-linear system that displays SDIC, and its phase-space trajectory is a strange attractor. It is the when that we don’t know. This is a case where we do understand, but cannot predict with reasonable accuracy.

Categories Understanding & Prediction Weather & Climate

Chaos & Complexity in the Evolution vs. Intelligent Design Debate


The 12/20/2005 ruling by Judge John E. Jones 3d in the Kitzmiller vs. Dover Area School District case may be a landmark one. In his far-reaching decision, Jones rules that:

The overwhelming evidence at the trial established that ID is a religious view, a mere re-labeling of creationism, and not a scientific theory.
More to the point about whether ID is really science:
… we find that, while ID arguments may be true, a proposition on which the court takes no position, ID is not science. We find that ID fails on three different levels, any one of which is sufficient to preclude a determination that ID is science. They are: (1) ID violates the centuries-old ground rules of science by invoking and permitting supernatural causation; (2) the argument of irreducible complexity, central to ID, employs the same flawed and illogical contrived dualism that doomed creation science in the 1980’s; and (3) ID’s negative attacks on evolution have been refuted by the scientific community

(Click here for the complete judicial ruling)While Jones’ ruling is a wonderful relief (which may only be temporary) from the problem of religion driving what can be taught, regardless of facts, I have been disappointed by the lack of explanations in the mainstream media about the contributions of chaos and complexity theory to our current understanding of evolution.As Judge Jones clearly understood, the ID fight against evolution was based on a “contrived dualism” - i.e. one either believes in evolution, or believes that a Supreme Being initiated the first living organism(s) via design. ID proponents attempt to manipulate opinion to accept this dualism by claiming that Darwinism (as ID supporters always refer to evolution - itself a way designed to insinuate a cult-like “ism”-status to the science of evolution) can not explain the origin of life. Indeed, evolution does not try to do this - a fact pointed out by Charles Darwin himself, whose work is on the evolution of species. Further, ID states that living organisms are “irreducibly complex”, meaning that they cannot arise via random mutation and natural selection; instead they bear the unmistakable signature of an “intelligent designer.“This is where chaos and complexity come into the picture. Complexity theory, which has its basis in chaos theory, has shown great promise in explaining how spontaneous order, often referred to as self-organization is a universal path for any biological system that reach a minimal level of complexity. Thus evolution’s natural selection is aided and abetted by self-organization in such a way that random mutations are effectively channeled into a much smaller space of possible outcomes. This restriction does two important things: (1) it makes it much more likely that a complex organism will develop via mutation and natural selection; and (2) the time needed for the complex organism to self-organize and evolve is not so long as to be longer than the accepted age of the

This argument is a direct counter to the ID claim that random mutation and natural selection cannot explain the complexity of a living organism. (The Origins of Order, by Stuart Kauffman, is the seminal work in this field. With a title clearly an homage to Darwin, Kauffman definitely sees his work as not just an extension of evolutionary theory, but an effort to put evolution in a larger box of “complexity.” Kauffman is a theoretical biologist who has written extensively on the edge between chaos and disorder. )A search of the full judicial ruling (139 pages!) does not show any reference to Kauffman’s work. I don’t know if it was brought up at all by the plaintiffs during the trial, nor do I know whether Jones is familiar with the self-organization argument. Luckily for all of those who believe in the process of scientific inquiry, Judge Jones was able to see through the ID arguments. I believe that his task would have been easier if there had been more public knowledge of the complexity theory results of Kauffman and others.Here’s a wish that popular media accounts of scientific/cultural/political debates won’t leave out the pertinent science under the assumption that the viewing/reading public won’t understand, or will lose interest. (Note that a discussion of chaos theory is never present in mainstream media accounts of global warming and weather.) Including the science might not deter those who wish to remove certain scientific facts and theories due to religious conviction, but it will go a long way towards ensuring that the general public has a better understanding of what science is, and what it isn’t.Click here for more information and provocative articles from both sides of the evolution vs. ID debateThe Intelligent Design image at the top of this post is from a Portland interior design firm. The photo was taken by Molly Holzschlag

Categories Evolution Religion Science

Modeling, Understanding, and Prediction


Sean has raised a couple of interesting points in his comments to Joe’s presentation on earthquakes and natural disasters, and Tom and Meridyth’s presentation on the outbreak of war. I was going to respond to his response, but, on further thought, believe that it is more of a general topic, and I invite responses.Sean seems to be comfortable with the fact that models exist for natural disasters, and for the onset of war, but is skeptical (his word) because of the inaccuracy in using these models for prediction. I wonder if this is a general feeling among some, or all, of you? If so, what does this say about the modeling process in these cases?For me it suggests that the modeling is yielding some deeper, or at least different understanding of the phenomena (disasters/war) as opposed to prediction. This is the complete opposite of how we started the semester - where prediction of random fractal patterns was much easier to come by than understanding why the images were produced.  

Categories Modeling Understanding & Prediction

Classroom Chaos Redux Final

Originally Posted by Sean Houlihan

My final project was a computer program that dealt with recreating the classroom chaos concerning the Logistic Growth map and producing new a various forms of analysis for it. I felt I was successful in producing my goal. However, some aspects are not as accurate as i would have like them to be. Hopefully, I will be able to work on it in the future in order to create a truly refined program. I would have like to put up a download of the program for everyone, however, I do not know how to do that. So if anyone could tell me how, it would be greatly appreciated.Maybe, one-day, RAD will use my program in his class.Please let me know what you (the class) thought of my project!

Categories Chaos Student Post