Cephalapod Fractals

Complex SutureSteve LaMonte, a student in my Fall 2007 version of Chaos and Fractals, has noted the fractal-like shapes that are formed by suture lines in ancient cephalopods. He points out the correlation between fractal structure and the ability of the cephalopod to withstand extremes of water pressure. He writes:

One often pictures fractals as consisting of pretty pictures generated by computer programs, but they are quite prevalent in nature. A notable example can be found in the fossils of ancient cephalopods, specifically nautiloids and ammonoids. Nautoloids and ammonoids are the ancient ancestors of modern squids, octopi, and the nautilus. The ancient organisms looked like modern squids and octopi with shells, some elongated and some coiled like a snail. These shells had internal chambers that the organism filled with gas for buoyancy. Each chamber is separated by a wall, or septa. The contact line between the septa and the inner shell wall is called a suture line. The structure of the suture line determines how well the organism can resist water pressure and adjust its buoyancy. The evolution of suture lines follows an increasingly fractal-like pattern from straight sutures to highly undulated sutures. In complex sutures, the dips and folds in the undulations are called lobes and saddles, respectively.

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A chaotic test for Parkinson's

Originally Posted by Matt Venanzi

In an article from New Scientist - 11 April 2006

CHAOS theory could help monitor the effectiveness of treatment for Parkinson's disease and aid in earlier diagnosis, according to physicists who have developed a method to monitor how much sufferers tremor.

There is still no definitive test to identify Parkinson's disease in its onset. Now Renat Yulmetyev at Kazan State University in Russia and colleagues have adapted a statistical technique based on chaos theory, and used to study earthquake vibrations, to monitor the distinctive progression of symptoms such as tremors.

Sixteen people in Canada who had Parkinson's disease held their index fingers in the path of a laser beam for measurements of tremor frequency in their fingers and the team analysed the results. In patients in the early stages of the disease, the tremor pattern is more chaotic, says Yulmetyev. As the disease takes hold, the tremors not only become more pronounced, but they become much more periodic and regular. Medication with the drug L-dopa causes the tremor patterns to become more chaotic again (Physica A, DOI: 10.1016/j.physa.2006.01.077).

Click here for the full 20-page PDF of the report.
(Physica A, DOI: 10.1016/j.physa.2006.01.077)

For more information on Parkinson's disease, click here.

Update to Reflections of Chaos

Originally Posted by Jeremiah Noll

Click on image to enlarge

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 the stock market is predictable to a small 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 imaginary 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.

I still use fractal software today and it is still interesting and novel even though I have been exposed to it for some time. I still have my own fractals on my website so that I can, hopefully, get others interested in math and science too. My friends never thought Math could be so intriguing. Since the Fall semester I have progressed in the ability to create lifelike landscapes from fractal software.

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!

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!

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.

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.