During today’s spirited discussion following Sean and Patrick’s presentation on Fractal Art, I mentioned that Marshall McLuhan’s quote “Art is anything you can get away with” seemed to capture a way of defining art that would include the use of fractal software.When looking up McLuhan on the web, I came across another art quote that might be more pertinent to this discussion -

Art at its most significant is a Distant Early Warning System that can always be relied on to tell the old culture what is beginning to happen to it.

Stephen Wolfram’s short article The Origins of Randomness in Physical Systems is like a black hole, or at the very least a neutron star. Incredibly short for the ideas elaborated - only 4 pages - the manuscript is incredibly dense. Each line is a distillation of entire courses of study.

How is this possible? The endnotes: there are 28, and many of them are full paragraphs of further information, and contain their own set of references.

How did you react when reading this paper? Did you recognize any of the ideas/statements as any that we have covered this semester? Or perhaps ideas you are familiar with from past study?

And what is your personal belief of randomness in physical systems? Does randomness arise from the interaction between your system and the outside world, or does randomness arise from deterministic processes within the system? If you answer that randomness comes from outside the system, how do you explain randomness if the system is the universe?

And, if the system is the universe, does this mean that all random processes are deterministic?

Are there no random processes?

(Image drawn using the EdgeOfChaosCA Java applet.)

Originally Posted by Tom Plick

Even after Feigenbaum discovered the constant that bears his name, I find it amazing that he could use it to predict the bifurcations of all sorts of different functions - in our experiments, the numbers did not converge quickly, and so there was likely a lot of error involved there.

We talked a lot about the similarities between different types of physical phenomena - for instance, between phase transitions and the onset of turbulence - and we said how the mathematics at the boundary is similar in both cases. This may be, but even if the equations for the two phenomena match up, I still don’t see how the wide-reaching concept of universality can pair them. Phase transitions involve changes among the solid, liquid, and gaseous states of matter; turbulence involves a multiplicity of frequencies in an oscillating fluid, resulting in “turbulent or not turbulent”. I see no way to parallel these two problems, between three states and two states.

I agree a lot with Goethe’s views on color, in particular that perception of color is subjective. I read an article two months ago, talking about linguists’ analysis of color words in different languages. In ancient societies, some say, the people had fewer words for colors - the Greeks had only identified a handful. It is interesting to think that perhaps, back then, they actually couldn’t see as many colors as we can today. That would be very subjective, no?

Information about the linguistic analysis of color words can be found at the Straight Dope; there is also a nice chart here.

Originally Posted by Meridyth Mascio

In the personal struggle to determine whether my mind has been created to think mathematically, I have started to look at my general feelings of disdain for computers. In actuality, I realize their usefulness, but I struggle with my one-on-one daily encounters with them. It has always bothered me that if a program stopped running or an error occurred, then I could not obtain a direct response to my questioning what had caused the error. I think that I now see that I find it unnatural to communicate with this entity as it does not possess consciousness (a term described frequently in Non Serviam).

A computer does not perceive the contradictions that qualify humanity. Logic is not the only component of a person; beings are diverse. As mathematics underlies the structure of the world, the "chaotic" element are the perspectives of human beings. We see things as we perceive them, and so the world is defined intrinsically to each individual as such. However, are these decisions and perspectives "programmed" into us? If so, can is it possible for us to ever replicate this program?

So many other questions remain in my mind from this reading: Why have we been "created"? What is the general purpose of humanity? What is my individual purpose? Are humans guided by some unknowing force? How do religious beliefs factor into our existence?

I had never before considered the idea of an "intermediate" God, where the one who has directly created us has been created by yet another "higher" power. Is this chain infinite? Can a process then ever be broken down to one?…

These questions plague my mind. I feel as though humans are trapped in the midst of incomprehensible infinities, never able to grasp the full extent of everything in the universe and yet not ever quite able to break down one situation, one entity completely (some sort of fractal nature).

What I do realize is that the scope of the world is beyond me; more specifically, I think that the reality of one’s individuality is beyond any limited human perspective.

Originally Posted by John Sehi

When I first thought of dimensions, I thought about the usual 1-D, 2-D, and 3-D. I did not examine it farther, did not think that there would be a need for any more thought. A fractal dimension was something I didn’t know about. I didn’t really question their existence, but more of something that’s not talked about. Since learning about the Middle Thirds Cantor Set in which is less than one dimension and the Sierpinski Triangle that is between 1-D and 2-D, this just fascinated me. Especially the Menger Sponge which has infinite area but no volume.

I am surprised at what can be done with fractals. There are so many possibilities with fractals that it is sometimes hard to comprehend all that can be accomplished with mathematics.

Originally Posted by Matt Venanzi

It never ceases to amaze me, even before "Chaos & Fractals," how random, yet logical, strange, yet sensible, intricate, yet explicable the world is. Could God just be watching us discover every little intricacy he has made about the world, like a proud parent, or an accomplished inventor, marveling at their "creation"? What, then, is still left to be discovered? And I mean that, not in that we have discovered it all, but in that we have barely touched the tip of the iceberg!

Is it laughable, then, that we think we are so far "advanced"?

But how could God, Him Herself, have done all of this? It just seems like too much! Then again, is it all relative? Do we have the capacity to fathom all that there is? Is that "capacity" expanding ~ with each generation"? "Evolutionarily"? Do some have more "capacity" than others? This all seems so esoteric, yet comprehensible—– I guess its just another one of those woes of God… I’ll keep trying…

Originally Posted by Rachel Hensey Answer: NO.

It is true that God created the universe which is perfect in every way. The universe is without a doubt mathematically correct. Many people would argue that because God is the creator and the universe is mathematically correct then God is a mathematician. However, the definition of a mathematician is someone who studies how things work. Since God is all knowing and already knows how everything works, there is no need for him to study. There is nothing for him to solve. Thus, God is above mathematics. He is NOT a mathematician.

Originally Posted by Pat Rafferty

Click here for interactive NASA site It is incredible to see the wide variety of subjects that chaos has an affect on. Things that could not normally be understood are simplified under the study of chaos and fractals. For example, the giant red spot of Jupiter is explained as a calmness that can be found within the disorder of the gaseous system. This perfectly explains the idea of chaos, a structured order found within seeming randomness. Chaos theory even has conceptions on other areas such as weather prediction, population growth, and turbulence.

Originally Posted by Sean Houlihan

The concept of “personoids” from Non Servium is really interesting. What does this mean about God, life, etc? Could we be living in a fractal-like, self-similar world? Could our decisions be governed by mathematics?

In essence I hope nope. The reality portrayed in this work is a little scary. I would hope certain aspects of the fractal nature of personoids is not present in reality.

Pressure front predictions. Click to enlarge. Frederick G. Shulman, an early pioneer in mathematical weather forecasting, died earlier this summer ( July 29, 2005). Shulman was the first director of the National Meteorological Center, and is credited with helping “to transform numerical prediction from an idea to a functional reality.”

Given all of the chaos materials that I have read that highlight the work of Edward Lorenz in furthering our understanding of weather prediction, I am surprised that I have not come across a reference to Shulman’s importance.

It would be interesting to look into the models developed by Shulman and compare them to Lorenz’s, and to modern-day forecasting models.

Click here for more info on Frederick Shulman’s career.

Also check out the Climate Prediction Center of the National Weather Service for a look at current modeling efforts, and the linkage between weather and climate. There are a number of excellent animated simulations at this site.