a non-linear space for students of chaos and fractals....


DNA in Series and Parallel

quattro-colored dna pasta
Modeling biological systems sure seems to be radically different from modelling something such as a chain of balls on springs. For balls on springs, Newton's 2nd law is written for each mass, yielding a pretty straightforward system of differential equations. The positions and velocities of the masses in time are the solutions to the system. Each variable x(t) in any of the differential equations refers directly to the actual position of a specific mass. For bio models, however, modeling is done more at a meta-level using a systems approach. For example, you wouldn't normally see Newton II applied in pharmokinetic modeling; instead a compartment model where the compartments are body systems such as blood stream, gut, etc. is typically used, often with great predictive power.

What about population modeling, and especially modelling of interacting species? Is this closer to balls on springs, or a compartment model? The differential equations that are typically used to predict the behavior of these populations employ mathematical expressions of the interactions chosen to produce a desired population behavior. Predator-Prey, mutual competition, and cooperation models are really the same with just minor changes to the terms in a differential equation system. I then think of this type of modeling more like compartments - the interaction terms are plugged into the differential equations in a manner analogous to building models with compartments.

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Aphorismically Consistent

Oscar Wilde Action Figure
One of the wonderful things about words and sentences is the ability to create paradoxical statements that are more clever than they are troubling. No, verbal paradoxes aren't typically offensive in the same way that mathematical ones are. Mathematical statements are usually not labeled "paradoxical;" instead, they are described as contradictions, or, even worse, inconsistent statements. (Try to think of a paradoxical mathematical statement that isn't a contradiction - go ahead, I dare you)

Mathematical inconsistencies are often nasty, unwanted, theorem-killers, and their presence is usually a sign of things gone awry. Actually, inconsistency and/or incompleteness is built into all of the mathematics we do - sort of. After all, Gödel showed that mathematics cannot be both consistent and complete.

Ahh, but words are a different story. Words are the atoms of sentences, and if somehow the sentence doesn't make "sense", well, consider metal atoms that are frozen into a glassy state, but in fact want to be in a nice, regular matrix. These metallic glasses are then "inconsistencies."

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Purpose in the Universe

Storm in the Omega Nebula
The Templeton Foundation is sponsoring a fascinating, ongoing debate among scientists, humanists, and theologians concerning the ultimate question of who and what we are. Titled Does the Universe have a purpose, a diverse group of figures such as Elie Wiesel, David Gelertner, and Jane Goodall , weigh in with their opinions. Answers range from Unlikely, to Not Sure to Indeed, to I Hope So.

Even though a faith-based organization, Templeton's mission is remarkably secular:

The mission of the John Templeton Foundation is to serve as a philanthropic catalyst for discovery in areas engaging life’s biggest questions. These questions range from explorations into the laws of nature and the universe to questions on the nature of love, gratitude, forgiveness, and creativity.

Our vision is derived from John Templeton’s commitment to rigorous scientific research and related scholarship. The Foundation’s motto “How little we know, how eager to learn” exemplifies our support for open-minded inquiry and our hope for advancing human progress through breakthrough discoveries.

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Eine Kleine Nachtfractal

Listen to Mozart by Renata Spiazzi
The melodic recursiveness of most music, especially as manifested in the crystalline, almost-mathematical purity of Mozart compositions, suggests the presence of fractal-like structures that exist both in time and the frequency domain - structures that are both solid and ephemeral, logical and otherworldly.

Some may hear (even if they don't articulate) a richness that is reminiscent of fractal construction. Marin Alsop, Music Director of the Baltimore Symphony Orchestra, describes Mozart's popularity as the result of "the depth of the music and ... the fact that Mozart makes contact with our inner selves. Maybe it's because of his organic approach to composition - taking a small, cellular idea and developing it into something beautiful, he takes you by surprise but also comforts you."

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Niels Bohr To Be Played by James Gandolfini

Niels and Werner in HBO's new QMech
I always knew that, even if one couldn't get a job doing physics in a lab or university, at least you'd be able to do anything else on the planet, and do it well.

So it's great to know that Hollywood continues to lure physicists, albeit only two so far (that I know of).

I used to think that the best physics job was that held by Andre Bormanis, who served as the science consultant for Star Trek for many years. While recognizing his envied stature, he does seem nonplussed by the incongruity of his role, remarking that " Like physics, storytelling involves problem solving, formulating hypotheses, exploring unexpected connections between phenomena, and seeking a solution"

Now there is Big Bang Theory ... the wacky new sit-com from CBS. The nano:

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No Comment(s) - Cool vs Hot Media

cool_media.jpgI've been writing FractaLog posts for almost two years now, and I've wondered why the number of comments per post is about as close to zero as you can get. Now I know that something is being read, or at least stared at (if I can trust the Squarespace statistics reports, or those produced by Google Analytics), so I'm left to wonder what's up. Are the posts written in such a way that discourages comments?

I read an interesting take on this situation recently by Phil Ford in his Dial M for Musicology blog. In his post Blog media hot and cool he writes that "I've often noticed how it's the posts that I took most seriously ... the ones that take 90% of my blog-related time, that end up being ignored in the blogosphere, while the ones I spend 5 minutes putting together get the mad blog love." Ford's post interests me because in it he brings up Marshal McLuhan's ideas of hot and cool media, with Ford's thesis that blogs are "cool media" , in the sense that "They're less "dense," less "high-definition," and offer a wider variety of ways one can react to them; they have open pockets, lots of interstitial spaces that others can fill in for themselves." Ford is writing his piece as a cautionary tale about the future of academic writing if too much becomes "cool" b/c of the growing market share of the blogosphere. (For an even more dire view, see Cool Media and the Virgin Mary by K. P. Hawes, the site that produced the image at the top of this post)

But how does this translate into whether comments to posts appear or not? It might be too simple to conclude that the actual posts are too long, filled with too many links, too discursive, i.e. they are definitely NOT COOL. 

So I should definitely try to shorten my


Wada Wada Wada

Wada - minus the reflecting balls
Normally, symmetric fractals doen't possess the type of arresting, intriguing beauty of the asymmetric ones. Wada basin fractals, which are often symmetric, do have a unique visualization that I find more aesthetically appealing.

Perhaps the best thing about Wada basins is that they can be simulated with easy-to set-up lighting and symmetric arrangements of reflecting balls - e.g. Christmas ornaments. Some instructions for doing this are at The Optical Gasket Lab , a fun module from the Yale Fractals course designed by M. Frame and B. Madelbrot himself.

The real instantiation of the basin boundaries allows for interesting comparisions of these real optical images and computer-generated graphics. See the Secret Inner Life of Mirrored Spheres: Wada Basins by for a set of beautiful images. (The image at the top of this post is from this site)


Sci-Fi Economics and Intelligent Design

invisiblehand.jpgOver two years ago John Allen Paulos, Temple mathematician famous for his book Innumeracy, wrote a provocative op-ed that pointed out the analogies between evolution of organisms and the evolution of the free-market system. His point is that the complexity of a free market economy is not designed by anyone, and ism, in fact reasonably described by Adam Smith's Invisible Hand model. Moreover, in a delicious irony, it is often the adherents of such a free-market system that deny the possibility of biological evolution without Intelligent design.

Paulos does point out the limitations of the analogy, but his piece raises an intriguing question for me: starting from scratch, how would one re-design the free-market system? Is there any other imaginable way to do it? I ask this because, as a science fiction reader, I often read about alternate life forms that may exist on other planets. (e.g. silcon-based) While the fundamental molecular bases of life as we know it seems to lend itself to another structure, I can't imagine a different free-market economy. What would it look like?

Of course, the fact that I can't imagine an alternate free-market economy may only be a result of my woeful lack of knowledge of economic theory and models.

Or a lack of science fiction writing that focuses on the economics of alien civilizations, which is a perfect excuse for a new Ursula K. LeGuin novel to be titled The Invisible Left Hand of Darkness...


Modeling The Universe: The History of Cosmology

cosmology.jpgSo much of the history of mathematics and science is encapsulated in the study of the heavens. One can argue that the first modeling may have been early views of the universe, and the planets riding on their celestial spheres in a cascade of epicycles. The American Institute of Physics has set up a wonderful site devoted to the history of cosmology that is a terrific resource for learning more about these models, and how our ideas of the solar system and universe have matured.

Titled Cosmic Journey: A History of Scientific Cosmology, the site is ddivided into two broad , complementary areas - History (e.g. The Greek Worldview, The Mechanical Universe, Big Bang) and Tools (The Naked Eye, The First Telescopes, Spectroscopy).

For good reason, the site devotes ample space to Harlow Shapley, whose pioneering work in 1916 on globular clusters and the real size of the heavens exploded our view of the universe and caused us to reappraise our position in it. Shapley write eloquently about how his discoveries, and the work of all of those before him, have necessarily changed our position as observers within the physical universe, and hence the way we model the universe and ourselves:

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Flaming Symmetric Fractals

Apophysis Fractal Flame (click to enlarge)
I've seen the flames - fractal flames - and they are amazing. Originally an outgrowth of the ideas developed in Symmetry in Chaos by Field and Golubitsky, fractal flames are related to the Chaos Game because they are created by tracking the iterates of starting points as they are mapped into other points. As in the Chaos Game, the iterates reveal, over time, the structure of the attractor associated with the map. Flame fractals use a combination of non-linear maps and very inventive approaches to coloring/visualization.

Flame mathematics is interesting enough on its own, but the intrigue of flames is the ability to generate images of surreal organic beauty. In the following excerpt from The Fractal Flame Algorithm by Scott Draves for the Cosmic Recursive Fractal Flames site, aesthetics are essential:

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