Press "Enter" to skip to content

DNA in Series and Parallel


quattro-colored dna pastaModeling 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.

Modeling with discrete maps seems even more systems-like, because most maps don’t have explicit time behavior instead the population is often just a function of the previous population value. Sometimes the interactions are not even explicit. Because of this, chaotic behavior can be observed in a simple 1-D map, with the logistic map being the most famous. A 1D map is about as far from the actual physical situation as you can get!!

To me, the differential equation and discrete map approaches seem more mathematical than physical. They allow one to predict without an awful lot of understanding. Using Newton 2 to model a chain of balls on a spring is somehow different. Even though a differential equations system is used, the terms in the equations have a real immediacy because they refer to the fundamental concepts of position and velocity.

With population modeling being one of the mainstays of chaos theory (thanks to the work of Robert May and many others), it’s nice to see a new approach to population modeling that is likely to provide more understanding without skimping on the prediction.

In an exciting crossover between disparate disciplines, ecologists are now investigating the use of elementary electrical circuit models in modeling the flow/change of DNA as species migrate.Apparently they are finding very good results, i.e. the predictions of the models are quite accurate. The explanation for the results is straightforward: genetic "flow" is hindered when migration routes are blocked or crowded in an analogous way to electrons in a resistive material. Decrease the resistance by opening up more "corridors" for migration, and you have a set of parallel resistors - and overall lower total resistance. More genetic flow.

This work is spear headed by Brad McRae who works at the National Center for Ecological Analysis and Synthesis in Santa Barbara, California.

Does this new modeling make it easier to understand how species migrate? For me, it does. I can picture an animal as an electron, moving through a circuit I understand as well as predict because rich contextual words such as corridor and crowded carry such real-world heft.

I’m fascinated by the irony here - in order to get an understanding via a model of a biological species, the members of the species are modeled as inanimate objects.

You could say that I get a charge out of it.

Categories Evolution Mathematics Modeling

Aphorismically Consistent


Oscar Wilde Action FigureOne 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."

So words that don’t quite form sensible sentences are like the atoms in a metallic glass..

Which brings me to aphorisms, about which I am prompted to write because of an interesting piece in the Dec. 7, 2007 Chronicle in which Jay Parini attempts to define and classify aphorisms (even though Umberto Eco claims that "there is nothing more difficult to define.")

One often-present feature of aphorisms is paradox, which often occur because of a sneaky self-referential quality. (e.g. "This sentence no verb"or "This sentence has six words" are classic paradoxical constructions - but they ain’t aphorisms.)

Whatever an aphorism is or isn’t, my image of the King of Aphorisms is Oscar Wilde, who skewers paradoxically: "An ethical sympathy in an artist is an unpardonable mannerism of style," a haughty comment that has its inverse aphorism in Groucho Marx’s classic "I don’t care to belong to any club that will have me as a member".

Note that the Wilde and Marx quotes are not really self-referential because they don’t refer to them selves as sentences in the way that "This sentence no verb" does. Nevertheless, there is a whiff of paradox that makes the aphorisms resonate.

Like panes in a metallic glass…

Categories Literature & Poetry Mathematics

Purpose in the Universe


Storm in the Omega NebulaThe 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.

I have written before about the Foundation’s continued efforts to support even-handed research and scholarly writing about the religion science interface. This latest effort is most welcome, and great reading.

From Lawrence Krauss, the director of the Center for Education and Research in Cosmology and Astrophysics at Case Western comes a quote which crystallizes for me the systemic impedance mismatch between science and religion. If faith is used to interpret the science there can never be a testable hypothesis that is agreed to by all:

One is always free, as some people do, to interpret the laws of nature as signs of purpose, as for example Pope Pius did when Belgian physicist-priest George Lemaitre demonstrated that Einstein's general theory of relativity implied the universe had a beginning. The Pope interpreted this as scientific proof of Genesis, but Lemaitre asked him to stop saying this. The big bang, as it has become known, can be interpreted in terms of a divine beginning, but it can equally be interpreted as removing God from the equation entirely. The conclusion is in the mind of the beholder, and it is outside of the realm of scientific theory and prediction.

Read Krauss’ full essay here, as well as those of the other contributors The Templeton site contains other "conversations" about Big Questions.

Categories Religion Science

Eine Kleine Nachtfractal


Listen to Mozart by Renata SpiazziThe 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."

The starting musical "cell ", feeding back on itself through multiple recursions/variations, grows into an exquisitely beautiful musical "structure", with layer-upon layer of filigreed nuance present in every measure.

If mathematics and music can’t be separated, what then of art, which despite all efforts of artists to break free of figurative representation, still must be beholden to the mathematics of the 2-D plane? The infinite richness of fractals can break this canvas constraint, impelling creativity. Digital artist Renata Spiazzi writes in Why Fractals of the power of fractals to inspire her work: it was not the mathematics of the science that was interesting, but the fascination of the shapes, the colors and the illusion of space that was achieved in the images.

Spiazzi proclaims a fascinating mission that seems eminently achievable: to continue working towards a fractal that because of its beauty will bring tears to your eyes.

And no one should be surprised that Spiazzi is aided in her work by listening to music - none other than the Mozart whose cellular ideas spring forth into artistic creation … in time, frequency, and the 2-D palette: When creating fractals I like to have the music playing. I think it puts me in a high feeling mood, and it allows me to see things in the fractals I would not see with different sounds surrounding me.

See Spiazzi’s website for some beautiful images. Her Listening to Mozart is included at the top of this post.

Categories Art Fractals Music

Niels Bohr To Be Played by James Gandolfini


Niels and Werner in HBO’s new QMechI 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:

Meet two brainiacs with a lot to learn. Leonard and Sheldon can tell their quarks from their quantum physics, but have no clue how women add up. Leave it to their pretty new neighbor, just off a messy breakup, to teach them a thing or two.

Obviously this scenario is much more intractable than formulating hypotheses and exploring unexpected connections between phenomena. Luckily, there is a physics consultant for the show from UCLA, although he/she is as yet unidentified.


Wolfgang Pauli Walnuts illustrates the "operator" method of solving Schroedinger’s EquationSo what’s next? Clearly HBO has to get into the act and create a series titled The Quantum Mechanics. The high concept here is that QMech (the logo that will adorn all the associated merchandise) is the prequel to Michael Frayn’s surprisingly successful play Copenhagen. This one-hour weekly drama will be a no-holds barred look at Niels Bohr, Werner Heisenberg, and the Sopranos-like rule they held over the physics world between the world wars. Much of the action will take place at the Niels Bohr Institute, the Copenhagen version of Satriale’s Pork Store.

Niels and Werner and their proteges live their lives in strict obedience to their version of omerta: The Copenhagen Interpretation of Quantum Mechanics.

One major arc in Season 3 will be the aborted attempt of underboss Wolfgang Pauli Walnuts to establish his own harsh code of conduct: the Pauli Exclusion Thing, where no more than two physicists are ever allowed in a closed room, or permitted to be co-authors on a paper.

In a brilliant cross-linking similar to the finale of the Bob Newhart show, the Heisenberg Uncertainty Principle is used to explain the final episode of The Sopranos.

Where do I sign up to be a script consultant?

Or an extra?

Categories Physics

No Comment(s) - Cool vs Hot Media


I’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

Categories Blogging

Wada Wada Wada


Wada - minus the reflecting ballsNormally, 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)

Categories Fractals

Sci-Fi Economics and Intelligent Design


Over 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

Categories Modeling

Modeling The Universe: The History of Cosmology


So 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:

The physical universe was anthropocentric to primitive man. At a subsequent stage of intellectual progress it was centered in a restricted area on the surface of the earth. Still later, Ptolemy and his school, the universe was geocentric; but since the time of Copernicus the Sun, as the dominating body of the solar system, has been considered to be at or near the center of the stellar realm. With the origin of each of these successive conceptions, the system of stars has ever appeared larger than was thought before. Thus the significance of man and the earth in the sidereal scheme has dwindled with advancing knowledge of the physical world, and our conception of the dimensions of the discernible stellar universe has progressively changed.

See the Cosmology site for much more.

Categories Mathematics Modeling Science

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:

The Fractal Flame algorithm is a member of the Iterated Function System (IFS) class of fractal algorithms. A two-dimensional IFS creates images by plotting the output of a chaotic attractor directly on the image plane. The fractal flame algorithm is distinguished by three innovations over text-book IFS: non-linear functions, log-density display, and structural coloring. In combination with standard techniques of anti-aliasing and motion blur the result is striking image variety and quality. The guiding principle of the design of the algorithm is to expose and preserve as much of the information content of the attractor as possible. We found that preserving information maximizes aesthetics.
             <img alt="Image hosted by" src="" />

There’s plenty of software available to download for creating flame fractals, including UltraFractal as well as a lot of freeware. Some interesting UltraFractal images are located at the blog site Where Art and Math Collide, which is apparently the name of flame fractal created by the blog’s author. The most interesting images I have seen are at the Fractal World Gallery of Cory Ench and the webshot site of Roger Johnston. (click on the image at left and view Johnston’s work). It seems that the software Apophysis , a freeware fractal flame editor, is the main rendering tool of choice for these beautiful images. (The image at the top of this post was created using Apophysis).

To go back in time a bit (circa 1995), read the Field and Golubitsky article Symmetric Chaos How and Why from the notices of the AMS.

I want to credit Ian Stewart for turning me on to symmetric chaos. I originally read his Christmas in the House of Chaos piece in Scientific American back in 1992, which is a clever piece featuring the Tractor family (John Tractor, Pointer Tractor, and Lorenza Attractor) making Christmas ornaments iteratively. (Unfortunately I have not been able to find this article online).

Categories Art Fractals Mathematics