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Playing a Mandelbrot Set - Hum a Few Iterations and I'll Fake It


    Eventually by Sergio Lazo     

In the annals of how-to songs one will find everything from delicate interactions with a weapon of mass destruction - e.g. How to Dismantle an Atomic Bomb by U2 - to an even more dangerous and potentially deadly situation - How to Handle a Woman from Camelot comes to mind.

(Yes I know that How to Dismantle an Atomic Bomb is not the name of an actual song! So suggest a better title to match up with How to handle a Woman.)

And now comes the ultimate how-to for the fractally-minded: Mandelbrot Set by Jonathan Coulton, the Contributing Troubadour for Popular Science Magazine and composer of "well-crafted geek folk-pop. Hilarious but heartbreaking songs about mad scientists, robot armies and self-loathing giant squids." His is the voice of "every one of us who has ever sat despairingly on the floor, surrounded by parts of an Ikea endtable, weeping over our Allen wrenches.

Coulton’s work is truly unique. Check out his site for all of his tunes. And there are plenty of them: In 2005-2006 he recorded and published a new song podcast every week. All of the lyrics are there, as well as guitar tabs and videos.

Mandelbrot Set is funny, incredibly clever, and mathematically correct! Some excerpts…

mandelbrot-1.jpg Pathological monsters! cried the terrified mathematician Every one of them is a splinter in my eye I hate the Peano Space and the Koch Curve I fear the Cantor Ternary Set And the Sierpinski Gasket makes me want to cry And a million miles away a butterfly flapped its wings On a cold November day a man named Benoit Mandelbrot was born His disdain for pure mathematics and his unique geometrical insights Left him well equipped to face those demons down He saw that infinite complexity could be described by simple rules He used his giant brain to turn the game around And he looked below the storm and saw a vision in his head A bulbous pointy form He picked his pencil up and he wrote his secret down mbrotsong.jpg Take a point called Z in the complex plane Let Z1 be Z squared plus C And Z2 is Z1 squared plus C And Z3 is Z2 squared plus C and so on If the series of Z’s should always stay Close to Z and never trend away That point is in the Mandelbrot Set Mandelbrot Set you’re a Rorschach Test on fire You’re a day-glo pterodactyl You’re a heart-shaped box of springs and wire You’re one bad*ss f***ing fractal And you’re just in time to save the day Sweeping all our fears away You can change the world in a tiny way

Then there’s That Spells DNA, a song that should be the theme song of the Human Genome Project…

If it says TGGTCGAAC Then you might get the cancer If it says GTCACGACAGG Then you shouldn't’t eat shrimp or nuts If it says TATACACATATCCTCGT Then you’ll probably wish that you didn't’t know

If you are inclined to pick up a guitar and try out Mandelbrot Set on your own, be sure to follow the Guitar Tab notes exactly:

  • E E Esus4 Esus4 and other fun noodling things as you recite mathematical formulae, and so on.
  • then more pre-chorus bit: A A B B Ab Ab C#m C#m, A B E
  • Then the chorus (I encourage dancing while playing it: take no pity on fragile objects near you, unless they are fragile relatives) B E B E E7, F#m Amaj7 F#m Amaj7 F#m A E
  • Repeat various bits until achieving mathematical enlightenment.

Achieve musical and mathematical enlightenment at the same time by spending some time with Coulton. Whether it’s Tom Cruise Crazy, Furry Old Lobster, or The Town Crotch, Coulton is one bad*ss f*****g troubador!

Categories Fractals Music

M.C. Escher: Chaos and Order and Back Again


Order and Chaos. Click to enlarge.
In an earlier post I commented on the ability of M.C. Escher to capture the infinity of time and space in his woodcuts, drawings, and lithographs.

Another common thread in many of Escher’s works is the sense of order arising out of chaos - or is it chaos out of order? A master of graphic puns that simultaneously emphasize and blur the demarcation between figure and ground, Escher does seem to lean more in the direction of order out of chaos. In The Magic of M.C. Escher , a fabulous compendium of Escher’s works that includes unpublished sketches from his workbooks and excerpts from his correspondence with family and colleagues, this direction is clear:

The consistency of the phenomena around us, order, regularity, cyclical repetitions and renewals, have started to speak to me more and more strongly all the time. The awareness of their presence brings me repose and gives me support. In my pictures I try to bear witness that we are living in a beautiful, ordered world, and not in a chaos without standards, as it sometimes seems.

For a very good intro to a broad selection of Escher’s most famous works, visit the M.C. Escher - Life and Work on-line tour at the National Gallery of Art. From the program notes for the Order and Chaos lithograph pictured in this post:

Escher described this print as a symbol of order and chaos: order represented by the polyhedron and the translucent sphere; chaos depicted by the surrounding broken and crumpled cast-off objects of daily life. The artist believed the polyhedron (a solid figure with many sides) symbolized beauty, order, and harmony in the universe. Yet, he rendered chaos with equal care, as in the exquisitely drawn sardine can at upper left.

But was Escher so one-sided, or should I say unidirectional in his rendering of order and chaos? Consider Development I - an early woodcut dating from 1937. From the National Gallery of Art notes:

This is one of Escher’s earliest prints to demonstrate his theory of the regular division of the plane," which he described as follows: "A plane, which should be considered limitless on all sides, can be filled with or divided into similar geometric figures that border each other on all sides without leaving any empty spaces. This can be carried on to infinity according to a limited number of systems."

Development I. Click to enlarge.
Even if his original motivation for this woodcut was more an exercise in the geometry of dividing space, I can’t help but see Development I as the lizard figures in the middle, frozen lock-stepped in a chaotic dance arising out of the ordered regularity of the chessboard at the edges of the woodcut. The light shading of the chessboard border is what gives me this impression, probably because the figure/ground resolution of the lizards/chessboard is not ambiguous. (Although it must be pointed out that Escher’s works rarely succeed on one visual puzzle alone. In Development I the figure/ground resolution of the white and black lizards is itself highly ambiguous.) 583047-691196-thumbnail.jpg

 <em>Reptiles</em>. Click to enlarge.  

Or maybe Escher had a more ambiguous view of the order-chaos continuum, one that could incorporate a symmetry in the paths that could be taken. Which brings me to Reptiles, his 1943 lithograph. (This image was my initial introduction to Escher - a colorized Reptiles was used as the cover of the inaugural Mott the Hoople album, which I first listened to in 1969.) In Reptiles the lizards emerge from the ordered 2-Dimensional world of the journal page to venture out into the 3-D world, marching around desktop items snorting smoke out of their snouts. After a brief foray into this 3-D world of freedom, they reenter the 2-D page.

So have the lizards learned their lesson? Is the chaotic world of 3-D too much for the paper lizards? Or do they never learn, repeating the same procession from order-to-chaos-to-order forever? (Note that I could just as well have written chaos-order-chaos-order. There is no designated starting point here.)

Escher’s image is wonderful in its implied motion. The action in Reptiles suggests an ongoing infinite process as the reptiles leave, then re-enter 2-D. But this endless loop then suggests that the lizards are moving in the 2-D world of the journal page. As he does so often, Escher again depicts the infinity of time in a way that is impossibly clever, meaningful, orderly and chaotic.

Categories Art Chaos

Modeling Acts of God, Part 1


Hurricane structure. Click to enlarge.   

This post is the first in a series of the state of modeling very powerful, nasty natural phenomena such as hurricanes, earthquakes, and tsunami.

Depending on where you live you may have trouble acquiring or affording insurance protection form these so-called "acts of God. " Whatever these AOG’s might be, they are often home- and life-threatening. Can these AOG’s be modeled successfully, i.e. how well are they understood and can their generation, properties, and behavior be predicted with any accuracy?

Along with the understanding and prediction provided by standard models for these event, it is natural to wonder how global warming contributes to these phenomena. Do models for these AOG’s compensate for GW’s effect? This is a very pressing question. Anecdotally, many claim that the events are both more frequent and more severe, with natural tendency to assign blame to global warming.

Consider hurricanes. Hurricanes are pretty well "understood" in terms of how they start and move. Proto-hurricanes start as small tropical vortices , which themselves originate because of rotational effects of the earth on atmospheric gases. Kerry Emanuel, Professor of atmospheric sciences at MIT explains this in Hurricanes: Tempests in a Greenhouse:

In the part of the tropics where the sea surface is warm enough and the projection of Earth's angular velocity vector onto the local vertical axis is large enough, random small-scale convective currents sometimes organize into rotating vortices known as tropical cyclones. In computer models of the tropical atmosphere, such organization can happen spontaneously, but usually only if a combination of ocean temperature and rotation is somewhat higher than those observed in nature. In subcritical conditions, some trigger is necessary to initiate the vortices, and in the terrestrial atmosphere tropical cyclones only develop from preexisting disturbances of independent origin. In mathematical parlance, tropical cyclones may be said to result from a subcritical bifurcation of the radiative convective equilibrium state. About 10% of them develop in the Atlantic Ocean, where the disturbance is often a 100-km-scale "easterly wave" that forms over sub-Saharan Africa and then moves westward out over the Atlantic. When its maximum wind speed exceeds 32 m/s, it, by definition, becomes a hurricane.

Note the description of random small-scale convective currents organizing spontaneously. This description reminds me of the descriptions of the Great Red Spot of Jupiter - which is a classic case of order arising out of chaos, and which is featured in Gleick’s Chaos: Making a New Science

In addition to how hurricanes start, Emmanuel describes how hurricanes can be modeled thermodynamically using the idea of the Carnot cycle, i.e. a heat engine. (Click here for a nice Carnot Cycle applet).

The mature hurricane is an almost perfect example of a Carnot heat engine whose working fluid may be taken as a mixture of dry air, water vapor, and suspended condensed water, all in thermodynamic equilibrium. The engine is powered by the heat flow that is possible because the tropical ocean and atmosphere are not in thermal equilibrium. This disequilibrium arises because, thanks to the greenhouse effect, the ocean must lose heat by direct, non-radiative transfer to the atmosphere to balance the absorption of solar radiation and back radiation from the atmosphere and clouds. The heat transfer is accomplished mostly by evaporation of water, which has a large heat of vaporization. To maintain substantial evaporation rates, the air a short distance above the sea surface must be much drier than would be the case were it in equilibrium with the sea.

Modeling hurricanes as Carnot cycles is not a far extrapolation from Sadi Carnot ’s own views about the general atmosphere, and what drives it:

We must attribute to heat the great movements that we observe all about us on the Earth. Heat is the cause of currents in the atmosphere, of the rising motion of clouds, of the falling of rain and of other atmospheric phenomena ...


Carnot hurricane cycle. Click to enlarge.

The heat engine model for hurricanes is an excellent example of a model that has been extrapolated well beyond its original intent. While the Carnot model works very well for a very special system, in this case heat engines, it is quite amazing that it’s features and predictive power can be transferred to a totally different system - in this case a hurricane. This model transferability is an indication that Carnot’s model is more than robust - it is capturing a very real property of the universe. In this case it is the primacy of the 1st and 2nd Laws of Thermodynamics.

Any thermodynamic model must take entropy into account. The intensely rapid swirling of air near a hurricane’s eye, with trees, houses, and vehicles sucked off of the earth’s surface is the poster-child of the massive disorder that marks a literal entropic explosion:

In equilibrium, the planet must generate entropy, and the vast majority of that entropy is produced in the atmosphere, mainly through the mixing of the moist air inside clouds with the dry air outside them and through frictional dissipation by falling raindrops and snowflakes. Were it not for moisture in the atmosphere, the entropy would have to be produced by frictional dissipation of the kinetic energy of wind. The resulting air motion would be too violent to permit air travel.

Because the idea of entropy production at equilibrium may seem counter-intuitive, I recommend reading The Second Law of Thermodynamics and the Global Climate System: A Review of the Maximum Entropy Production Principle. This 2003 Geophysics Review by Ozawa, Ohmura, Lorenz, and Pujol does an excellent job of explaining the reasons and causes of entropy production. This paper also displays the excellent success of modeling the atmosphere in the Carnot fashion.

Now back to hurricanes. Emmanuel does indicate that the Carnot cycle model applied to hurricanes yields a pretty accurate prediction for maximum wind speed that depends on ocean and atmospheric temperatures. But what about global warming? How does it play a role? As described above, the greenhouse effect leads to a more pronounced thermal disequilibrium between the tropical oceans and atmosphere. The more the greenhouse effect, the higher the surface temperatures, and the greater the disequilibrium, with the net result that more entropy is produced. The hurricane vortices serve as big entropy channels, and thus these must grow more intense to handle the larger load. So there you have it - more global warming leads to more entropy which leads to bigger, nastier hurricanes.

So where are we? With a basic understanding of hurricanes as provided by a Carnot model, and the prediction of maximum wind speed, it would seem that not much more needs to be done.

Not so fast - can scientists predict WHERE and WHEN hurricanes will spontaneously generate? Or how severe they will be, and how fast they will move? Hurricane models are still frustratingly inexact, relying on many assumptions as well as statistical measurements. But the state of prediction is much better than it was in the past.

See the Feb. 2007 NOAA article on Predicting Hurricanes: Times Have Changed for much more on hurricane modeling and prediction.

Categories Modeling Understanding & Prediction Weather & Climate

As the World Turns - Foucault's Eco


Feb. 3rd marked the 156th anniversary of Léon Foucault’s demonstration that the earth revolved on its axis. The experiment was brilliantly simple: let a large pendulum swing back and forth over the course of a day, with a sharp tip hanging off of the pendulum bob making patterns in a bed of sand below it. As the pendulum bob moves back and fort in a vertical plane the earth rotates beneath it, and this rotation can be measured by the marks in the sand. (For a nice intro to Foucault’s Pendulum, check out the scriptographic booklet on the subject at the California Academy of Sciences.)

Foucault did not stop with simply showing that the earth rotates. He was able to predict how fast the pendulum would appear to rotate because of the fact that the rate at which the marks carve out a pattern depends on the latitude of the pendulum. So, for example, at one of the poles the earth will spin once in 24 hours, while at the equator the earth does not spin with respect to the plane of the pendulum. Anywhere in between the amount of time it takes for the pendulum to make one complete rotation is given by

T = 24/sin L

Here T is the time for the pendulum to return to its original position (in hours), and L is the latitude at which the experiment is carried out.

The pendulum demonstration was a huge success, showing as it did in a very simple way what could not be felt by earthly inhabitants - the rotation of the earth they were standing on. (It was also a majestic demonstration, being conducted in the the Panthéon in Paris.) The fact that Foucault could actually predict the exact timing of the rotation made him akin to the explorers who mesmerized remote villagers with abilities to predict eclipses, and he achieved physics rock-star status - maybe the most until Einstein.

There is a wonderful passage that starts Umberto Eco’s novel Foucault’s Pendulum, an abridged version of which I often read to my classes at some point during a semester, depending on the class (physics, differential equations, or chaos and fractals). I don’t believe that there can be any better homage to Foucault, in writing any more soaring than Eco’s:

That was when I saw the Pendulum. The sphere, hanging from a long wire set into the ceiling of the choir, swayed back and forth with isochronal majesty. I knew -- but anyone could have sensed it in the magic of that serene breathing -- that the period was governed by the square root of the length of the wire and by pi, that number which, however irrational to sublunar minds, through a higher rationality binds the circumference and diameter of all possible circles. The time it took the sphere to swing from end to end was determined by an arcane conspiracy between the most timeless of measures: the singularity of the point of suspension, the duality of the plane's dimensions, the triadic beginning of pi, the secret quadratic nature of the root, and the unnumbered perfection of the circle itself. ... The copper sphere gave off pale, shifting glints as it was struck by the last rays of the sun that came through the great stained-glass windows. Were its tip to graze, as it had in the past, a layer of damp sand spread on the floor of the choir, each swing would make a light furrow, and the furrows, changing direction imperceptibly, would widen to form a breach, a groove with radial symmetry -- like the outline of a mandala or penticulum, a star, a mystic rose. No, more a tale recorded on an expanse of desert, in tracks left by countless caravans of nomads, a story of slow, millennial migrations, like those of the people of Atlantis when they left the continent of Mu and roamed, stubbornly, compactly, from Tasmania to Greenland, from Capricorn to Cancer, from Prince Edward Island to the Svalbards. The tip retraced, narrated anew in compressed time what they had done between one ice age and another, and perhaps were doing still, those couriers of the Masters. Perhaps the tip grazed Agarttha, the center of the world, as it journeyed from Samoa to Novaya Zemlya. And I sensed that a single pattern united Avalon, beyond the north wind, to the southern desert where lies the enigma of Ayer's Rock. ... I knew the earth was rotating, and I with it, and Saint-Martin-des-Champs and all Paris with me, and that together we were rotating beneath the Pendulum, whose own plane never changed direction, because up there, along the infinite extrapolation of its wire beyond the choir ceiling, up toward the most distant galaxies, lay the Only Fixed Point in the universe, eternally unmoving. So it was not so much the earth to which I addressed my gaze but the heavens, where the mystery of absolute immobility was celebrated. The Pendulum told me that, as everything moved -- earth, solar system, nebulae and black holes, all the children of the great cosmic expansion -- one single point stood still: a pivot, bolt, or hook around which the universe could move. And I was now taking part in that supreme experience. I, too, moved with the all, but I could see the One, the Rock, the Guarantee, the luminous mist that is not body, that has no shape, weight, quantity, or quality, that does not see or hear, that cannot be sensed, that is in no place, in no time, and is not soul, intelligence, imagination, opinion, number, order or measure. Neither darkness nor light, neither error nor truth.

The power of Foucault’s idea will continue to echo through the centuries, reminding us that, even though we believe ourselves at the unmoving center of all things, it is we who all move as one.

Categories Literature & Poetry Science Understanding & Prediction

When Scientists Argue About Religion and Science


God the GeometerIn his July 2006 Addressing the public about science and religion opinion piece for Physics Today, Murray Peshkin - a theoretical physicist at Argonne national Lab - argues passionately about the benefits of scientists addressing the boundary between religion and science in a public forum. Peshkin describes how his public appearances have led to some interesting give-and-take - learning experiences for him and his audiences.

Peshkin’s position is summarized by the title and opening line of an essay published in the Chicago Tribune magazine:

SCIENCE AND RELIGION: CAN THEY LEARN TO LIVE WITH EACH OTHER? The answer is that they can and they must, or we will all suffer the consequences.

Given the nature of the polarized debates surrounding Intelligent Design, and particularly the Dover court case, it’s clear that two topics need a good deal of elaboration: describing how a scientist uses the word theory, and the need for a theory to be falsifiable in order to count as science.

Peshkin uses an interesting, and effective hypothetical situation to drive home the falsifiable idea, which leads to his main point about the inherent differences between a scientific and religious world view:

Science is based entirely on experiment. To illustrate what that means, I raise the possibility that the world was created three hours ago with all our memories and everything else in place, and I encourage discussion of that possibility. Science cannot refute it. That leads into the notion that a proposition is not a scientific theory at all unless it's falsifiable in principle. Absent a possible experiment, science does not even know the meaning of the proposition. Nobody is surprised when I confess that I'm really not thinking of three hours ago, but of a few thousand years. Science and religion have different assumptions, different rules of inference, and different definitions of truth or reality. The fence that surrounds science is the test by experiment. That fence is both the greatest strength and the most fundamental limitation of science, and it needs to be respected from both sides. Scientists may have opinions about religion, but they cannot honestly invoke the authority of science when they try to apply the logic of science on the other side of the fence. Similarly, creationists and advocates of intelligent design should not pretend to be conducting a scientific argument.

Peskin’s appearances are positive steps towards reconciling the deep chasm between science and religion that exists in the vacuity of our media, and his account of these efforts in the public sphere is illuminating.

Just as illuminating has been the response to Peshkin’s article. In the recent issue of Physics Today (Feb. 2007), a number of scientists have written letters to the editor of Physics Today protesting Peshkin’s arguments and methods. While disappointing, this response is not surprising given the unimaginably entrenched feelings - of faith, reason, and scientific "turf" that color so many of our beliefs and interactions with others.

The letters, and Peshkin’s response are themselves important documents in the creation of the dialogue that Peshkin is calling for.

Click here to read these letters and Peshkin’s response. A visit to Peshkin’s Religion and Science resources Page is also recommended, as it contains links to mainstream religious sources that stress that evolution and religious belief are not mutually exclusive.

Categories Religion Science

The 10th Carnival of Art


Carnival by Elaine Normandy , who uses Gnofract 4D - freeware fractal software for Linux.I found out earlier today that my post on Art and Science Transvergence: Glowing Bunnies was listed in the latest Carnival of Art, a blog carnival that has been hosted by Australian artist Jennie Rosenbaum since June 2006.

I really appreciate the link there - so thank you Jennie!

I urge all readers here to check out Carnival of Art - #10 . With categories in Art History, Art News, Art Philosophy, Artworks, Creating Art, and The Biz, there are a lot of intriguing, compelling, beautiful, and informative posts.

And don’t stop there - read the Carnivals of Art #1 through #9.

This is my first experience with a blog carnival, and I am very impressed with the ability to reach more readers, and find more amazing blog posts than I can imagine finding by other routes, including Technorati.

For more on blog carnivals, including how to find them, submit a post, or even host one, check out

For a detailed analysis of the ability of blog carnivals to be important engines of social networks, and their potential to "become online equivalents of not just TIME magazine, but also GQ, Vogue, Parenting, National Geographic, People, and, why not, Science and Nature" see Blog Carnivals And The Future Of Journalism, an excellent post by Bora Zivkovic in his Science and Politics blog.

Categories Art Blogging

Luminiferously Aethereal Dark Matter and Energy


      Star Traveler from Angelbert Metoyer's Dark Energy Splitting the Universe . Click to enlarge.At the end of the 19th Century century many physicists were still searching for the luminiferous aether - the mysterious, invisible substance that permeated the universe - the medium that (it was presumed) needed to exist in order to support light waves.

At that time, waves were understood to be mechanical disturbances in a medium - in effect, shapes that propagate through the medium transporting energy from source to receiver with a speed dependant on the medium itself . Water waves, waves on a string, sound waves in a column of air - all of these phenomena relied on matter to support these moving shapes.

The Aether was supposed to be the invisible stuff that somehow oscillated as the shapes of light waves passed from point A to point B. It had to be really odd stuff - invisible, for one thing, and also incredibly resilient, because it had to support an unbelievable speed - the speed of light.

Nevertheless, by this point the mechanical qualities of the aether had become more and more magical: it had to be a fluid in order to fill space, but one that was millions of times more rigid than steel in order to support the high frequencies of light waves. It also had to be massless and without viscosity, otherwise it would visibly affect the orbits of planets. Additionally it appeared it had to be completely transparent, non-dispersive, incompressible, and continuous at a very small scale. (wikipedia)

In other words, aether couldn’t be seen, weighed, touched or tasted, but it was stronger than anything known to man. And it was everywhere - in your shoes, hair, mouth, nose, and subway tunnels. A really strange thing for physicists to believe in.

The Michelson-Morely experiment in 1887 shattered the idea that the aether existed. M & M were looking for evidence of the aether, and truly expected to see this evidence in their carefully-designed and high-precision experiment. As landmark experiments go, this was earth shattering for a number of reasons, not the least of which was that it was a failed experiment - it yielded a "null result." It’s hard not to rhapsodize about an effort that failed so mightily that it revolutionized not just the theory of electromagnetic waves, but of the nature of the universe - ultimately leading to Einstein’s Special Theory of Relativity in 1905.

Why was the aether believed to exist with absolutely no direct evidence before M&M? This is a case of predictions being so successful that scientists come to totally believe the components of the model that is used for the prediction. For the aether, it was a fact that incredible success had been achieved by modeling waves as mechanical disturbances. So, if light was a wave, it needed to be supported by a mechanical medium, even if we couldn’t possibly tell that it was there.

This phenomenon of belief that a model is in fact the reality of a situation is certainly not the first time something like this has happened - just consider the flawed models of heat as a bubbling caloric, for example. But it may be the biggest case of a model requiring that something exist - something that by it’s very name is so aethereal that it can’t ever be sensed.

Which brings be to the reason I am writing this post in the first place. Jump ahead a century and we are now in an eerily similar place in our view of the physical universe. We believe in magical, mystical, mysterious stuff that, according to our models, are not just present (though invisible), but dominant in terms of amount and effect. Even crazier, there’s two different types of this nutty stuff. I am referring, of course, to Dark Matter, and Dark Energy.

The reason for belief in these mystery items appears similar at first to the luminiferous aether. In order to be able to use our current models of the universe, and physics in general, to explain experimental results, something extra and invisible has to be postulated to exist. Dark Matter naturally arises as a way to make sure that our standard theory of gravitation coincides with observations of galactic dynamics. Dark Energy arises as a way to explain observed acceleration in the rate of expansion of the universe..

Of the two, Dark Matter is a little bit easier to understand, once one gets beyond the fact that it is believed to comprise approx. 21% of the total matter & energy of the universe. The belief in the existence of Dark Matter, which by its definition can’t be observed visually, is high. A recent experiment claims`the first direct observational evidence of dark matter. (Read A Direct Empirical Proof of the Existence of Dark Matter, by a team of researchers using NASA’s Chandra X-ray Observatory.) The title of the paper makes it very clear that this finding is not direct observation - just what does "empirical proof" mean? It would seem then that the belief in dark matter is itself based on a model that may be as ontologically-challenged as the mechanical wave model of light that lead to the luminiferous aether.


Colliding galaxies showing spatial separation of baryonic and non-baryonic matter. Click to enlarge.Actually, the "proof" is an intriguing one, and brings in clear relief the difference between the aether of the late 19th century and dark matter today - it is the triumph of physical laws over model. The idea of the aether was dictated by the model of what a wave was - and, at that time, the model was clearly that of a mechanical wave. On the other hand, the idea for dark matter is dictated by the belief in the law of gravitation being a universal law. The dark matter is postulated in order to make sure that gravitation still works. In other words, we just have to believe that there is different stuff in the universe that obeys the law of gravitation, but, conveniently, can’t be seen. The proof is complicated, but does rely on showing that a change in the law of gravitation cannot explain the experimental results, and therefore dark matter must exist. Quite a stunning argument. (Not surprising, there is a vocal anti-dark matter contingent who don’t necessarily by the "proof," and who believe that gravitational theory must be tweaked. For a gentle introduction to the proof, as well as a description of the position of those who claim that gravity must be modified, and why they are probably incorrect, see Bedeviling Devil’s Advocate Cosmology.)

It’s still not clear what dark matter is made of - there are ideas out there that include MACHOS (Massive Compact Halo Objects), and WIMPS (weakly interacting massive particles) - which sound like rival gangs in a universal West Side Story - but it appears that dark matter is here to stay - even if we can only infer its existence.

Now on to Dark Energy. Suffice it to say that this is the most mysterious stuff in the universe, where I am using the word "stuff" deliberately to stress the equivalence between matter and energy. Why is is postulated to exist? Again, like Dark Matter, it is needed to preserve a physical law - in this case the conservation of energy (See the Scientific American piece on Dark Energy for full details.) The Conservation of Energy is a law that is so ingrained that any suggestion that we should tweak it seems absurd (Actually, some physicists did consider violations of conservation of energy before sanity was regained in the heady early days of Quantum Mechanics).


Energy distribution of the universe. Click to enlarge.But consider that Dark Energy is thought to be everywhere. Like the luminiferous aether it is everpresent in our shoes, hair, mouth, nose, and subway tunnels. There’s (supposedly) even more of it than Dark Matter, it represents 75% of the universe’s matter & energy. (If you are counting, then, dark matter + dark energy account for 96% of the entire universe!) It’s a very strange energy - a source of repelling force causing the entire universe to expand at an accelerating rate, but, like aether, we can’t see, weigh, touch or taste it.

Will Dark Energy fade into the dustbin of history to join the luminiferous aether? I don’t think so. I expect that we will someday, in the not-too-distant future, see an "empirical proof" of its existence.

You see, what’s really different about the late 19th century and now is that we are much more inclined to consider outlandish ideas - new states of matter and energy, each new one weirder than the last. This willingness to "think outside the galaxy" has been quite successful. Consider how quasars and pulsars have come to be accepted as matter-of-fact. The idea of a "black hole" is now so accepted that it is the universally agreed-upon reason/destination for missing socks in drawers.

Meanwhile the laws of gravitation, and conservation of energy keep on ticking, and we stay legal by inviting the most outrageous characters into our party and our models - The Dark Lords of Matter and Energy. Modify the models, not the law. Let the darkness in.

But darkness can illuminating. As Jesse Colin Young sang:

Darkness, Darkness, be my pillow, Take my head and let me sleep In the coolness of your shadow, In the silence of your deep Darkness, darkness, hide my yearning, For the things I cannot see Keep my mind from constant turning, To the things I cannot be Darkness, darkness, be my blanket, cover me with the endless night Take away the pain of knowing, fill the emptiness with light Emptiness with light now

Categories Modeling Philosophy Understanding & Prediction

One Degree of Separation - Global Hot Air


Sterling Hot-Air Engine. Click to enlarge.The world, and especially all of blogville has exploded with the release of the IPCC’s executive summary of its 2007 Climate Change: The Physical Basis. A web search for anything to do with global warming seems to yield just as many blogs that talk about the "myths" of global warming (e.g. see JunkScience) as there are "myths about myths" about global warming (e.g. see The Environmental Defense Fund site). How can anyone navigate through this stream of flotsam/jetsam? How do these blogs do anything but attract those who already believe in that point of view, or are leaning heavily that way?

Not even political persuasion is a good predictor of what you’ll find out there. For example, read Facts and Myths about Global Warming: A Conservative Perspective at the The Green Elephant site. Green Elephant being the Republicans for Environmental Protection, of course.

Then there is the godfather of the anti-warming crowd - none other than the king of science fact/fiction media success - Michael Crichton (see my previous post on Crichton’s role in the GW debate). His recent lecture The Impossibility of Prediction is a good example of why Crichton is listened-to and often quoted, especially by members of the Bush administration who, at least until this week’s IPCC report, were quite sure that there was a good deal of controversy about all of this global warming hullabaloo. In his speech, Crichton comes across as pushing for fact-based science, when in fact he dismisses the very tool that all scientists must rely on when making predictions: the use of models for the prediction.

What in the world could possibly be causing such a globally heated debate? There are as many answers to this question as there are blogs currently writing about the IPCC report, including this one.

One answer came to me as a bit of serendipity in the form of a brief e-mail from Jim - an acquaintance who believes that the Democratic Party ceased to exist after FDR:

I heard this just a few days ago on a talk show. Someone whose father was a meteorologist for all his life and had average daily temperature records since 1894 claims the average daily temperature for most areas where records were kept has changed only 1 degree since the records were kept. While one may ask why the 1 degree change, that sounds a whole lot less alarming than what Al Gore is claiming.

So, did I write Jim back with what was wrong with this argument? That 1 degree over a century is a whole lot more than what should have happened, and if we don’t do something about it, models predict that we’ll see another 2.4 degrees in just 50 years? That the temperature change is just one of many events - the rapid disappearance of arctic ice, e.g. - that are harbingers of global warming? That 1000’s of scientists aren’t in a grand conspiracy to cripple the US economy by forcing us to adhere to Kyoto-agreed-to levels of CO2 emissions?

Nah, I didn’t write any of this. Once he dropped the Al Gore reference, I knew it was too late. Boy, could there ever be a more inconvenient messenger to deliver such an Inconvenient Truth than Al? The political overtones of message and messenger are just too much for most to tolerate, let alone disentangle. And this is just the beginning, because the politics of global warming will soon turn out to be the hottest of hot political benchmarks - already Democratic presidential hopefuls are staking out some of the greener landscape.

So I told Jim to start his own blog (and that I would help him do this). And I can see a post along the lines of "myths about myths about myths" about global warming.

It’s already getting more than 1 degree warmer….

Categories Politics Weather & Climate

Coping with Chaotic Climate Models


Model prediction of Arctic sea ice loss - 2000 (L) vs. 2040 ®. From BBC news. Click to Enlarge.Now that the IPCC has released the summary of its upcoming study report Climate Change 2007: The Physical Science Basis , strongly stating that global warming is man-made, it is still important for scientists to clearly enumerate any issues surrounding the accuracy and reliability of their models.

Because if they don’t, global warming deniers, or those who believe that we need still more time for convincing proof will focus on the slightest inaccuracy in a model’s prediction in classic red herring fashion. (In a way similar to anti-evolutionists, they will neglect the hundreds of accurate predictions and claim that the one that doesn’t quite fit calls for total abandonment of a theory.)

Climate models are by their very nature prone to chaotic behavior. This behavior must be accounted for when using climate models for any type of prediction. An excellent article on how chaotic models are handled has been provided recently by Cecilia Bitz of the Univ. of Washington. In her Real Climate article Arctic Sea Ice decline in the 21st Century, Bitz describes work she did with colleagues Marika Holland and Bruno Tremblay, which culminated in a paper for the Dec. 2006 Geophysical Research Letters. I want to focus on two aspects of Bitz’s commentary: the presence of chaos in the climate models, and the overall accuracy of the modeling process. Bitz recounts:

It is common practice to run climate models multiple times with slight variations to the initial conditions. Because the system is chaotic, the natural variability in each run is random and uncorrelated from one run to the next. When an ensemble of runs is averaged, the natural variability is reduced in the ensemble mean, and it is easier to detect a significant trend.

With a chaotic system there is sensitive dependence on initial conditions (SDIC). This is the cause of the "random and uncorrelated" variability in each model run. Note that this variability is known to be present in the models, and therefore a methodology of handling this variability must be adopted, which has typically been accomplished by the ensemble averaging. Any trend that survives this averaging is then by definition "significant." Such a trend in the output of a climate model could be a rapid temperature change, or, in the case described by Bitz and her colleagues, the rapid disappearance of arctic sea ice.

The natural question then is how accurate a prediction is the noted "significant trend"? The only way to establish accuracy for a prediction is to wait and see if it comes true. So the operative way to describe a prediction is more on the lines of "reliability" or "confidence". Perhaps the best way to develop confidence in a modeling technique is to compare model results with previous history. Bitz’s climate model does a good job of matching past observations:

The trends in the seven ensemble members for 1979-2006 span the trend in the observations: Some members retreat a little faster and some a little slower, as expected from the random natural variability in the runs. The model also reproduces the mean and variance of the observations with good fidelity.

Given the care in which the modeling has been handled, what about future predictions of the model? Here is Bitz’ response. Note how circumspect she is about exactly what will happen when, as opposed to the confidence in something that probably will happen during a given time frame:

There is considerable uncertainty in future model projections, and Figs 2 and 4 illustrate why it would be better not to focus too much on the year 2040, which to our dismay was highly publicized. The more important message from models is that all but a few outliers predict enormous sea ice retreat this century. At least a few respectable models predict a nearly ice-free Arctic by midcentury, with a retreat that may be punctuated by rapid events.

Read Bitz’s full article for all of the details, including whether anything can be done to reverse the trend in loss of sea ice. I recommend doing this before tackling the full IPCC report on the physical basis for Climate Change, where there should be ample references to model ensemble averages. (Note: The full report of the 2001 IPCC study on the Physical Basis of Climate change is available online. Find out more about climate model methodology by following the Model Evaluation link.)

And definitely keep Bitz’ explanations in mind when you read the inevitable complaints against the IPCC report as being purely political as opposed to scientific.

Categories Chaos Modeling Weather & Climate

Pi-Brained Schemas & Savants: Who's Normal Now?


Meesh Pi. Click to enlargeA most peculiar feat was reported in yesterday’s Philadelphia Inquirer: Marc Umile, a filing clerk from the Philly area, was able to recite the first 12,887 digits of π from memory - an American record. (The current verified world record is for 43,000 digits by Krishan Chahal of India.)

In an odd twist, Umile performed his prodigious feat at a law office, and not for the Guiness Book folks. In these litigious times, it is obviously prudent to be prepared for intellectual property infringement in any activity such as this, which in this case comes under the heading of π’s and torts …

While Umile’s feat is incredible, I am more intrigued by the physical/mental issues involved in the data entry, storage and retrieval of these digits. The actual amount of data is not the issue - 12,000 is a very small number of single digits when compared to the potential of the human brain. Data entry is not hard to comprehend, either, with Umile spending two-plus years memorizing the digits. (I am not commenting on motivation or sanity here. See the Inquirer piece for this!)

How in the world are the values recalled/retrieved?

Umile was able to produce 1,000 digits at a time typing them on a keyboard. What’s not mentioned is whether or not he can start at any number position in π and proceed from there. i.e. if he were asked for the 5000th digit, could he do it? Or does he have to start from the beginning (or perhaps the previous 1000-number cutoff). My guess is that it’s the latter - Umile recalls a number stream in a serial fashion, and probably relies on some prior quasi-pattern to get him off and running.

(I am just speculating, of course, and may be relying too much on my experience with piano. I play a fair amount, but I don’t play from sheet music - everything I play is embedded in a sort of "muscle memory" that allows me to play without thinking - and just play with feeling. As a result, if I mess up a note or chord I often have to backtrack to the beginning of the song, or to some natural break, in order to play the rest of the song through to the end.)

Now compare the Umile feet with the legend of Kim Peek - a savant who was the inspiration for Rain Man. Peek has memorized 12,000 books! To top that off, he knows every zipcode in the US, and supposedly can provide travel directions within any large US city, or between them. Again, the amount of information here, while sizeably larger than the first 12,000 digits of π, does not sound like it should tap the potential storage capability of the human brain. However, it is the entry and retrieval activity that is staggering.

Peek can memorize a page in 8-10 seconds! How else to load those 12,000 books?

It is the data retrieval that sets Peek apart from Umile. Peek has the ability to access the data in a random access fashion, a sort of Ram Man. In a 2005 Scientific American account titled Inside the Mind of a Savant by Darold Treffert and Daniel Christenson, Kim is described as being able to recall incredible details about the books he reads some 4 months later, and can name the page(s) on which the details are written, as well as other complete passages.

Peek has a severe brain abnormality, with no corpus callosum linking his left and right hemispheres. Treffert and Christensen speculate on how this defect may lead to some of Peek’s abilities, and their article presents some provocative ideas about memory, savants, and brain.

The reason for this post, in addition to pointing out the differences between savant ability and the ability to memorize a huge number of digits π, is to focus on the unique way that models and understanding need to be developed in brain studies. The approach is quite different from the typical inductive/deductive approach more associated with the basic sciences: brain research requires pathologies in order to understand normal behavior. It is like saying that the only way to understand how a pendulum works is to investigate pendulums that behave abnormally!

I don’t know any Pendulums Behaving Badly, but it sure sounds like a good name for a band.

The message is clear, though - without RamMan and others with similar abilities/developmental issues, our normal is too normal to ever yield the understanding, prediction, and ultimate clinical applications to keep it normal.

Although I’m reasonably sure that Marc Umile, while wishing that he could memorize much more, much faster, would choose a corpus callosum over a colossal corpus of records.

For more on the π records, see the PI World Ranking List for rules, regulations, and how to enter.

Categories Consciousness Modeling Understanding & Prediction