Chaos, fractals, and the dripping faucet

Physicist Arthur Eddington said, “the Universe is not only stranger than we imagine, it’s stranger than we can imagine.”

Right off the bat are two possibilities: either it always existed, or had a beginning. Either one blows fuses in my brain. (Note: the God idea doesn’t help. The same problem applies to him.)

Mandelbrot

Mandelbrot

Which brings me to chaos.

Religionists imagine God organized creation from primordial chaos; in common parlance that word connotes a state of complete disorganization. But in science its meaning is more subtle, and much more interesting, as famously pioneered by mathematician Benoit Mandelbrot starting in the 1960s.

Take the weather. It can’t be forecasted very far because there are so many interacting factors; a tiny change in one cascades into ever bigger changes over time. images-1Thus the proverbial “butterfly effect” – one flapping its wings in Brazil can ultimately cause a storm in Canada.

Mandelbrot posed the seemingly simple question: how long is Britain’s coastline? But it’s not so simple. Measuring it on a map of course can’t account for all the little crenellations. You could take a yardstick and walk the coast, getting a much more accurate answer. unknownBut the coast between two ends of the yardstick is not exactly a straight line, so you’re under-measuring. A foot-ruler would do better, but still won’t capture irregularities within each foot. No matter how finely you measure, the true coastline will always be longer. (Does that mean it’s infinite?)

Coastline irregularities are a kind of seemingly patternless phenomenon found throughout existence. But Mandelbrot’s startling discovery was that there is a pattern. The kind of coastal irregularities you see on a world map are exactly replicated when you focus on a smaller area. No matter how small. unknown-1And this paradigm of like patterns repeating at different scales of examination occurs again and again in nature. The word for this is fractal. It is order hidden within seeming randomness, seeming chaos.

Look at the illustration. No matter the scale, no matter how much you magnify, the pattern persists. If the picture reminds you of a snowflake, it should, because snowflake formation is a good example of the phenomenon.

Environmentalists romanticize a “balance of nature,” an ecosystem in harmonious equilibrium. It turns out no such thing exists. An ecosystem works like the weather, one small perturbation sending it on an unpredictable and quintessentially chaotic path.

Chaos can also affect a system close to your own heart. In fact, it is your heart. Its normally regular beating can sometimes become chaotic in the textbook sense. That calls for attention.

images-3I read James Gleick’s book Chaos hoping for a better understanding. Frankly much of it was way too deep for me. But it described one illuminating experiment, conducted by Robert Shaw at the University of California at Santa Cruz. It involved the most mundane thing: a dripping faucet.

Shaw found that certain flow rates produced chaotic drips, with no regular intervals between them. Then all he did was measure those intervals and plot those numbers on a graph. Actually he used pairs of intervals to produce a graphing in three dimensions. Now, you might expect a truly random distribution, with the dots falling all over, patternlessly. But that’s not what Shaw found. The pattern of dots took on a distinct shape (“resembling loopy trails of smoke left by an out-of-control sky-writing plane”).

Strange attractor

Strange attractor

A shape thusly revealed is called a “strange attractor.” I was puzzled by that term until I realized it’s as though the shape attracts the data points to itself, keeping them from falling elsewhere.

This is extremely weird. While the shape acts like a magnet for data points, of course a magnet is a physical object, but the “strange attractor” is not, it’s just a concept. So what is going on here? What makes the seemingly random, chaotic drip intervals form a certain distinct shape when graphed? unknown-2The hand of God?

Of course not. Surely God wouldn’t bother to carefully regulate the dripping to produce the pattern. Yet it’s as if he did.

But why? That’s what I really wanted to understand. The book doesn’t tell me; Gleick writes as though the question never occurred to him. He even quotes John von Neumann: “The sciences do not try to explain, they hardly even try to interpret, they mainly make models . . . [which describe] observed phenomena.” In other words, science reveals what happens, but not why.

With all respect to the great von Neumann, I disagree. Why the Universe exists may be a meaningless question, but why Shaw’s faucet dripped the way it did is not. Another scientist Gleick quotes answered Einstein’s famous line by saying God does play dice with the Universe, and the dice are loaded; “the main objective of physics now is to find out by what rules were they loaded and how can we use them for our own ends.”unknown-3

Science is humanity’s great quest for understanding. Through that understanding we can control our destiny. But that’s almost a mere side effect of the real motivation: we just want to know.

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2 Responses to “Chaos, fractals, and the dripping faucet”

  1. Anonymous Says:

    Blasted Eve and the Tree of Knowledge. Now we have the ability to judge and the desire to understand things we can’t comprehend.

    Please don’t misinterpreted Religion and Theology. Religion is the study of the “social” effects of a particular faith or worship. Theology is the study of God.

    If you want an exceptional book on the topic, see if you can find Scientism, Man, and Religion by Fr. D.R.G Owen, 1952. This might be a hard title to find. Looking at my local libraries here in Minnesota they only show up in all the religious college libraries.

  2. Rashad Saleh Says:

    This is very interesting! I was thinking along similar lines recently. Here are my two cents:

    Science relies heavily on observation made by the senses. But there is no such thing as observation without interpretation — I agree with you when you disagree with von Neumann.

    To know this is true is to study how animals (including humans) react when they see their image in a mirror. A sophisticated enough brain will make the interpretation that what it is seeing is only a reflection and not a real object. Children below a certain age can’t do that, as do many animals and bugs, but grown up humans and chimps can.

    A somewhat harder problem from the same domain for our brains to solve is a mirage, which we did solve. One can think of science as the process of interpreting the input from the senses.

    Taking this point to a philosophical extreme, idealist philosophers (think Descartes) concluded that the world of the senses is the product of our minds. This is crazy enough for most people hopefully for a good reason, but it is not absolutely unthinkable.

    The opposite extreme, which does find its way into science, unfortunately, is to let go of all interpretation and consider sensory material to be the only reality and interpretation added to sensory material to be at best a useful tool and at worse an unnecessary error.

    This consciously or unconsciously, I suspect, does crawl into many scientists’ heads. For example, since we can check with observation, more or less, that everything is made up of atoms, one can get an inclination to explain everything using atoms.

    For example, many scientists might get an inclination to explain human behavior using atoms, which of course a very hard if not impossible problem to solve (just too many goddamn atoms to keep them all in mind somehow!)

    But the fractal idea you elaborate on is refreshing and liberating! One day, I predict, we will find the idea of “everything is made out of atoms” to have been more harmful than useful in the history of science.

    If the same pattern exists at different “layers” of reality, what right do we have to say that the “basis” of reality is the “sensory things” and not the pattern itself?

    If you ask the question this way, the atom model immediately comes into question as the big monster of reality based on “things” not “patterns”.

    Thanks for a well written article.

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