Posts Tagged ‘mathematics’

Chaos, fractals, and the dripping faucet

January 4, 2017

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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How Big is a Googolplex?

December 30, 2014

K.C. Cole is an award-winning science writer, whose 1998 book The Universe and the Teacup—The Mathematics of Truth and Beauty, I typically found at a used book sale. UnknownMy wife chided me that it could now have only antiquarian interest. But I figured mathematics can’t have changed that much in 16 years. Two and two still make four, no?

The book broadly (and somewhat poetically) talks about the intersection between mathematics and life. It has some good stuff. One chapter discusses how goofy our risk perceptions can be. People worry about pesticide residues on fruit (annual U.S. death toll: zero) but not going for a drive (death toll: 30,000). Similarly, those terrified of child abduction drive kids to school – exposing them to the vastly greater auto accident risk. (All this echoed the “Freedom from Fear” chapter in my own very excellent book, The Case for Rational Optimism.)

However, not only did I also find some things I disagreed with, but some major bloopers.

Unknown-1Cole brings up one of my favorite paradoxes: “This sentence is false.” It contradicts itself. If the sentence is true, that means it’s false, so it can’t be true; but if it’s not, then it is true. However, Cole concludes this is no more paradoxical than the conflict between an American who thinks June is a summer month and an Australian who calls it winter. But that paradox is resolved with just a little more information. No additional information will resolve “this sentence is false.”

imagesAnd how about this: “Those of us reared on Euclid swallowed without thinking all those axioms about the obviousness of such propositions as: two parallel lines never meet. Yet one only needs to look at the lines of longitude – which are parallel at the equator – to see that they do.” Hello? That’s non-Euclidean geometry! Euclid’s geometry applies only to flat surfaces, not curved ones (like the Earth’s).*

Then Cole says a googolplex is “a googol multiplied by itself a hundred times.” I’m no award-winning science writer, but even I knew this is wrong. (To confirm that, I googled it, of course.)

Unknown-2A googol is the number 10 to the hundredth power; i.e., 10 multiplied by itself a hundred times; i.e., 1 followed by 100 zeroes. A googolplex (contrary to Cole) is the number 10 to the googol power; i.e., 1 followed by a googol zeroes.

These are very big numbers. Cole observes that we have trouble grasping how much bigger a billion is than a million, or a trillion than a billion. A billion is 1 followed by nine zeroes; a trillion by 12 zeroes; a quadrillion by 15 zeroes, and so on, for every three zeroes, through quintillion, sextillion, septillion, etc., each a thousand times bigger than the last. But we run out of those “illion” names long before reaching the end of all hundred zeroes in a googol.

Unknown-3(NOTE: The following has been modified, from what I originally posted, based upon helpful comments from my friend Professor Judy Halstead).

Now, I asked myself, might Cole’s definition of a googolplex – a googol to the hundredth power – actually equal (the correct) 10 to the googol power? I didn’t think so, but how can one do this math? Not on a calculator! Too many zeroes. Indeed, there literally would not be enough space in the Universe for all the zeroes. But one can do it using exponents. (Since I don’t know how exponents might be displayed on your screen, I will use the notation “10^100” to stand for ten to the hundredth power).

Ten to the googol power (a true googolplex) can be written as 10^(10^100). Cole’s false googolplex, a googol to the hundredth power, would be (10^100)^100. images-1To multiply a googol by itself once, you add the superscripts; 100+100=200; that is, you get a number with 200 zeroes. Twice, and it’s 300 zeroes. So a googol to the hundredth power would be 1 followed by 10,000 zeroes. And that, of course, is way less than 1 followed by a googol zeroes!

 

By the way, yes, Google was named for googol, to evoke the vastness of the information accessible. But they inadvertently (?) got the spelling wrong!

images-2When my daughter Elizabeth was eight, I explained googol to her. She was fascinated. Then she asked if the Universe would last a googol years.

Not a simple question. So I answered, “possibly.”

“Well,” she said, “if I’m eight now, then I’ll be a googol and eight.”

Now there’s an optimist for you.

* In fairness, Cole later does discuss non-Euclidean geometry.