Fractals

Scientists are supposed to give us answers.  After all, that’s what they do, right?  We ask them questions and they give us answers.  Although I suppose we should give a special nod to those odd people who come up with answers to questions no (sane) person ever thought to ask.

But while we’re giving nods to people who help progress along by giving us answers, let’s not forget those annoying people who not only don’t give us answers to our questions, but also prove that our questions were all wrong in the first place.

Consider Benoit Mandelbrot.

Born in Poland and raised in France, Mandelbrot finally moved to the U. S. and spent most of his career working for IBM at the Thomas J. Watson Research Center in New York.  There his curious mind led him to work on Information Theory, Fluid Mechanics, and Cosmology, among a host of other odd digressions. [Parenthetically, another nod to IBM for allowing gifted lunatics like Mandelbrot to wander where their interests took them rather than demanding clear, profitable, short-term returns.]

In 1967 Mandelbrot put out a paper with the innocuous (not to say boring) title of How Long is the Coast of Britain?  Statistical self-similarity and fractional dimensions.

If you are curious, you can look up the answer to his question on the Web.  It is there, a nice, precise, 12,429 kilometers.  What makes that fact queer is that in his paper Mandelbrot proved that this question could not be answered. The coast of Britain, or any coastline for that mater, has no determinate length.

Say what, I hear someone ask?  Surely you can simply pick up a map and with one of those little map wheels or a good pair of dividers carefully measure the coastline of Britain?  It’s not only possible, it’s easy.

There’s lots of ways to explain the problem, but let’s start here: Imagine you are looking down on a satellite picture of Britain.  Through some strange magic, when the picture was taken the sky over Britain was completely clear (c’est impossible).  You can see the entire coastline.  And you can see, even with a casual glance, that the coastline is a bit ragged, with lots of ins and outs to foil the intrepid measurer.

But you give it a shot, anyway, stepping off the coastline as carefully as you can, until finally you get a number for the entire coast.

But you are still not satisfied.  You can see that as you step off the margin, you are skipping over lots of tiny irregularities.  They are too small to be measured with your tools, but aren’t they important?  You need another method to give you a length to compare to.

So you zoom in a bit, focusing your attention on one conspicuous feature, say the Cornwall Peninsula (that’s the sharp bit on the bottom, pointing west).  On the enlarged view, you notice that it, too, is a bit ragged, with lots of inlets, coves, and promontories.  But, correcting for scale, you step off all the big features you can see, ignoring the little jaggies too small for your measuring tool.  Then you move up the coast to the next county, zoom in and repeat the process.  Once you’ve gone all the way around the island, you total up all of your numbers and compare to your previous total.

Unfortunately, the two numbers don’t compare.  The length you calculated from your enlarged views is much bigger than your first effort.

So you try again.  You zoom still further in, noticing in passing that this closer-in view is still rough and jagged.  Once again you trace out the outline as well as you can, moving to the next section of coast, repeating the process.  And once again you total up all of the lengths for a grand total of the whole island.  The you compare it to your previous number.

Rats!

This time the total is not only bigger than you initial try, it’s even bigger than your second.

So you try again, and again, and again.  And each time you can still see finer jaggies and each total is larger than the last.

Finally, your closeup view is in so tight that you are tracing your way around the erose margins of individual grains of sand.  And you are having to make philosophical decisions about whether wet grains of sand are part of the shoreline or only the dry ones.  Or perhaps it’s only grains wet on one side and dry on the other.

Clearly the process cannot go on infinitely.  Eventually you’re going to get down to individual atoms and their component parts, although what that has to do with the concept of a “coastline” is pretty unclear.

But if the possibilities are not endless, a couple of things are clear.  First, each close-in view is as ragged and uneven as the last.  From the point of view of your scaling, each view is essentially identically hard to measure.  Second, nowhere do you find some natural stopping point where you can say, “Here.  This is the limit of the shoreline.  This is where the land ends and the sea begins.”

The strange property of looking the same no matter how you zoom in, Mandelbrot called self-similarity.  He set out to study the family of mathematical shapes that shared their properties.  In 1975 he coined a name for them: fractals.

By the early 80s, fractals were getting famous and giving birth to Chaos Theory.  And it was not too long after that I first encountered Mandelbrot.

It was pretty much by accident.  Around the end of the 80s, two things converged: I upgraded my PC to a VGA monitor (state of the art at the time) and I first heard of something called the Mandelbrot Set (he didn’t invent it – it was named in his honor).

You might have seen the Mandelbrot Set.  If you plot it, it generates a pretty cardioid shape with a dark center and peculiar, knobby borders.  If you zoom in to get a closer look at the knobby borders, you find yourself looking at a miniature version of the larger shape.  Zoom in on that and you find a still smaller version, and so on ad infinitum.  Today, you can buy sheets and tee shirts with the Mandelbrot Set on them.

I thought it looked pretty cool.  But what was really exciting for me was finding out that the mathematics to generate the Mandelbrot Set were pretty straightforward.  Which meant, in turn, that I could write a program to generate a colored version of the Set right on my new VGA monitor.

Too cool.

Of course, with the relatively primitive processing of my little computer, the process was slow.  Really sloooow.

Still, that had an excitement of its own, watching the picture glacially grow on the screen.  It was sort of rattle-clank, rattle-clank, pixel.  Rattle-clank, rattle-clank, pixel.  It was slow and it was ultimately pointless, but I was immensely proud of it.

At which point I made my mistake.

I was dating a lady who happened to drop by my house one day while my computer was crunching away.  All of us like to show off before our beloveds.  Proud as a new father, I showed her the half-completed picture and tried to explain to her how really cool the whole thing was.

Her eyes glazed.

Have you ever fallen into this trap?  There is some nerdy thing that you really love that you just know if someone else truly understood they could not help but love it, too.  So when their eyes begin to glaze and they start edging for the door, instead of dropping the subject and moving on (like a sensible person), you stubbornly try harder and more earnestly to make them see the wonderfulness of some phone app or your model trains or collecting Barbie dolls or whatever.  Your eyes probably glow with the light of a fanatic and your voice probably grows shrill.

Trust me, this is not only not the way to get some civilian to adopt your favorite enthusiasm, I’ve proved it doesn’t do a heck of a lot for your love life, either.

Mind you, I still think Benoit Mandelbrot is a genius.  With a mind that works like nobody else’s, he has made major contributions in a staggering number of fields.  Fractals and Chaos Theory alone would be enough to mark him as a giant.  And the Mandelbrot Set is gorgeous and a fitting monument to him

But as an aphrodisiac, well, not so much.

0 thoughts on “Fractals

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