Most of us encounter mathematics only as a school subject, and most

of us, thanks to uninspired teaching, find it the most numbingly dry of

all school subjects at that. But when you consider it, basic mathematics

is so neatly arranged, so precise: everything has a right answer and

there’s always a correct way to get there. It can give you a remarkably

comforting outlook on life — in grade school I found math a much more

satisfying pursuit than, say, social studies.

Alas, once I hit the turbulent waters of adolescence, the murkier and

more psychological world of the novel or the poem suddenly seemed to

reflect the way things worked better than the old cut-and-dried

mathematical logic.

Math lost its sense of fundamental relevance, although it was still

vaguely interesting, in an escapist, jigsaw-puzzle kind of way. What I

didn’t realize as an emotionally buffeted teenager and young adult was

that math isn’t just about simplification. Most high school math

involves working out geometrical proofs and reducing equations, that is,

solving problems by simplifying them — a tactic that, frankly,

translates poorly into the real world. But it turns out some aspects of

mathematics are as baffling, as complex, and as elusive as a good

three-volume novel.

If you or your bored-in-math kids want a look at the sort of math

that really (in my humble opinion) has a shot at reflecting the deeply

screwed-up quality of real life, I suggest you check out fractals — the

most famous amongst which is the Mandelbrot set.

The actual math underlying this fabulous monster isn’t all that

difficult if you’ve got a decent foundation in high school algebra and

can understand a simple graph. (It’s actually graphed in what’s called

the complex plane, meaning that complex numbers, e.g., “three plus the

square root of minus two,” are involved, but it’s not all that different

from the basic x-axis, y-axis graph of, say, the line y = 3x that you

learned in eighth grade.) The best simple description of fractals for

the mathophobic on the Web, including background on complex numbers, is

here. For those

already founded in high school algebra and essentially comfortable with

the way math

works in general, this is also

a

good, though dense, explanation of the math behind the Mandelbrot set;

(it

includes instructions on using Pascal to program your computer to draw

one,

though if I were you I’d just

download a piece of

Mandelbrot-generating software).

Anyhow, take it from an English major that the math is essentially

simple — if admittedly a bit off the beaten track for most of us. The

way the thing looks when you see it graphed, though … well, that’s not

so simple. Happily, a number of websites have now made it possible for

you to investigate and play with the Mandelbrot set without any special

software or arcane knowledge, generating your own wild and savage images

of this strange beast of the mathematical realm.

As you can see by clicking

here, there’s basically one big

shape dominating the picture, something like a dented circle or a heart.

When you zoom in on it (just click on the image to

zoom in — if you’re having trouble, try

this similar

site), you see it’s made out of a lot of little shapes just like that

big one. Zoom in again, and you see the little shapes are made out of

even tinier copies of themselves. That’s pretty intricate, and even

beautiful. So much precise repetition seems comforting and predictable:

a tremendous complexity, but one that follows a simple rule, like a

snowflake — in short, just the sort of neatness you expect from

mathematics.

But that’s just because you haven’t been told the scary part yet. The

scary part about fractals, like the Mandelbrot set or the similar Julia

set, is that all of that comforting, predictable stuff turns out to be

exactly false. All those repetitions are close — but it turns out

they’re not exact. The little shapes are NOT precisely the same as the

big ones. The pattern looks like it’s repeating, but it isn’t, not

quite. That dented-circle shape on which you’re zooming in is actually

slightly deformed compared to its predecessor. (I’m not even

over-interpreting: actual mathematicians use the word “deformed” to

describe this.) Horrible, isn’t it? Ominous and sinister and a betrayal

of everything that’s clean and comforting about math? Yes, and it only

gets worse.

The length of a line should stay the same whether you measure it in

inches or in centimeters, right? Well, the length of the line in that

graph doesn’t stay the same at all. It changes depending on what unit

you measure it in. Think of a coastline, which when measured one

kilometer at a time might turn out to be 100 kilometers long, but when

measured one meter at a time comes out to be, say, 250 kilometers.

That’s how a fractal behaves (in fact, it turns out coastlines are

described by fractals).

Sit thinking about all this for ten minutes while contemplating a few

images of the Mandelbrot set — which starts looking more and more like

a sort of misbegotten, gaping black hole of the mind — and you will, if

you are anything at all like me, come to experience a kind of profound

and existential panic. (For horror-movie fans and other panic

aficionados: this sense of dread can be enhanced for sound by actually

listening to the Mandelbrot set. Click

here. The site provides sound files generated from the numbers in the

Mandelbrot set mapped into simple pitches. Obviously I’m not the only

one who finds the whole business spookily obsessive.)

The point is: life resembles the Mandelbrot set a hell of a lot more

than

It resembles the sterile and simplistic perfection of your basic Algebra

101 straight line or parabola, doesn’t it? Think about it: It seems to

make sense, but is actually deeply screwy. It starts out with a simple

idea, but it turns out to generate unimaginable, unbearable, massively

multiplying amounts of chaos. The chaos keeps looking like it’s about to

make sense, it ought to make sense, it seems to make sense, it’s just

groping for meaning … but no.

You know in your bones that it’s really just getting weirder the more

you look at it. Maybe I’m insane, or maybe my personal life is just a

mite peculiar, but I have to say this is what life feels like to me a

lot of the time.

As it turns out, in fact, fractals really do seem to describe a lot

of the hardest-to-graph real-world phenomena out there, such as

coastlines, or turbulence (e.g., the shapes cream makes swirling into

your coffee). All I know is, if someone had showed me a Mandelbrot set

when I was fourteen and first reeling with the complexity of it all, it

seems to me I might have looked at mathematics with a certain renewed

respect.

Now I think about it, I might even have majored in math, become an

engineer, and been making real money by now. Such, indeed, is life.