Math really is scary … here’s why

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.