Math Mutation 53: Is Flatland Doomed?
As you may recall from earlier podcasts, I've always been
fascinated by imagining two-dimensional worlds, like in Edwin Abbott's
"Flatland" or A.K.Dewdney's "Planiverse". These are worlds that exist
entirely on a flat surface, like a gigantic sheet of paper or floating
on top of a giant lake, whose inhabitants can move north, south, east,
or west, but have no concept of additional directions like "up" and
"down". One interesting aspect of these worlds to think about is:
what would be the effect of fewer dimensions on things like chemistry
and physics? How would the laws we're familiar with be subtly warped
by the lack of dimension?
One example Dewdney points out is the familiar inverse-square law
that applies to the strength of forces like gravity over distance.
You may recall from high-school physics class that in our
three-dimensional universe, if you multiply the distance between two
objects by a factor of (n), the gravity between them diminishes by a
factor of (n^2). Probably like me, you accepted this as a fact, used
it in formulas, and just moved on. But is there a reason why it works
like this? Why is it inverse-square, and not inverse-cube, or just
inversely proportional? There is actually a reasonable explanation,
that is intimately tied with the dimensionality of our universe.
Think about a point of light on the inside of, say, a large
radius beach ball. The area it illuminates is the inside surface area
of the beach ball, proportional to the square of the radius. Now
imagine that the beach ball's radius expands by a factor of three.
The inner surface area increases by a factor of nine, or three
squared. But the point in the center hasn't changed-- the same amount
of light must now be spread across nine times the surface area. In
other words, the inverse square law happens because as you move
farther away, the same amount of energy must spread over an area that
squares with the distance.
So what does this mean in two dimensions? Well, our
two-dimensional "beach ball" would be a circle, and the light would
just have to illuminate the outer perimiter, which is directly
proportional to the radius rather than its square. So rather than an
inverse-square law, the two-dimensional universe has a simple-inverse
law. This means that forces and energy diminish much less rapidly
over distance in the two-dimensional universe. But does this
simple-inverse law have more drastic consequences?
You may recall from calculus class that an integral, or continuous
summation, from 1 to infinity of an inverse-square function converges:
that is, values diminish so rapidly that the sum total of all the
energy it would take to separate two objects to an infinite distance
has a finite sum. This is why we have the concept of "escape
velocity", a finite surface speed an object can have to take off of
the earth's orbit. It also helps explain why a finite amount of
energy can permanently sever an electron from its atom, and similar
small-scale effects that facilitate chemical reactions.
But the integral from 1 to infinity of a simple inverse function
does NOT converge. This means that in our two-dimensional universe,
there is no escape velocity, and our two-d-NASA has a much harder time
getting off the ground to explore space. And it also means that
chemical elements are much more stable. In fact, much of our
chemistry and physics would be impossible, so our poor two-dimensional
Planiversans would find themselves in a much less rich universe than
ours, even after you take into account their lack of dimension.
Actually, taking this information into account, we should seriously
doubt whether our two-dimensional beings can exist at all, or their
world is just too limited by this simple-inverse law.
However, all it not lost-- let's reexamine our original
reasoning. Could it be that somehow our 2-D universe will still obey
an inverse square law? One possibility is that it's a 2-D subset of a
truly 3-D universe: so our hypothetical point of light has to
illuminate not just the circular beach ball, but a spherical area
above and below the plane, that the inhabitants are not aware of. Or,
alternatively, there is no reason to assume thw 2-D space is empty:
maybe there is some kind of space-filling aether, like people used to
once hypothesize in our own universe, that partially absorbs energy
and causes it to more rapidly diminish. You can probably think of
other ways our two-dimensional universe might rescue the
inverse-square law, and still have lifelike chemistry and physics
despite its lack of dimension.
And this has been your math mutation for today.
References:
Flatland at Wikipedia
The
Planiverse at Wikipedia
Harmonic
series (sum of 1/n) at Wikipedia
Escape
Velocity at Wikipedia