For most of us, or perhaps all of us, it's impossible to imagine a world consisting
of more than three spatial dimensions. Are we correct when we intuit that such a world couldn't exist? Or is it that our brains
are simply incapable of imagining additional dimensions—dimensions that may turn out to be as real as other things we
can't detect?
String theorists are betting that extra dimensions do indeed exist; in fact, the
equations that describe superstring theory require a universe with no fewer than 10 dimensions. But even physicists who spend
all day thinking about extra spatial dimensions have a hard time describing what they might look like or how we apparently
feebleminded humans might approach an understanding of them. That's always been the case, and perhaps always will be.
From 2D to 3D
An early attempt to explain the concept of extra dimensions came in 1884 with the
publication of Edwin A. Abbott's Flatland: A Romance of Many Dimensions. This novel is a "firstperson" account of
a twodimensional square who comes to appreciate a threedimensional world.
The square describes his world as a plane populated by lines, circles, squares,
triangles, and pentagons. Being twodimensional, the inhabitants of Flatland appear as lines to one another. They discern
one another's shape both by touching and by seeing how the lines appear to change in length as the inhabitants move around
one another.
One day, a sphere appears before the square. To the square, which can see only
a slice of the sphere, the shape before him is that of a twodimensional circle. The sphere has visited the square intent
on making the square understand the threedimensional world that he, the sphere, belongs to. He explains the notions of "above"
and "below," which the square confuses with "forward" and "back." When the sphere passes through the plane of Flatland to
show how he can move in three dimensions, the square sees only that the line he'd been observing gets shorter and shorter
and then disappears. No matter what the sphere says or does, the square cannot comprehend a space other than the twodimensional
world that he knows.
Only after the sphere pulls the square out of his twodimensional world and into
the world of Spaceland does he finally understand the concept of three dimensions. From this new perspective, the square has
a bird'seye view of Flatland and is able to see the shapes of his fellow inhabitants (including, for the first time, their
insides).
Armed with his new understanding, the square conceives the possibility of a fourth
dimension. He even goes so far as to suggest that there may be no limit to the number of spatial dimensions. In trying to
convince the sphere of this possibility, the square uses the same logic that the sphere used to argue the existence of three
dimensions. The sphere, now the shortsighted one of the two, cannot comprehend this and does not accept the square's arguments—just
as most of us "spheres" today do not accept the idea of extra dimensions.
From 3D to 4D
It's difficult for us to accept the idea because when we try to imagine even a
single additional spatial dimension—much less six or seven—we hit a brick wall. There's no going beyond it, not
with our brains apparently.
Imagine, for instance, that you're at the center of a hollow sphere. The distance
between you and every point on the sphere's surface is equal. Now, try moving in a direction that allows you to move away
from all points on the sphere's surface while maintaining that equidistance. You can't do it. There's nowhere to go—nowhere
that we know anyway.
The square in Flatland would have the same trouble if he were in the middle of
a circle. He can't be at the center of a circle and move in a direction that allows him to remain equidistant to every point
of the circle's circumference—unless he moves into the third dimension. Alas, we don't have the fourdimensionsal equivalent
of Abbott's threedimensional sphere to show us the way to 4D. (In mathematics, moving into ever higher dimensions is a walk
in the park. See Multidimensional Math.)
How about 10D?
In 1919, Polish mathematician Theodor Kaluza proposed that the existence of a fourth
spatial dimension might allow the linking of general relativity and electromagnetic theory. The idea, later refined by the
Swedish mathematician Oskar Klein, was that space consisted of both extended and curledup dimensions. The extended dimensions
are the three spatial dimensions that we're familiar with, and the curledup dimension is found deep within the extended dimensions
and can be thought of as a circle. Experiments later showed that Kaluza and Klein's curledup dimension did not unite general
relativity and electromagnetic theory as originally hoped, but decades later, string theorists found the idea useful, even
necessary.
The mathematics used in superstring theory requires at least 10 dimensions. That
is, for the equations that describe superstring theory to begin to work out—for the equations to connect general relativity
to quantum mechanics, to explain the nature of particles, to unify forces, and so on—they need to make use of additional
dimensions. These dimensions, string theorists believe, are wrapped up in the curledup space first described by Kaluza and
Klein.
To extend the curledup space to include these added dimensions, imagine that spheres
replace the KaluzaKlein circles. Instead of one added dimension we have two if we consider only the spheres' surfaces and
three if we take into account the space within the sphere. That's a total of six dimensions so far. So where are the others
that superstring theory requires?
It turns out that, before superstring theory existed, two mathematicians, Eugenio
Calabi of the University of Pennsylvania and ShingTung Yau of Harvard University, described sixdimensional geometrical shapes
that superstring theorists say fit the bill for the kind of structures their equations call for. If we replace the spheres
in curledup space with these CalabiYau shapes, we end up with 10 dimensions: three spatial, plus the six of the CalabiYau
shapes, plus one of time.
If superstring theory turns out to be correct, the idea of a world consisting of
10 or more dimensions is one that we'll need to become comfortable with. But will there ever be an explanation or a visual
representation of higher dimensions that will truly satisfy the human mind? The answer to this question may forever be no.
Not unless some fourdimensional lifeform pulls us from our threedimensional Spaceland and gives us a view of the world
from its perspective.


Creatures living in Flatland see triangles and other twodimensional
objects as lines.  

From a twodimensional point of view, a sphere passing through a
plane appears as a line that initially gets longer, then, as this image shows, gets shorter and shorter until it disappears
altogether.  
The circles represent an additional spatial dimension that is curled
up within every point of our familiar threedimensional space.  
Sixdimensional CalabiYau shapes can account for the additional
dimensions required by superstring theory.   