The Unimaginable Mathematics of Borges' Library of Babel (23 page)

On a sphere,
on a torus, or in the Euclidean plane, any journey the flag might take would result
in, at worst, a rotation. There is no possible path that allows for the flag to
be mirror-reflected—a journey into the fourth spatial dimension is required.

Let's generalize these
2-manifolds, the torus and the Klein bottle, to their three-dimensional
equivalents, the 3-torus and the 3-Klein bottle. To do so, we start with a
solid cube, instead of a square, and identify opposite sides, and mimicking
what we did in two dimensions, we'll begin by creating a 3-torus. Again, we'll
take advantage of working in three dimensions to bend the cube to identify the
sides; the natural space for the identifications is 51^-dimensional Euclidean
space: in six dimensions, the 3-torus is flat. If we are willing to have a
curved, distorted representation akin to the 2-torus in three dimensions, a
"mere" four Euclidean dimensions suffices to hold the 3-torus. For
the 3-torus, arrows are insufficient to specify an orientation of a face of the
cube, but spirals will serve. (Think about why this should be so.)

Figure 32 shows the initial
solid block inscribed with appropriate spirals. This time, we bend the sides of
the cube around, identifying the left and right faces while taking care that
the spirals being glued together spin in harmony (figure 33). After this first
identification, we are confronted with a millstone, whose top and bottom must
be identified. (The inside and outside are, of course, also identified. We'll
discuss that afterwards.) Turn the millstone sideways—and shrink it—to make it
easier to visualize this step (figure 34).Proceed by identifying the visible
gray ring on the right-hand side of the millstone with the hidden gray ring on
the left-hand side (figure 35).

Now, we are presented with a
donut that has a smaller donut drilled out of the middle of its interior; a
donut waiting for a filling, as it were. A donut with a non-donut inside. The
surface of the exterior donut is a 2-torus, while the surface of the interior
non-donut is also a 2-torus. These two tori correspond to the front and back
square faces of the initial cube, and they sprang into being when, in the
process of identifying the other four faces of the cube,
only the edges of
these squares were identified.
Now, the (invisible) interior 2-torus must
be identified with the (visible) exterior 2-torus. By tugging them both into
the fourth dimension, where they no longer divide space into an
"inside" and "outside," they may be glued together,
producing the 3-torus.

Before
moving on, let's look at one more way to visualize a 3-torus. Once again, we'll
proceed by analogy with the eminently imaginable 2-torus. If we take a 2-torus
and intersect it with a plane (as in figure 36), the result is a circle (figure
36). Another way to see this is to take a slice of the square that becomes the
2-torus (figure 37). If we think of a 2-torus in this way and flatten it out
onto the plane, we may represent it as a circle of circles (figure 38).

If we take a
three-dimensional slice of a 3-torus, we get a 2-torus. One way to see that is
to look at a slice of the solid cube we started with (figure 39). Consequently,
each slice of the 3-torus is "flattened" out into 3-space as a
2-torus. Since the two sides of the cube are identified, we get a circle of
2-tori (figure 40).