her pitch for help. We know the holographic image of Princess Leia is
just a holographic image, but we can conjecture about whether the im-
age, if somehow it were imbued with consciousness, would realize that it
is only that, a holographic projection. If this is the way the universe is,
and we are but holographic projections of some sort, how would we ever
know?
What Mathematics Has to Say About This
From a mathematical standpoint, it is easy to put lower-dimensional ob-
jects into one-to-one correspondence with higher-dimensional objects.
To see a simple example of how to put the points on a line segment into
one-to-one correspondence with the points in a square, take a number
between 0 and 1 and write out its decimal expansion, then simply use the
odd-numbered digits (the tenths, thousandths, and hundred thousandths,
etc.) to define the x coordinate and the even-numbered digits (the hun-
dredths, ten thousandths, and millionths, etc.) to define the y coordinate.
Thus, the number .123456789123456789123456789... would correspond
to the point in the square (.13579246813579... , .2468135792468.... ).
To map points in the square to the line, simply interleave the digits alter-
nately—the exact reverse of what was done above. T he point (.111111... ,
.222222... ) would correspond to the point .12121212... on the line.
The problem here is that these transformations are discontinuous.
Points close to one another can end up widely separated, analogous to the
baker’s transformation we saw when we were discussing chaos. In fact, it
can be shown in topology that every one-to-one transformation of the por-
tion of the real line between 0 and 1 onto the square in the plane must be
discontinuous.^12
This has interesting consequences for Princess Leia, as well as for us, if
we are holographic projections. One would expect that the recipe for con-
structing the holographic Princess Leia would be continuous in the fol-
lowing sense: just as all the points in Princess Leia are close to one another
(at least, they are all within Princess Leia), the recipe for constructing her
would consist of instructions close to one another. But mathematics shows
this cannot be the case. The recipe for constructing Princess Leia (or, if
not Princess Leia, some other holographic projection) must be widely
scattered. Instead of a single block of instructions on how to construct
Princess Leia, the instructions for doing so may appear all over the book
that represents the totality of all the holographic recipes. We might expect
that pages 5–19 of the holographic recipes are devoted to construction of
Princess Leia, but what might actually happen is that one line of page 5
198 How Math Explains the World