The Unimaginable Mathematics of Borges' Library of Babel (37 page)

in other words, the equation
of the standard unit 2-sphere. If we fix
w
= 1 (or -1) we are at the top
or bottom of the unit 3-sphere, and the equation becomes

which implies that

The only way that three
nonnegative numbers can add up to 0 is if they themselves are all 0. In other
words, the three-dimensional slice at the coordinate
w
= 1 yields only
the point (
x
,
y
,
z
) = (0, 0, 0).

On the other
hand, let
w
be any number strictly between —1 and 1. For a concrete
example, let
w
be 1/2. Then the equation becomes

, which implies that

By taking the square root of
both sides, we arrive at

and this is equivalent to the
statement "the set of all points in three-dimensional Euclidean space
located at a distance
 
from the origin (0, 0, 0)."
In other words, the equation specifies a 2-sphere of radius
.

In the above
argument, there was nothing special about letting
w
be 1/2. We could
have chosen any number strictly between —1 and 1, and we would again end up
with an equation specifying a sphere.

In general,
let
w
=
R
where
-
1 <
R
< 1. Then the examination of the three
dimensional slice at
w
=
R
is facilitated by the equation

, which implies that

By taking the square root of
both sides, we arrive at

,

which is the equation for a
2-sphere of radius
.

The Pythagorean theorem states
that for a right triangle with legs of lengths
x
and
y
and with
hypotenuse of length
h
contained in a Euclidean plane, the equation
 always holds (figure 71). (There are dozens, maybe hundreds, of
proofs of this theorem.) If the length of the hypotenuse is, say, 1/2, then the
equation becomes