The Unimaginable Mathematics of Borges' Library of Babel (40 page)

The unit 3-sphere is the set
of all points in 4-space uniformly a distance one from the origin. This is
equivalent to the set of all points (
w
,
x
,
y
,
z
)
satisfying the distance equation

and, after squaring both
sides, we end up with

The twentieth-century artist
who uses symbols is alienated because the system of symbols is a private one.
After you have dealt with the symbols you are still private, you are still
lonely, because you are not sure anyone will understand it except yourself

—Louise
Bourgeois, quoted in
Lives and Works

: This says that for our intents and purposes, we can think of
a
as being roughly equivalent to
b
: "
a
is approximately
b
."

: One way of signifying the product of
a
with
b
;
that is, a way of notating the act of multiplying
a
by
b
.

:
Another way of signifying the product
of
a
with
b
; that is, another way of notating the act of
multiplying
a
by
b
.

 Read "n factorial," and
.

: The integer
a
raised to the
b
th power, for example,
.

: The closed interval between
a
and b. All numbers between
a
and
b,
inclusive.

: A symbol for infinity. It's important to note that although
 connotes a kind of magnitude, and is sometimes shorthand for the
idea of "arbitrarily large number," the symbol
 
is not a number.