The Unimaginable Mathematics of Borges' Library of Babel (43 page)

factorial
A useful notation for positive integers that often crops up in
combinatorial formulas. Generalized by the gamma function. Approximated by
Stirling's formula. See the section "Notations" for a formal
definition and an example.

Fibonacci sequence
Counts, for each successive generation, the number of immortal
rabbits living in an infinite universe. Has terms whose ratios converge to the
Golden Mean. Arises in surprising places in nature. Is related to logarithmic
spirals. Is the object of study of entire books.

fixed point
A fixed point of a function that maps from a space to itself
doesn't move. For example, if we map the real numbers to themselves by the
function f(x) = 3x, then f(0) = 3-0 = 0, entailing that 0 is a point fixed by
the function.

 
flat
Describes a
geometric object equipped with a notion of distance which may be precisely the
same as in Euclidean space. For example, although curved, the surface of a
cylinder is flat, because to find the distance between two points, the cylinder
may be "cut open and unrolled" and "laid flat." Then the
two points may be connected by a Euclidean straight line and then
"rerolled."

fractal
An object with a dimension that is not an integer. An object that
continues to present visual complexity under increased magnification. Clouds,
bark, lungs, leaves, coastlines, strange attractors, the Koch snowflake curve,
the Cantor set,. ..

function
A way of relating two spaces. A way of relating a space with
itself. A way of corresponding elements of one set with elements of another
set. A systematic process that inputs real numbers and outputs real numbers.

Funes-like
Ireneo Funes is a character in a remarkable Borges short story
who is gifted and afflicted with essentially perfect memory. Funes spends more
than a day reliving every detail of a day.

gamma function
An elegant way of generalizing the concept of the factorial of a
positive integer to that of all real numbers.

glossary
See "definition" or "self-referential."

great circle
An equator of an
n
-sphere. A circle of maximal size that
can be contained in an
n
-sphere.

hexagonal prism
A hexagon is a symmetric six-sided object contained in a plane. A
hexagonal prism is a three-dimensional object whose horizontal slices are
filled-in hexagons.

homomorphism
A function maps elements of one set to elements of another set. A
homomorphism is a function that also preserves algebraic relations during the
mapping; for example, we can think of integers as points in the set of real
numbers, but we can also think ofnumbers as things that do algebraic stuff,
such as addition and subtraction. A homomorphism maps integers both as elements
and as algebraic objects. If this intrigues, see Gallian's
Contemporary
Abstract Algebra.

hyperreal number
An infinitesimal affiliated with any nonzero real number.

hyperreal number line
The real number line, combined with all the hyperreals affiliated
with each real number.

illegitimate deduction
See "circular logic."

infinitesimal
An idea used by Euler, Newton, and Leibniz in thinking about
calculus. Infinitesimals may be thought of as actual numbers in a logically
consistent way, and may loosely be thought of as entities that have
"magnitude" greater than 0 but are smaller than every positive real
number. Every real number
x
can logically be thought of as being
surrounded by infinitesimal hyperreal numbers that are closer to
x
than
any other real number.

initial position
The starting point for a Turing machine.

integers
The set of whole numbers { . .. -2, -1, 0, 1, 2,... }.

internal state
A particular set of instructions for a Turing machine, telling
the Turing machine what should be done in response to each possible input.

irrational numbers
The set of real numbers that can't be written in the form of a
fraction
p/q,
where
p
and
q
are both integers. When an
irrational number is written out in decimal form, the digits in the expansion
neither terminate nor turn into a repeating pattern.

Klein bottle
A torus that has lost it's way in 4-space. A boundaryless
nonorientable two-dimensional object.

Koch snowflake curve
A pleasantly symmetric example of a fractal that appears in a
math book that Borges reviewed. One unusual property that it possesses is that
the distance between
any
two points is infinite. The more closely we
look at any portion of the snowflake curve, the more detail emerges.

lemma
A lemma is an assertion not quite important enough to be called a
theorem.

libit
Short for "library unit." A library unit is a
collection of contiguous hexagons sufficiently large to hold all 25
^1,312,000
distinct volumes and sufficiently symmetric that copies of it are able to tile
the infinite 3-space model of the Library.

locally Euclidean
A space is locally Euclidean if at every point of the space, a
severely myopic individual is convinced that they are, in fact, inhabiting a
Euclidean space. As an example, consider the circle. It is clearly not a
Euclidean line, but if you have access to a math program or drawing program
that allows you to zoom in on an object, and you zoom in on any point, you'll
find that what began as looking like a curve looks a lot like a straight line.
Thus, a circle is locally Euclidean.

logarithm
The logarithm is a function characterized by several remarkably
useful properties. All of these stem from the fact that it is the inverse
function to the exponential function in base 10. (An inverse function cancels
the effect of its corresponding function.)

lower bound
A minimal estimate. "At least thus-and-such."

manifold
A shorter name for a locally Euclidean space.
map
Another
name for a function, for we can think of a function not only as taking inputs
and returning outputs but also as taking a point and mapping it, or moving it,
to another point.
median
The median of a finite set of numeric data is a
kind of a middle number: half of the data will be larger than the median, and
half will be smaller.

Mobius band
A Mobius band is a nonorientable, one-sided surface with one
boundary circle. Taking two Mobius bands and gluing them together along their
boundary circles produces a Klein bottle! (This is not an obvious
construction.)

non-Euclidean
Any space that is not a Euclidean space. For example, all
manifolds, including spheres, tori, and Klein bottles, are non-Euclidean. A
cylinder, a figure eight, and a spiral are all non-Euclidean. Typically,
though, we'd only refer to a space as non-Euclidean if it's everywhere locally
Euclidean.

nonorientable space
Best defined in opposition to an orientable space: in an
orientable two-dimensional manifold, at any point we may choose a definition of
"up" and "right," and then, regardless of the path we
navigate through the space, when we return to our beginning point our notions
of "up" and "right" will agree with those that we
originally chose. By contrast, in a nonorientable two-dimensional space, after
making choices of "up" and "right," there are circuitous
paths we may follow such that when we return to the starting point, either
"up" will look like "down" or "right" will appear
"left." In a three-dimensional manifold, we'd also have to choose a
"front" to make a legitimate definition.

nonstandard analysis
Logically sound mathematics done with infinitesimals and
hyperreal numbers.

numerator
The numerator of a fraction is the number being divided by the
denominator. The top of the fraction. The attic of the fraction. In the
expression 3/5, the numerator is 3.

one-to-one correspondence
This is a map between sets A and B such that every element in A
is sent to a distinct element in B and every element in B has exactly one
element of A mapped to it. If A and B are finite sets, it means that they each
have the same number of elements. If A and B are infinite sets, the implication
is that they have the same cardinality.

origin
The point in coordinatized
n
-space such that all
coordinates are 0. The point where the axes all intersect. An arbitrarily
chosen point that serves as the center of the space.

periodic
A pattern is periodic if it repeats over and over. For example,
the pattern of letters MCVMCVMCVMCV is periodic of period 3. The earth orbiting
the sun is an example of periodic motion. A wallpaper pattern may be periodic.

power of 10
An exponential expression with a base of 10. Examples include 10
³
,
10
¹⁰⁰
, and, more abstractly, 10
ⁿ
, which signifies
"some power often."

prime number
A number
p
whose factors are limited to 1 and
p
. No
other positive integer may divide a prime number.

product
Another name for the act of multiplication.

raised to a power
Another phrase for raising a base by an exponent. Another way of
saying that a number is being multiplied by itself a specified number of times.

rational numbers
All numbers of the form
p/q,
where
p
and
q
=
0 are both integers.

real numbers
The set of all rational and irrational numbers.

real number line
Euclidean 1-space. The set of real numbers identified with points
on the Euclidean line.

self-referential
See "self-referential."

set
A collection of objects, usually called elements. If memory
serves correctly, "set" is the word in the English language with the
most definitions—at any rate, the 2nd edition of the Oxford English Dictionary
runs to 23 pages of definitions and citations for the word "set."

set of measure
0 An inconsequential set. A set that essentially occupies none of
the ambient space that it lives in. Pick an arbitrarily small number
c:
a set of measure 0 can be covered by—contained in— countably many sets whose
diameters sum to a number smaller than
c.
(The
diameter
of a set
may be thought of as the maximal distance across it.)

set theory
One of the foundations of modern mathematics. One of the
underlying languages of modern mathematics. A collection of seemingly
unassailable intuitions about objects in our world.

space
A collection of points, often equipped with some notion of
distance between points.

Stirling's approximation to
the factorial
A way of approximating
n
!
using Euler's constant
e
, exponentials, square roots, and 2w. See, for
example, page 616 of Apostol's
Calculus, Volume II
for a derivation of
the formula.

subset
A subcollection of a set. A subset of a set can be the whole set,
some of the set, or none of the set. The subset consisting of no elements is
called the
empty set.
See "empty set."