\addtocontents{toc}{\contentsline {section}{The Square Root of 2}{\thepage}}
\clearpage
\vspace*{0.5in}
\begin{flushleft}
{\huge \bf Appendix A \\ \bigskip The Square Root of 2 \par}
\end{flushleft}
\bigskip
In order to do mathematics it is important to understand
the idea behind proofs and how to construct them. I have chosen
to demonstrate a proof that the square root of 2 is irrational.
This may not seem to be an important result, but it was to the Greeks, who
were horrified by the idea of irrational, or illogical,
numbers. The existence of such numbers conflicted with their concept of
numbers as ratios. The Pythagoreans believed in a
link between number, the soul, and the universe. Thus, the illogical in
numbers implied the illogical in the universe and a disruption of harmony
in the soul. This was hardly something to be taken lightly. Ultimately,
however, they had to accept the irrational numbers because they proved that
some numbers could not be represented by ratios.
What is a proof? A proof is a step-by-step justification of a
statement. But, says Ian Stewart, it is more than this:
\begin{quotation}
Textbooks of mathematical logic say that a proof is a
sequence of statements, each of which either follows from the
previous statements in the sequence or from agreed axioms --- unproved
but explicitly stated assumptions that in effect define the area of
mathematics being studied. This is about as informative as describing
a novel as a sequence of sentences, each of which either sets up an
agreed context or follows credibly from previous sentences. Both
definitions miss the essential point: that both a proof and a novel
must tell an interesting story (Stewart, {\it Nature's Numbers}, 39).
\end{quotation}
And so we end with a proof --- a mathematically rigorous story.
The proof that $\sqrt{2}$ is irrational is fairly straightforward
and serves as a good example of a proof. It is what mathematicians
call an elegant or beautiful proof. Beauty is of the utmost
importance to mathematicians; as Godfrey H. Hardy said, ``Beauty is the first
test: there is no place in the world for ugly mathematics.''
The first step in a proof is to state precisely what is to be proven.
{\it Theorem:} {\bf The number $\sqrt{2}$ is irrational.}
What does this mean? Irrational numbers are, by definition,
those which are not rational. Rational numbers are fractions. Any
number which can be written as the ratio between two integers is a
rational number. An irrational number, then, is a number which
{\it cannot} be written in the form $\frac{p}{q}$.
Proofs are puzzles, and a good puzzle solver has several tricks and
tactics for solving puzzles. A mathematician doesn't rely only on
these techniques but uses them as tools to tackle a problem. To prove
a statement, it must follow as the logical conclusion from a
series of valid premises and inferences. In our proof the premises are
the axioms. If the premises are the pieces of a puzzle, the inferences
are the way these pieces fit together. A mathematician knows how to put
the pieces together, using standard methods such as induction, the
pigeonhole principle, and reduction. It is sometimes difficult to know
which method to use and how to begin approaching a problem.
We will use the method of contradiction. This method stems from a form in
logic known as {\it modus tollens}, which states that ``if A
implies B, then not-B implies not-A.'' Our
theorem reads: ``If $x=\sqrt{2}$, then $x$ is irrational.'' The reversal,
or contrapositive, is ``If $x$ is rational, then $x$ cannot
be $\sqrt{2}$.'' The idea behind a proof by contradiction is to begin with
an assumption that negates the desired conclusion. When absurd statements
that contradict the real hypothesis follow from this assumption,
mathematicians can conclude that because impossible, or contradictory,
conclusions follow, the assumption must be invalid. Then, the
original hypothesis must lead to the original conclusion. This approach is
like playing a game of make-believe. We pretend that instead of wanting to
prove our theorem, we want to disprove it. For our theorem, we start out by
considering that $x$ is, in fact, rational, and we want to show that because
$x$ is rational, $x$ cannot equal $\sqrt{2}$.
We begin our proof:
{\it Proof:} {\bf Suppose our theorem is false. That is, suppose
$\sqrt{2}$ is rational.}
We know that a rational number can be written as $\frac{p}{q}$ where
$p$ and $q$ are integers. In fact, any rational number can be
written not only as a fraction, but as a fraction in lowest terms.
This means that it can be written so that the numerator and the
denominator have no common factors. We should be convinced that statement
is true, because given any fraction representation
for a number, we can write the fraction in lowest terms by dividing out
common factors.
In a proof it is important to question and verify everything. A proof
with gaps, falsities, or vagaries is open to failure. We continue
our proof:
{\bf We write $\sqrt{2}=\frac{p}{q}$, where $p$ and $q$ are integers
with no common factors greater than 1.}
Now we have an equation on which we can perform simple algebra.
{\bf From which it follows by multiplication by $q$ that $\sqrt{2}q=p$, and
by squaring both sides, that $2q^{2}=p^{2}$.}
What do we know about this quantity that is represented on both sides
of the equation? We know that it is an integer
and that it is even because $2q^{2}$ is a multiple of 2.
{\bf Thus, $p^{2}$ is an even number.}
Does anything follow from this? Can we determine anything about
$p$? It turns out that we can. If $p^{2}$ is even, then $p$ must be
even as well. We can convince
ourselves that the statement is true by considering
several examples. The number 2 is even; $2^{2}=4$ which is even.
Another even square number is 16; the square root of 16 is 4 which is
even. On the other hand, 25 is an odd square number and its square
root, 5, is not even. Concrete examples often help to make
generalizations easier to internalize. To be certain that this is true, we
consider the generic even case $2n$ and the generic odd case $2n+1$.
The square of $2n$ is $4n^{2}$, which is even, and the square of $2n+1$
is $4n^{2}+4n+1$, which is odd.
{\bf And so $p$ must be even.}
At the beginning of our proof we used the fact that any rational
number can be written in the form $\frac{a}{b}$. Again, we find it
useful to write an equation for $p$ which tells us something about
it. To say that a number is even means that it is divisible by 2, so
it is true that $p=2a$ for some integer $a$.
{\bf We write $p=2a$.}
Now we substitute $2a$ where we previously had $p$, so the
equation $2q^{2}=p^{2}$ becomes $2q^{2}=(2a)^{2}$. We can simplify
this to $2q^{2}=4a^{2}$ or $q^{2}=2a^{2}$.
{\bf By substitution, $2q^{2}=(2a)^{2}$ or $q^{2}=2a^{2}$.}
Notice how similar this equation
$q^{2}=2a^{2}$ looks to our earlier $2q^{2}=p^{2}$. We can apply the
same logic as before to conclude that since $2a^{2}$ is even, $q^{2}$ is
even, and $q$ is also even.
{\bf Just as before, we see that $q$ is even.}
Let's look again at precisely what we assumed. We started out by
supposing that we could write $\sqrt{2}=\frac{p}{q}$, for some $p$ and
$q$ with no common factors. But --- this is the part that's key to the
idea of contradiction --- we have shown that $p$ is even and that $q$
is even. That means that both $p$ and $q$ are divisible by 2; 2 is a
common factor of $p$ and $q$. Oh no! That's a contradiction. We said that
$p$ and $q$ didn't have any common factors, and we showed that our
assumptions imply that they do.
{\bf Both $p$ and $q$ are even numbers, so they are both
divisible by 2. This contradicts the assumption that $p$ and $q$
have no common factors.}
Our assumption has been shown to be invalid. It falls to the ground
with the big, loud thump of its own absurdity. We now know that we
can't write $\sqrt{2}$ as $\frac{p}{q}$, where $p$ and
$q$ have no common factors. And that's what it means to say that
$\sqrt{2}$ is rational.
{\bf So our assumption that $\sqrt{2}$ is rational is false. Then,
$\sqrt{2}$ is irrational, which is what we wanted to prove.}
At the end of proofs, mathematicians often write Q.E.D., an abbreviation for
{\it quod erat demonstrandum}, which is Latin for ``which was to be
demonstrated.''
{\bf Q.E.D.}
\addtocontents{toc}{\contentsline {section}{Cantor and Infinity}{\thepage}}
\clearpage
\vspace*{0.5in}
\begin{flushleft}
{\huge \bf Appendix B \\ \bigskip Cantor and Infinity\par}
\end{flushleft}
\bigskip
The Yellow Pig introduces Alice to cardinality, and to rational and irrational
numbers. He tells her that there are no more rational numbers than there
are integers or natural numbers, but he does not tell her that there are
more real numbers than rationals, so many more that the real numbers cannot
be listed. The proof of this fact
--- credited to Georg Cantor --- relies on the techniques of contradiction and
enumeration. Suppose we {\it do} have a list
with all of the real numbers. Or, for the sake of making a ``shorter''
list, suppose we have a list with just all of the real numbers between 0 and 1.
This would be a sublist of our list of all real numbers.
We consider such a list.
(Actually, we consider only the representations of reals between 0 and 1 that
do not contain any 9's as digits. We must be careful because
0.99999 \ldots is equal to 1. This is the only way for two different
decimals to represent the same real number.
It is okay for us to be looking at representations of such a
subset of the real numbers, because we need only to show that this subset
cannot be listed to conclude that all of the reals cannot be listed.)
The list could begin like this:
\vbox{
\ssp
$$
0.01357 \ldots \\
0.12714 \ldots \\
0.10625 \ldots \\
0.34712 \ldots \\
0.01470 \ldots
$$
\dsp
}
To show that this list does not contain all of the reals between 0 and 1, we
need to find a
real number which is not already on the list. Cantor outlines a method for
doing precisely this. We consider the digits along the diagonal.
\ssp
\begin{center} $ \begin{array}{ccccccc}
0 & . & {\bf 0} & 1 & 3 & 5 & 7 \\
0 & . & 1 & {\bf 2} & 7 & 1 & 4 \\
0 & . & 1 & 0 & {\bf 6} & 2 & 5 \\
0 & . & 3 & 4 & 7 & {\bf 1} & 2 \\
0 & . & 0 & 1 & 4 & 7 & {\bf 0} \\
\end{array} $ \end{center}
\dsp
The first number contains a 0 in the first digit to the right of the
decimal place. To construct a new number that is different from this
first number, we make sure the first digit in our new number is different
from the first digit in the first number. Say it is 1.
Now, we look at the second number in the list. Again, we want to
insure that our new number is different from this number. We see that the
second digit to the right of the decimal
place is a 2, so we can tack a 3 on after the 1 in our new number.
Again for the third number in our list, we consider the third digit, 6. The
third digit of our new number can be 7. And so on for all of the numbers
in our list. Our new number might begin 0.13721 \ldots.
This completely outlines one procedure for constructing a ``new'' number;
that is, the $n$th place in the new number is 1 more than the $n$ place in
the $n$th number, unless the $n$th place in the $n$th number is 8, in which
case the $n$th place in the new number is 0.
Now, we just need to be certain that this number is not already on the
list. But (if each representation is unique) how could it be? We
constructed it to differ in the $n$th place from the $n$th number,
so it must be different from all of the numbers in the list!
We conclude that we can't list all of the real numbers. There are
infinitely many rational numbers, but there are even more reals ---
{\it uncountably} many.
To be more mathematically precise, we say that the cardinality of
the set of real numbers is greater than that of the set of the rationals.
What does cardinality mean? One way to look at cardinality is to
consider correspondences between two sets. We say the
cardinality of two sets is the same if there
is a one-to-one correspondence, or {\it bijection} between elements of the
two sets. Cantor proposed to call two sets {\it equipollent} if there exists
such a correspondence between them. This terminology is useful because it
allows us to distinguish various kinds of infinity.
A set $D$ is {\it denumerable} or {\it countable} if there exists a bijection
with the natural numbers.
It is true that any subset of a denumerable set is denumerable, that the
product of denumerable sets is denumerable, and that the set of finite
subsets of a denumerable set is denumerable. Similarly, ${\bf R}$, the set
of the reals, and the set of finite subsets of ${\bf R}$ are equipollent.
So, is there any set with greater cardinality than ${\bf R}$? The answer is
yes.
Given a set, there always exists another set whose cardinality is greater.
Just as the collection of all subsets --- not just finite subsets --- of the
natural numbers is not countable, the set of all subsets of the reals has a
greater cardinality than that of the reals. There is no set with greatest
cardinality. We can always create a set with greater cardinality than a set
$S$ by taking the set of all subsets of $S$.
Onward to another question: Are there any sets with cardinality greater
than that of the natural numbers but less than that of the real numbers?
When does infinity cease to be countable or denumerable? The result of this
tricky question has been stated as the Continuum Hypothesis:
Every infinite subset of the reals is equipollent with either the naturals or
the reals.
Is this hypothesis true? There is no proof or disproof. I do not mean that
no proof or disproof has been found; there is a proof that there can be no
proof or disproof of this statement. In 1938 Kurt G\"{o}del
showed that the Continuum Hypothesis is both irrefutable and unprovable.
This means that there is a problem in mathematics for which mathematics is
at a loss to solve.
A paradox is a statement that cannot be either true or false. For example,
if the statement ``This sentence is false'' is considered true, then it must
be false. But if it is false, then it is true that ``This sentence is
false,'' and this process continues. Just as ``This sentence is false'' is
paradoxical, some
problems in mathematics are undecidable. At the beginning of the twentieth
century, David Hilbert and most mathematicians believed ``true'' and
``provable'' to be the same thing. That is, they believed that every
statement in mathematics was either provable or disprovable; there was
no such thing as unprovable.
G\"{o}del, however, said that either the axiomatic system of mathematics
was incapable of producing some results, or the system must contradict
itself. In his
terms, mathematics cannot be both {\it complete} and {\it consistent}. He
showed this by constructing (via a system of encoding and manipulating
mathematical statements)
what is basically the self-referential
mathematical theorem ``This theorem cannot be proven.'' Because of its
paradoxical nature, this theorem has no proof or disproof.
What G\"{o}del asserted in this mathematical madness was that if ``true''
and ``provable'' are the same thing, then such a
self-contradictory statement exists. In other words,
either our system is not complete and there are some things that we
cannot prove or disprove, or our system is inconsistent and we are left with
absurd conclusions. Because of this so-called Incompleteness Theorem,
mathematics can no longer be viewed as a field in which everything
can be proven or disproven. G\"{o}del's results revolutionized the
philosophy of mathematics.
\addtocontents{toc}{\contentsline {section}{The Golden Angle in Nature}{\thepage}}
\clearpage
\vspace*{0.5in}
\begin{flushleft}
{\huge \bf Appendix C \\ \bigskip The Golden Angle in Nature\par}
\end{flushleft}
\bigskip
The Pig points out much mathematics to Alice in the golden garden, but he
doesn't explain
why the Fibonacci numbers $(1,1,2,3,5,8,13\ldots)$ and the golden ratio
$\phi=(\frac{\sqrt{5}+1}{2})$ appear so frequently in nature.
Recently St\'{e}phane Douady and Yves Couder came to the conclusion that the
occurrence of Fibonacci numbers in plant matter arises from the
plant's central spiral. The tip of a growing plant contains
a central lump of tissue, known as the {\it apex}, from which the main
features of the plant --- leaves, petals, etc. --- develop. Tiny lumps
called {\it primordia} form around the apex. Then the apex moves away from
the primordia, causing the generative spiral to appear.
\centerline{\epsfbox{images/golden-gspiral.ps}}
We can measure the angle between successive primordia, and when we
do, we get roughly the same angle of divergence between any successive
primordia: $137.5^{\circ}$. Crystallographers
Auguste and Louis Bravais considered this angle in 1837 and realized that it
was $360(1-\phi)$. This angle is known as the golden angle.
How does this relate to the Fibonacci numbers? With the Pig's help, Alice
discovered that the ratio between successive Fibonacci numbers approximates
the golden ratio. Essentially, the Fibonacci numbers are contained in the
generative spiral.
Fibonacci numbers frequently appear in petals because petals represent the
outer primordia of such a spiral. In 1907 G. Van Iterson explained the
appearance of sunflower heads. Sunflowers seem to have two series of
interpenetrating spirals, one clockwise and the other counter-clockwise.
As a consequence of the golden angle in the generative spiral, these radial
spirals contain successive Fibonacci numbers of primordia.
We have explained {\it how} Fibonacci numbers appear in plants, but not {\it
why}. In 1979, H. Vogel performed experiments which suggested that a most
effective packing is obtained when primordia are placed along the generative
spiral using the golden angle. That's certainly a good reason why the golden
angle occurs in nature.
But why does it lead to the most efficient packing?
If a divergence angle of $90^{\circ}$ is used, the primordia
are arranged along four radial lines. This certainly isn't a tight
packing, and it offers the plant very little support. This is true for
other factors of $360^{\circ}$. The more irrational the angle
is, the tighter the packing will be. The most irrational number --- one
that is most poorly approximable by rationals, as considered in the study of
continued fractions --- is $\phi$. So, the tightest packing is achieved when
the divergence angle is $360(1-\phi)^{\circ}$, the golden angle.
At angles just less than (below left) or just greater than (below right) the
golden angle, a fairly tight packing is achieved; however, the tightest
packing (below center) occurs when the angle of divergence is the golden angle.
\centerline{\epsfbox{images/golden-divangle.ps}}
Douady and Couder offer another explanation for the occurrence of the golden
angle. They used silicone oil in a vertical magnetic field to demonstrate
that elements formed at equally spaced intervals of time on the rim of a
small circular apex repel and migrate radially at a specified initial
velocity. In this way they were able to conclude that the generative spiral
is a result of the principles of dynamics.
Another theory is that the golden angle was not always present in the
generative spiral, but that
over time plants have been fine-tuned by natural selection to favor
the tightest packing of primordia.
Evolution, genetics, geometry, and dynamics all involve this special number
$\phi$. The golden ratio --- and more generally, mathematics --- are truly
ubiquitous in beauty and nature.
\addtocontents{toc}{\contentsline {section}{Paul Erd\H{o}s}{\thepage}}
\clearpage
\vspace*{0.5in}
\begin{flushleft}
{\huge \bf Appendix D \\ \bigskip Paul Erd\H{o}s\par}
\end{flushleft}
\bigskip
Paul Erd\H{o}s, pronounced ``air-dish'', was one of the greatest
mathematicians. He not only studied and proved theorems in a variety of
branches of mathematics, but he also
encouraged and supported many other mathematicians. He co-authored papers
with 485 mathematicians. These mathematicians are said to have an Erd\H{o}s
number of 1, and those who collaborated with them are said to have an
Erd\H{o}s number of 2; it is believed that all currently collaborating
mathematicians have an Erd\H{o}s number of less than 8.
Erd\H{o}s helped to transform a mostly solitary study into one of an
open and cooperative community.
Erd\H{o}s, a self-proclaimed
``poor great old man, living dead, archeological discovery,
legally dead, counts dead,'' lived for mathematics. From his birth in
Budapest, Hungary on March 26, 1913 until his death 83 years later on
September 20, 1996, Erd\H{o}s produced
mathematics, often for over 12 hours a day, while sustained by a variety of
substances. He remarked: ``A mathematician is a machine for
turning coffee into theorems'' (Hoffman, 7).
In addition to such witty sayings, Erd\H{o}s invented
his own language in which wives were called ``bosses,'' God
was ``the supreme fascist,'' people who stopped doing mathematics had
``died,'' and children were called ``epsilons.'' Erd\H{o}s was eccentric;
he owned little clothing and had limited personal possessions, he had
obsessions with both death and cleanliness, and was almost completely
dependent on others to feed and chauffeur him. Another Hungarian mathematician,
Andrew V\'{a}zsonyi, recalls that even in his youth, Erd\H{o}s was odd.
When the two met in 1930, he immediately asked the 14 year old V\'{a}zsonyi
for a four-digit number and proceeded to find its square. He also announced
that he knew 37 proofs of the Pythagorean Theorem.
Erd\H{o}s had a strong liking for small children and had a special interest
in students of mathematics. He sought out child prodigies such as J\'{o}zsef
Pelik\'{a}n and Louis P\'{o}sa and supported their interest in mathematics.
He was a patron of mathematics, sponsoring
students and donating rewards for unsolved problems. He gave freely to
numerous non-mathematical charities and causes, keeping hardly any money
for himself.
He expended much of his energy in mathematics on combinatorics,
``the art of counting without counting,'' and graph theory. His
studies included that of Ramsey Theory, named after Frank Plumpton
Ramsey. The Yellow Pig explains one central question
of Ramsey Theory, which involves coloring a complete graph with two colors.
The Ramsey number of a graph is the minimum number of vertices needed to
force a monochromatic subgraph, or a set of $n$ points where all of the edges
connecting those points are the same color. More rigorously:
For two graphs $G$ and $H$, let the Ramsey number $R(G,H)$ denote the
smallest integer $m$ satisfying the property that if the edges of the
complete graph $K_{m}$ are colored in red or blue, then there is either a
subgraph isomorphic to $G$ with all red edges or a subgraph isomorphic to
$H$ with all blue edges.
As the Yellow Pig explains, $R(3,3)$ is 6. That is, a two-colored graph of
all edges connecting five vertices does not necessarily contain a
monochromatic triangle, but one with six vertices must.
Finding higher Ramsey numbers remains an open question. The table below
lists many of the known Ramsey numbers.
\ssp
\begin{center} $ \begin{array}{cccccc}
R & 2 & 3 & 4 & 5 & 6 \\
2 & 2 & 3 & 4 & 5 & 6 \\
3 & 3 & 6 & 9 & 14 & 18 \\
4 & 4 & 9 & 18 & 25 \\
5 & 5 & 14 & 25 \\
6 & 6 & 18 \\
\end{array} $ \end{center}
\dsp
Mathematicians are trying to find more Ramsey numbers. Although they have
not found an equation for obtaining Ramsey numbers, they have specified a
range for numbers of the form $R(n,n)$. The current lower bound (attributed
to Spencer) is $\frac{\sqrt{2}}{e}n2^{n/2}$, and the current upper bound
(from Thomason) is $n^{-1/2+c/\sqrt{\log{n}}}$ $2n-2 \choose n-1$ (Chung, 9).
This is just one of many intriguing open problems in mathematics.
\addtocontents{toc}{\contentsline {section}{Escher and Hyperbolic Geometry}{\thepage}}
\clearpage
\vspace*{0.5in}
\begin{flushleft}
{\huge \bf Appendix E \\ \bigskip Escher and Hyperbolic Geometry\par}
\end{flushleft}
\bigskip
Geometry, as we know it, is based on several postulates or axioms, rigorous
assertions, and definitions. One that has caused much discussion in geometry
is known as the Parallel Postulate. It states:
Given any line and any point not on the line, there is only one line
through the given point that never intersects the given line.
This is
something that we take for granted in Euclidean geometry, but it turns out
that many results in geometry follow without using this postulate.
Mathematicians have studied neutral geometry, without the postulate at all,
and several non-Euclidean geometries based on the negation of the postulate.
In hyperbolic geometry there are {\it infinitely many} parallel lines
through a given point, and the sum of angles in a triangle (left) is less than
$180^{\circ}$. Spherical geometry is based on the idea that on a spherical
surface any two lines intersect, and the sum of the angles in
a triangle (right) is
greater than $180^{\circ}$. In both of these alternative geometries, lines
are not straight lines in the Euclidean sense, but appear as curves like
arcs of circles or lines of latitude and longitude on a globe.
\centerline{\epsfbox{images/esch-triangles.ps}}
Many kinds of geometry, including hyperbolic geometry, can be seen in the
works of M.C. Escher, whose formal mathematical training was extremely
limited. Escher's grasp of mathematics, as seen by his independent
studies and artistic intuition, includes an understanding of isometries,
symmetry groups, crystallography, chromatic groups, and tesselation in
spherical and hyperbolic geometry.
The problem of regularly dividing the plane interested Escher greatly. He
wrote: ``I cannot imagine what my life would be like if this problem had
never occurred to me. One might say that I am head over heels in love with
it, and I still don't know why'' (Locher, 67). In the Euclidean plane there
are seventeen essentially different ``wallpaper'' patterns using combinations
of translations, rotations, reflections, and glide-reflections. Escher
discovered these on his own and used them in his art.
Escher was greatly influenced by a number of mathematicians, including G.
P\'{o}lya, R. Penrose, and H. S. M.
Coxeter, the geometer who introduced him to hyperbolic geometry.
Escher met Coxeter at the International Congresses of
Mathematicians in 1954 and soon after asked for an explanation of how to
construct a series of objects that decrease in size as they reach the
boundary of a circle. Escher came across the idea of a hyperbolic
plane in 1958 from a figure in ``A Symposium on Symmetry'' sent to him by
Coxeter.
\centerline{\epsfbox{images/esch-cox.ps}}
Many of Escher's works, included at the end of this section, make use of
hyperbolic geometry. Several of these are based on the Poincar\'{e}
disk model of hyperbolic geometry. In the Poincar\'{e} disk model, lines are
diameters and arcs perpendicular to the boundary of a circle at infinity.
This causes distances to appear distorted while angle measures are preserved.
\centerline{\epsfbox{images/esch-poincare.ps}}
Escher's ``Butterflies'' is one tessellation that employs the Poincar\'{e} disk model. Because
the dividing line between the front and the back wings of a butterfly is
perpendicular to its body, the framework of butterflies can be seen as
circles intersecting at right angles. Similarly, a net of circles with six
fold symmetry is used for ``Ringsnakes.''
It was in his ``Circle Limits,'' which the Yellow Pig stops to admire, that
Escher felt the greatest sense of achievement. He saw his use of the
Poincar\'{e} disk model in ``Circle Limits'' as a milestone in his career.
Escher created four ``Circle Limits'' pieces using lines in the
Poincar\'{e} disk model. (Actually, the third ``Circle Limits'' doesn't use
lines in the sense of hyperbolic geometry. The arcs of the backbones
of the fish meet the outside circle at angles of approximately $80^{\circ}$,
not $90^{\circ}$.)
Escher also experimented with hyperbolic tilings in rectangular regions and
spirals, using hyperbolic geometry to shrink his figures while maintaining
similarity, as in ``Smaller and Smaller I'' and ``Whirlpools.''
M.C. Escher was clearly an artist, but was he also a
mathematician? Escher wrote: ``\ldots I have often felt closer to people who
work scientifically (though I certainly do not do so myself) than to my
fellow artists'' (Locher, 71). Many Escher admirers suspect he had more
mathematical talent than he was willing to admit, but others, including
Coxeter, believe he was guided not by mathematics but by aesthetics.
Escher's description of his ``Circle Limits III'', shows that he was unaware
of the rigorous mathematical foundations. Surely he had no idea that
mathematicians would puzzle over the precise $60^{\circ}$ angles and their
implications to the hyperbolic geometric model.
Escher's works correspond in many ways with those of crystallographers,
scientists who study the structure of crystals,
but there is an important distinction to make in the motivation for their
studies. Escher saw crystallographers as interested in opening up the study
of symmetry for its application, whereas he was intrigued by symmetry
for the sake of beauty.
Escher was well aware that he was untrained in mathematics.
In his {\it Regelmatige vakverdeling} (Regular Division of the
Plane) Escher wrote that without the basic principles of mathematics and
symmetry it was difficult at first for him to design congruent shapes for
his work (Locher, 164). He did not consider himself to be a mathematician,
but, like a pure mathematician, he had strong concepts of beauty and symmetry
and realized the role that mathematics must play in achieving such symmetry.
He developed mathematical principles in an effort to understand their
beauty. Mathematics, for him, was not about formal training, but about
intuition, experimentation, and aesthetics, characteristics shared by
artists, mathematicians, and creative people in other fields.
\pagebreak
\begin{center}
{\large \bf Prints by Escher\par}
\end{center}
\centerline{\epsfbox{images/esch-page.ps}}
Top left: ``Butterflies''. Top right: ``Ringsnakes''. Bottom left:
``Circle Limits III''. Bottom right: ``Smaller and Smaller I''. Right:
``Whirlpools''.
\addtocontents{toc}{\contentsline {section}{Lewis Carroll and Logic}{\thepage}}
\clearpage
\vspace*{0.5in}
\begin{flushleft}
{\huge \bf Appendix F \\ \bigskip Lewis Carroll and Logic\par}
\end{flushleft}
\bigskip
Lewis Carroll was born Charles Lutwidge Dodgson on January 27, 1832. His
father, who had studied both the classics
and mathematics, always encouraged him and his eleven brothers and sisters
to learn. Dodgson attended Christ Church College where he
received first honors in mathematics in 1852 and was offered a
paid position.
He began teaching at age 23 and became a fully established member of
the community when he won a mathematical lectureship the
following year.
At this time Christ Church got a new dean --- Henry George
Liddell. Dodgson met Liddell's daughter Alice on April 25, 1856, just
before her fourth birthday. Like Erd\H{o}s, Dodgson was very interested
in children. He entertained them with his stories and
often photographed them and drew their portraits. Alice and her older
sisters, Lorina and Edith,
quickly became good friends with Dodgson. In June 1862
Dodgson, two of his sisters, their aunt, the three Liddell sisters,
and Robinson Duckworth, a friend of Henry Liddell, went on a picnic
at Nuneham. Dodgson wrote about such a picnic with a lory
(Lorina), an eaglet (Edith), a duck (Duckworth), and a dodo (Dodgson)
in his {\it Alice's Adventures in Wonderland}, published in 1865
under the pseudonym Lewis Carroll (his name translated into Latin and
back to English). He continued writing stories and mathematics until his
death on January 14, 1898. His works include his 1871 sequel {\it Through
the Looking-Glass}, his operetta of {\it Alice} in 1886, the
{\it Sylvie \& Bruno} stories, several volumes on logic, and many
mathematical puzzles and games.
The horse in Logicland is puzzled by one of Lewis Carroll's exercises from
{\it Symbolic Logic}:
\vbox{
\ssp
\begin{flushleft}
1. \\
Babies are illogical. \\
Nobody is despised who can manage a crocodile. \\
Illogical persons are despised.
\end{flushleft}
\dsp}
To ``solve'' this syllogism --- series of logical statements --- Lewis Carroll
proposes the following abbreviations:
$a$ = able to manage a crocodile; $b$ = babies; $c$ = despised; $d$ =
logical.
The syllogism can then be rewritten as:
\vbox{
\ssp
\begin{flushleft}
All $b$ are not $d$. \\
No $c$ are $a$. \\
All not $d$ are $c$.
\end{flushleft}
\dsp
}
This gives us three statements each containing two of our letters. We see
that the letters $c$ and $d$ both occur in two statements, once as the
subject and once as the object, while the letters $a$ and $b$ each only occur
once, as an object and a subject respectively. This suggests a reordering of
the statements:
\vbox{
\ssp
\begin{flushleft}
All $b$ are not $d$. \\
All not $d$ are $c$. \\
No $c$ are $a$.
\end{flushleft}
\dsp
}
Now our statements appear tied together. We want to conclude something
about $b$ and $a$, and there is a natural progression from $b$ to $a$ via
$d$ and $c$.
First we consider only the first two statements. Since all $b$ are not $d$,
and all not $d$ are $c$, we can conclude by a form of substitution that all
$b$ are $c$. We can represent this scenario with a diagram of three
overlapping circles. This gives us eight regions: one outside of all
of the circles, one inside all of the circles, three representing
intersections between two circles, and three containing only one
circle.
We then label the circles. For this pair of statements, we would label them
$c$, $b$, and $d$. Next, we shade the regions that represent
impossibilities. The first statement tells us that there are no $b$
which are also $d$, so we can shade the two regions that represent the
intersection of $b$ and $d$. The second statement says that anything
which is not $d$ is $c$. That is, there is nothing which is neither
$d$ or $c$, so we can shade the remainder of the circle $b$ which
does not overlap $c$. Our diagram is a visual representation of the
statement ``all $b$ are $c$.''
\centerline{\epsfbox{images/logic-1.ps}}
Similarly, we consider this statement with the third premise: ``no
$c$ are $a$.'' This time we draw circles $b$, $c$, and $a$. We shade
the region of $b$ which is not contained in $c$ to demonstrate that
all $b$ are $c$. Then we shade the area common to $c$ and $a$ to
show that no $c$ are $a$.
\centerline{\epsfbox{images/logic-2.ps}}
We want to conclude something about $b$ and $a$. What can we say
about them? We look at the intersection of $b$ and $a$ in our
diagram. The entire intersection has been shaded. That means that
there is no possible intersection of $b$ and $a$. That is, no $b$
are $a$; no babies are able to manage a crocodile. And that is the
solution to the horse's riddle.
While logic is not really math, the two are closely related.
Perhaps most importantly, logic and mathematics require the same type
of thinking. What I find most intriguing about Lewis Carroll is that
he was interested in mathematics and writing, two things which are
not often associated with each other but perhaps should be. After all,
both are ways of communicating ideas, of telling stories.