2jul11.

Overview of Lessons on Models, Part One
\textit{ $\C$  2010,2011 Prof. George K. Francis, Mathematics Department,
University of Illinois}

\begin{document}
\maketitle

\section{Introduction}
In this section of the course we introduce the complex numbers
as an efficient tool for studying both Euclidean and non-Euclidean
geometry of the plane. In particular, Euclid's primitive (undefined)
concept of \textit{ congruence} is made precise in terms of
length and angle preserving transformations, called \textit{isometries}.
Each kind of geometry of the plane has its own charericstic \textit{ group}
of isometries.

\subsection{Synthetic vs. Analytic Geometry}
In the first part of the course, we have seen how the axiomatic method,
begun by Euclid (-300) and completed by Hilbert (1900), applies to
plane geometry, both Euclidean and non-Euclidean. One feature of axiomatic
geometry is the need to keep track of the order in which theorem are
deduced logically from prior ones, and ultimately from the unproven axioms.
Avoiding logical gaps and circular arguments requires training and
concentration. This  contributes to the difficulty as well as the
pedagogical value of \textit{ synthetic geometry}, the more technical
and proper name for this traditional form of doing geometry.

A second, seemingly more experimental and less demanding method,
\textit{analytic geometry} now dominates. It is based on set theory
and the properties of numbers. Begun by Descartes and Fermat in the
seventeenth century, applied to physics and astronomy by scientists
since Newton, it is familiar to every high school graduate.

One important differences between analytic and synthetic geometry is that
in the former, algebra plays the role that purely logical reasoning
does in the latter. And direct analytical
"verification" replaces long, carefully maintained chains of
"proofs". Of course, the theory of numbers, which underlies analytical
geometry, is itself an axiomatic system.  And, as Hilbert
showed, the two are equi-consistent in that each has a model in the
other. So it is a practical choice between analytic and synthetic
geometry, not a fundamental one. Mathematics is one science, not
a collection of competing or complementing theories.

\subsection{Categorical Axiom Systems}
We finished the first part of the course by demonstrating how a
system of axioms for Euclidean geometry, the four by Birkhoff, has
a model (in the technical sense from axiomatic systems) in the
Cartesian plane (analytic geometry). When we investigated geometries
with but a small number of points and lines, we found it useful to
prove theorems in their axiomatic system by deducing them from the
model. This process is justified only in the case that all models
of a particular system of axioms are \textit{ isomorphic}, namely
that each model is structurally identical to any other. Such a system
of axioms is called \textit{ categorical}.

Although we will not prove the equivalence of the various models
of non-Euclidean geometry we have studied experimentally by
means of Hvidtsten's Geometry Explorer (GEX), we get a sense of
this from the fact that his QuadModel feature shows the same result
for all models, regardless of which model we started from.
The advantage of a categorical axiom system is precisely its property
that theorems can be deduced in an "analytical" way from (any convenient)
model, and does not have to be fit into an orderly, systematic arrangement
of proofs. And this fully justifies our studying plane geometry analytically
from now on.

\section{The Complex Plane}
Hvidsten introduces the essentials for the complex plane at the end of
his chapter on Cartesian geometry. They are developed and applied later
in Chapter 8, which is written at an almost
graduate-level and needs more careful exposition here. The reason for this
is again pedagogical. The usual approach to this subject in a college
course uses nothing beyong the analytic geometry at a high school level.
And the chapters prior to Chapter 8 are written at this level. It is similar
to what those of you who will teach some non-Euclidean geometry in the
high school can expect to see in the high school textbook.

The approach we take here is appropriate for college in that it teaches
at least two valuable lessons at the same time. The first lesson is
the elegance and utility of complex numbers and their functions
to understand the plane geometry.
The second is expository compactness, so we won't lose track of the
forest on a ccount of the trees. The cost is no more than a careful
application to complext numbers of the skills in logical thinking you
have mastered in the first part of the course.

\subsection{How far into Chapter 8 we will get.}
However, we do not have the time to complete Chapter 8 in this course.
It suffices to get you to a plateau of comprehension so that you can
profitably complete the chapter on your own at some later time. Perhaps
that time will be after you've nearly burned out teaching high school
geometry and need to revive your enthusiasm for geometry by doing something
harder, and therefore more rewarding.

\section{Moebius Transformations}
The principal tool of our study is the \textit{ Moebius Transformation},
which has many other names and origins. We shall devote several lessons
to its various definitions, properties and applications.

Felix Klein originated the modern concept that all geometry is based,
ultimately, on groups of transformations. This is idea is more properly
pursued in the sister course, netMA403. The astronomer, geometer and
physicist, August Ferdinand Moebius is famous for much more than the
strip of paper, glued together after a half-twist, that bears his name.
Indeed, he merely \textit{used} this artefact as an example.
It was known to the Greeks
and used by medieval farmers, and by engineers of the industrial age,
to make drive belts that wear evenly on "both" sides:
there was only one side to wear!

\subsection{Aside on Moebius Bands and Projective Geometry}
For the record, and this subject belongs into an elementary topology course,
M\"obius (as you would write the Umlaut in LaTeX) proposed gluing a flexible
disk to a (very) flexible Moebius band, thereby closing it up into a
one-sided "sphere". This object becomes a model (in four space) of
\textit{projective geometry}, which is the non-Euclidean geometry
introduced in netMA403.  The last undergraduate course in Projective Geometry
at the University of Illinois died in the '70s and the subject is rarely
presented in an elementary form anywhere in the world today.

\subsection{Aside on Elliptic Geometry}
Another reason for our approach of using complex numbers is the ease with
which all three kinds of plane geometry, the \textit{ hyperbolic,
parabolic, and elliptic} can be treated uniformly. Other versions of college
geometry now apply the word "non-Euclidean" to any geometry of the plane
which is not Euclidean. For the example, the geometry on a sphere (such as
our own earth and therefore well known from geography) is 2-dimensional but not
Euclidean. Indeed, not only is the angle sum of spherical triangles always
greater than $\pi$ (just as it is always less than $\pi$ for hyperbolic
geometry), but most theorems in absolute geometry, and even Euclid's
first postulate, are false on the sphere.

Recall from geography that "straight lines" on a sphere
are \textit{ great circles}. And any two points on a sphere are
joined by two segments, the short one and the long one on the same
great circle. That is, unless the two points are \textit{ antipodes},
i.e. on opposite sides of the sphere through its center. Then there
are infinitely many lines through them, just as in hyperbolic geometry.
Much more surprising is that in spherical geometry, there are no parallel
lines, all pairs of lines have two points in common. Thus there is no
nice axiomatic treatment of spherical geometry comparable to Euclid's.

Nevertheless, the angle sum of a triangle (the form of the exterior
angle theorem) actually goes three ways: less than, equal to and greater
than $pi$. For historical reasons, these are referred to a "hyperbolic,
parabolic and elliptic geometry" respectively. While there is a
relation to the eponymous conic sections, it goes well beyond the
present discusssion. It is based on the classification of all Moebius
transformations.

We will define and explore some geometrical properties of Moebius
transformations, particularly in connection with the isometry group
of the hyperbolic non-Euclidean plane. But, with little extra effort
we also describe the isometries of Euclidean and spherical (elliptic)
geometry.

\section{The Models of Hyperbolic Geometry}

We will finish this course with a study of the the three models of
hyperbolic geometry and a brief look at how GEX computes them. There
will be a subsequent, separate overview for this part.