\chapter*{Introduction}
\markboth{\quad\ Introduction\hfill}{}
\addcontentsline{toc}{chapter}{\numberline{\thechapter}Introduction}

\section{What is geometry about?}
\markright{\hfill \thesection\ What is geometry about?\quad} Geometry
``measuring the world'' attempts to describe and understand space around us
and all that is in it. It is the central activity and main driving force in
many branches of math and physics, and offers a whole range of views on the
nature and meaning of the universe. This book treats geometry in a wide
context, including a wealth of relations with surrounding areas of math and
other aspects of human experience.

Any discussion of geometry involves tension between the twin ideals of
intuition and precision. {\em Descriptive} or {\em synthetic} geometry takes
as its starting point our ideas and experience of the observed world, and
treats geometric objects such as lines and shapes as objects in their own
right. For example, a line could be the path of a light ray in space; you
can envisage comparing line segments or angles by ``moving'' one over
another, thus giving rise to notions of ``congruent'' figures, equal
lengths, or equal angles that are independent of any quantitative
measurement. If $A,B,C$ are points along a line segment, what it means for
$B$ to be {\em between} $A$ and $C$ is an idea hard-wired into our
consciousness. While descriptive geometry is intuitive and natural, and can
be made mathematically rigorous (and, of course, Euclidean geometry was
studied in these terms for more than two millennia,
compare~\ref{sec!history}), this is not my main approach in this book.

My treatment centres rather on {\em coordinate}
geo\-metry.\index{coordinate!geometry} This uses Descartes' idea (1637) of
measuring distances to view points of space and geometric quantities in
terms of numbers, with respect to a fixed origin, using intuitive ideas such
as ``a bit to the right'' or ``a long way up'' and using them quantitatively
in a systematic and precise way. In other words, I set up the $(x,y)$-plane
$\R^2$, the $(x,y,z)$-space $\R^3$ or whatever I need, and use it as a
mathematical model of the plane (space, etc.), for the purposes of
calculations. For example, to plan the layout of a car park, I might map it
onto a sheet of paper or a computer screen, pretending that pairs $(x,y)$ of
real numbers correspond to points of the surface of the earth, at least in
the limited region for which I have planning permission. Geometric
constructions, such as drawing an even rectangular grid or planning the
position of the ticket machines to ensure the maximum aggravation to
customers, are easier to make in the model than in real life. We admit
possible
% \begin{figure}[h]
% \renewcommand{\thefigure}{\arabic{chapter}.\arabic{section}}
% \vskip4cm
% \centering
% \caption{A coordinate model of space}
% \label{coor_model}
% \end{figure}
drawbacks of our model, but its use divides any problem into calculations
within the model, and considerations of how well it reflects the practical
world.

{\em Topology} is the youngster of the geometry family. Compared to its
venerable predecessors, it really only got going in the 20th century. It
dispenses with practically all the familiar quantities central to other
branches of geo\-metry, such as distance, angles, cross-ratios, and so on.
If you are tempted to the conclusion that there is not much left for
topology to study, think again. Whether two loops of string are linked or
not does not depend on length or shape or perspective; if that seems too
simple to be a serious object of study, what about the linking or knotting
of strands of DNA, or planning the over- and undercrossings on a microchip?
The higher dimensional analogues of disconnecting or knotting are highly
nontrivial and not at all intuitive to denizens of the lower dimensions such
as ourselves, and cannot be discussed without formal apparatus. My treatment
of topology runs briefly through abstract {\em point-set topology}, a fairly
harmless generalisation of the notion of continuity from a first course on
analysis and metric spaces. However, my main interest is in topology as {\em
rubber-sheet geometry},\index{rubber-sheet geometry} dealing with manifestly
geometric ideas such as closed curves, spheres, the torus, the
M\"obius\index{M\"obius strip} strip and the Klein bottle.

\section[Change of coordinates, motions, group theory and the \\ Erlangen
program]{Change of coordinates, motions, group \\ theory and the Erlangen
program}
\markright{\hfill \thesection\ Coordinate changes, motions, groups\quad}
\index{Erlangen program|(}
\index{coordinate!changes}

Descartes' idea to use numbers to describe points in space involves the
choice of a {\em coordinate system\/}\index{frame} or {\em coordinate
frame\/}:\index{coordinate!frame} an origin, together with axes and units of
length along the axes. A recurring theme of all the different geometries in
this book is the question of what a coordinate frame is, and what I can get
out of it. While coordinates provide a convenient framework to discuss
points, lines, and so on, it is a basic requirement that any meaningful
statement in geometry is {\em independent of the choice of coordinates}.
That is, coordinate frames are a humble technical aid in determining the
truth, and are not allowed the dignity of having their own meaning.

Changing from one coordinate frame to another can be viewed as a {\em
transformation\/} or {\em motion\/}\index{motion}: I can use a motion of
space to align the origin and coordinate axes of two coordinate systems. A
statement that remains true under any such motion is independent of the
choice of coordinates. Felix Klein's 1872 {\em Erlangen
program}\index{Erlangen program} formalises this relation between geometric
properties and changes of coordinates by defining geo\-metry to be the study
of properties invariant under allowed coordinate changes, that is, invariant
under a {\em group of transformations}. This approach is closely related to
the point of view of special relativity\index{special!relativity} in
theoretical physics\index{physics} (Einstein, 1905), which insists that the
laws of physics must be invariant under Lorentz transformations.

This course discusses several different geometries: in some case the spaces
themselves are different (for example, the sphere and the plane), but in
others the difference is purely in the conventions I make about coordinate
changes. Metric geometries such as Euclidean and hyper\-bolic non-Euclidean
geometry include the notion of distance between two points and angle between
two lines. The allowed transformations are rigid motions (iso\-metries or
congruences) of Euclidean or hyperbolic space. Affine and projective
geo\-metries consider properties such as collinearity of points, and the
typical group is the general linear group $\GL(n)$, the group of invertible
$n\times n$ matrixes. Projective geo\-metry presents an interesting paradox:
while its mathematical treatment involves what may seem to be quite arcane
calculations, your brain has a sight driver that carries out projective
transformations by the thousand every time you recognise an object in
perspective, and does so unconsciously and practically instantaneously.
\index{Erlangen program|)}

The sets of transformations that appear in topology, for example the set of
all continuous\index{continuous} one-to-one maps of the interval $[0,1]$ to
itself, or the same thing for the circle $S^1$ or the sphere $S^2$, are of
course too big for us to study by analogy with transformation groups such as
$\GL(n)$ or the Euclidean group, whose elements depend on finitely many
parameters. In the spirit of the Erlangen program, properties of spaces that
remain invariant under such a huge set of equivalences must be
correspondingly coarse. I treat a few basic topological
properties\index{topological!property} such as compactness,\index{compact}
connectedness, winding number\index{winding number} and simple connectedness
that appear in many different areas of analysis and geometry. I use these
simple ideas to motivate the central problem of topology: how to distinguish
between topologically different spaces? At a more advanced level, topology
has\index{algebraic topology} developed systematic invariants that apply to
this problem, notably the fundamental group\index{fundamental group} and
homology groups. These are invariants of spaces that are the same for
topologically equivalent spaces. Thus if you can calculate one of these
invariants for two spaces (for example, a disc and a punctured
disc) and prove that the answers are different, then the spaces are
certainly not\index{homeomorphism!problem} topologically equivalent. You
may want to take subsequent courses in topology to become a real expert,
and this course should serve as a useful guide in this.

\section{Geometry in applications}
\markright{\hfill\thesection\ Geometry in applications\quad}

Although this is book is primarily intended for use in a math course, and
the topics are oriented towards the theoretical foundations of geometry, I
must stress that the math ideas discussed here are applicable in different
ways, basic or sophisticated, as stated or with extra development, on their
own or in combination with other disciplines, Euclidean or non-Euclidean,
metric or topological, to a huge variety of scientific and techno\-logical
problems in the modern world. I discuss in Chapter~\ref{ch!lie} the quantum
mechanical description of the electron\index{electron} that illustrates a
fundamental application of the ideas of group theory and topology to the
physics of\index{physics} elementary particles. To move away from basic to
more applied science, let me mention a few examples from technology. The
typesetting and page layout software now used throughout the newspaper and
publishing industry, as well as in the computer rooms of most university
departments, can obviously not exist without a knowledge of basic coordinate
geometry:\index{coordinate!geometry} even a primary instruction such as
`place letter $A$ or box $B$, scaled by such-and-such a factor, slanted at
such-and-such an angle, at such-and-such a point on the page' involves
affine transformations.\index{affine!transformation} Within the same
industry, computer typefaces themselves are designed using Bezier curves.
The geometry used in robotics is more sophisticated. The technological aim
is, say, to get a robot arm holding a spanner into the right position and
orientation, by adjusting some parameters, say, angles at joints or lengths
of rods. This translates in a fairly obvious way into the geometric problem
of parametrising a piece of the Euclidean group;\index{Euclidean!group
$\protect\Eucl(n)$} but the solution or approximate solution of this problem
is hard, involving the topology and analysis of manifolds, algebraic
geometry and singularity theory. The computer processing of camera images,
whose applications include missile guidance systems, depends among other
things on projective transformations (I say this for the benefit of students
looking for a career truly worthy of their talents and education). Although
scarcely having the same nobility of purpose, similar techniques apply in
ultrasonic scan used in antenatal clinics; here the geometric problem is to
map the variations in density in a $3$-dimensional medium onto a
$2$-dimensional computer screen using ultrasonic radar, from which the human
eye can easily make out salient features. By a curious coincidence, 3 hours
before the senior author gave the first lecture of this course in January
1989, I was at the maternity clinic of Walsgrave hospital Coventry looking
at just such an image of a 16-week old foetus, now my third daughter
Murasaki.

\section{About this book}
\markright{\hfill \thesection\ About this book\quad}

\subsection*{Who the book is for}

This book is intended for the early years of study of an undergraduate math
course. For the most part, it is based on a second year module taught at
Warwick over many years, a module that is also taken by first and third year
math students, and by students from the math/physics course. You will find
the book accessible if you are familiar with most of the following, which is
standard material in first and second year math courses.

\paragraph{Coordinate geometry:}\index{coordinate!geometry}
How to express lines and circles in $\R^2$ in terms of coordinates, and
calculate their points of intersection; some idea of how to do the same in
$\R^3$ and maybe $\R^n$ may also be helpful.

\paragraph{Linear algebra:}
Vector spaces and linear maps over $\R$ and $\C$, bases and matrixes, change
of bases, eigenvalues and eigenvectors. This is the only major piece of math
that I take for granted. The examples and exercises make occasional
reference to vector spaces over fields other than $\R$ or $\C$ (such as
finite fields), but you can always omit these bits if they make you
uncomfortable.

\paragraph{Multilinear algebra:}
Bilinear and quadratic forms, and how to express them in matrix terms; also
Hermitian forms. I summarise all the necessary background material in
Appendix~\ref{app!q}.

\paragraph{Metric spaces:}
Some prior familiarity with the first ideas of a metric space course would
not do any harm, but this is elementary material, and
Appen\-dix~\ref{app!metric} contains all that you need to know.

\paragraph{Group theory:}
I have gone to some trouble to develop from first principles all the group
theory that I need, with the intention that my book can serve as a first
introduction to transformation groups and the notions of abstract group
theory if you have never seen these. However, if you already have some idea
of basic things such as composition laws, subgroups, cosets and the
symmetric group, these will come in handy as motivation. If you prefer to
see a conventional introduction to group theory, there are any number of
textbooks, for example Green~\cite{Gr} or Ledermann~\cite{Le}. If you intend
to study group theory beyond the introductory stage, I strongly recommend
Artin~\cite{Ar} or Segal~\cite{Se}. My ideological slant on this issue is
discussed in more detail in \ref{sec!gp_thy}.

\subsection*{How to use the book}

Although the thousands queueing impatiently at super\-markets and airport
bookshops to get their hands on a copy of this book for vacation reading was
strong motivation for me in writing it, experience suggests the harsher view
of reality: at least some of my readers may benefit from coercion in the
form of an organised lecture course.

Experience from teaching at Warwick shows that Chapters~1--\ref{ch!gp} make a
reasonably paced 30 hour second year lecture course. Some more meat could be
added to subjects that the lecturer or students find interesting; reflection
groups following Coxeter~\cite{Cx}, Chapter~4 would be one good candidate.
Topics from Chapters~7--8 or the further topics of Section~\ref{sec!titles}
could then profitably be assigned to students as essay or project material.
An alternative course oriented towards group theory could start with affine
and Euclidean geometry and some elements of topology (maybe as a refresher),
and concentrate on Chapters~3, 6 and~8, possibly concluding with some
material from Segal~\cite{Se}. This would provide motivation and techniques
to study matrix groups from a geometric point of view, one often ignored in
current texts.

\subsection*{The author's identity crisis}

I want the book to be as informal as possible in style. To this end, I
always refer to the student as ``you'', which has the additional advantage
that it is independent of your gender and number. I also refer to myself by
the first person singular, despite the fact that there are two of me. Each
of me has lectured the material many times, and am used to taking personal
responsibility for the truth of my assertions. My model is van der Waerden's
style, who always wrote the crisp ``Ich behaupte \dots'' (often when
describing results he learned from Emmy Noether or Emil Artin's lectures). I
leave you to imagine the speaker as your ideal teacher, be it a bearded
patriarch or a fresh-faced bespectacled young Central European intellectual.

\subsection*{Acknowledgements}

A second year course with the title `Geometry' or `Geometry and topology'
has been given at Warwick since the 1960s. It goes without saying that my
choice of material, and sometimes the material itself, is taken in part from
the experience of colleagues, including John Jones, Colin Rourke, Brian
Sanderson; David Epstein has also provided some valuable material, notably
in the chapter on hyperbolic geometry. I have also copied material
consciously or unconsciously from several of the textbooks recommended for
the course, in particular Coxeter~\cite{Cx}, Rees~\cite{Rees}, Nikulin and
Shafarevich~\cite{NSh} and Feynman~\cite{Fn}. I owe special thanks to Katrin
Wendland, the most recent lecturer of the Warwick course MA243 Geometry, who
has provided a detailed criticism of my text, thereby saving me from a
variety of embarrassments.

\subsection*{Disclaimer}
\begin{quote}
Wen solche Lehren nicht erfreun,\\ Verdienet nicht ein Mensch zu sein. \\[6pt]
\hbox{\qquad} From Sarastro's aria, The Magic Flute, II.3.
\end{quote}
This is an optional course. If you don't like my teaching, please
deregister before the deadline.
