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intro.tex
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\chapter*{Preface}
\addcontentsline{toc}{chapter}{Preface}
This book is meant to be
rigorous,
conservative,
elementary,
and minimalist.
At the same time, it includes about the maximum that students can absorb in one semester.
The present book is based
on courses given by the author
at Pennsylvania State University.
These lectures were oriented toward sophomore and senior university students who are familiar with real numbers and continuity.
It makes it possible to cover the material faster
and in a more rigorous way
than it could be done in high school.
Approximately one-third of the material used to be covered in high school, but not anymore.
\section{Prerequisite}
The students should be familiar
with the following topics:
\begin{itemize}
\item Elementary set theory:
$\in$,
$\cup$,
$\cap$,
$\backslash$,
$\subset$,~$\times$.
\item Real numbers: intervals, inequalities, algebraic identities.
\item Limits, continuous functions, and the intermediate value theorem.
\item Standard functions:
absolute value,
natural logarithm,
exponential function.
Occasionally, trigonometric functions are used.
\item Chapter~\ref{chap:trans} uses basic vector algebra.
\item To read Chapter~\ref{chap:sphere}, it is better to have some previous experience with the \textit{scalar product}, also known as the \textit{dot product}.
\item To read Chapter~\ref{chap:complex}, it is better to have some previous experience with complex numbers.
\end{itemize}
\section{Overview}
We use the so-called \textit{metric approach} introduced by Birkhoff.
It means that we define the Euclidean plane as a \textit{metric space} that satisfies a list of properties (\textit{axioms}).
This way we minimize the tedious parts
which are unavoidable in the more classical Hilbert's approach.
At the same time, the students have a chance to learn the geometry of metric spaces.
Here is a dependency graph of the chapters.
\begin{figure}[!ht]
\centering
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,scale=1.4,
thick,main node/.style={circle,draw,font=\sffamily\bfseries,minimum size=8mm}]
\node[main node] (1) at (-1.5,10/6) {\ref{chap:metr}};
\node[main node] (2) at (-.5,10/6){\ref{chap:axioms}};
\node[main node] (3) at (.5,10/6) {\ref{chap:half-planes}};
\node[main node] (4) at (1.5,10/6) {\ref{chap:cong}};
\node[main node] (5) at (2.5,10/6) {\ref{chap:perp}};
\node[main node] (6) at (3.5,10/6) {\ref{chap:parallel}};
\node[main node] (7) at (4.5,10/6) {\ref{chap:angle-sum}};
\node[main node] (8) at (5.5,10/6) {\ref{chap:triangle}};
\node[main node] (9) at (5,5/6){\ref{chap:inscribed-angle}};
\node[main node] (10) at (4.5,0) {\ref{chap:inversion}};
\node[main node] (11) at (3,5/6) {\ref{chap:non-euclid}};
\node[main node] (12) at (3.6,0){\ref{chap:poincare}};
\node[main node] (13) at (3,-5/6) {\ref{chap:h-plane}};
\node[main node] (14) at (5,-5/6) {\ref{chap:trans}};
\node[main node] (15) at (4.5,-10/6) {\ref{chap:proj}};
\node[main node] (16) at (4,-5/6) {\ref{chap:sphere}};
\node[main node] (17) at (3.5,-10/6) {\ref{chap:klein}};
\node[main node] (18) at (5.5,0) {\ref{chap:complex}};
\node[main node] (19) at (4,15/6) {\ref{chap:car}};
\node[main node] (20) at (5,15/6) {\ref{chap:area}};
\path[every node/.style={font=\sffamily\small}]
(1) edge node{}(2)
(2) edge node{}(3)
(3) edge node{}(4)
(4) edge node{}(5)
(5) edge node{}(6)
(6) edge node{}(7)
(7) edge node{}(9)
(7) edge node{}(8)
(7) edge node{}(20)
(10) edge node{}(14)
(14) edge node{}(15)
(7) edge node{}(19)
(9) edge node{}(10)
(10) edge node{}(12)
(10) edge node{}(16)
(10) edge node{}(18)
(5) edge node{}(11)
(11) edge node{}(12)
(12) edge node{}(13)
(15) edge node{}(17)
(16) edge[dashed] node{}(17)
(13) edge node{}(17);
\end{tikzpicture}
\end{figure}
In (\ref{chap:metr}) we introduce the concepts necessary to formulate the axioms;
it includes metric spaces,
lines,
angle measure,
continuous maps,
and congruent triangles.
Next, we dive into Euclidean geometry:
(\ref{chap:axioms}) Axioms and immediate corollaries;
(\ref{chap:half-planes}) Half-planes and continuity;
(\ref{chap:cong}) Congruent triangles;
(\ref{chap:perp}) Circles, motions, and perpendicular lines;
(\ref{chap:parallel}) Similar triangles and (\ref{chap:angle-sum}) Parallel lines
--- these are the first two chapters where we use Axiom~\ref{def:birkhoff-axioms:4}, an equivalent of Euclid's parallel postulate.
In (\ref{chap:triangle}) we give the most classical theorems of triangle geometry;
this chapter is included mainly as an illustration.
In the following two chapters, we discuss the geometry of circles on the Euclidean plane:
(\ref{chap:inscribed-angle}) Inscribed angles; (\ref{chap:inversion}) Inversion.
It will be used to construct the model of the hyperbolic plane.
Next, we discuss non-Euclidean geometry:
(\ref{chap:non-euclid})
Neutral geometry --- geometry without the parallel postulate;
(\ref{chap:poincare})
Conformal disc model ---
this is a construction of the hyperbolic plane,
an example of a neutral plane that is not Euclidean.
In (\ref{chap:h-plane}) we discuss geometry of the constructed hyperbolic plane --- this is the highest point in the book.
In the remaining chapters, we discuss additional topics:
(\ref{chap:trans}) Affine geometry;
(\ref{chap:proj}) Projective geometry;
(\ref{chap:sphere}) Spherical geometry;
(\ref{chap:klein}) Projective model of the hyperbolic plane;
(\ref{chap:complex}) Complex coordinates;
(\ref{chap:car}) Geometric constructions;
(\ref{chap:area}) Area.
The proofs in these chapters are not completely rigorous.
We encourage the use of visual assignments on the author's website.
\section{Disclaimer}
Most of the proofs in the book already appeared in Euclid's Elements.
I did not try to find original references; it is an impossible task.
\section{Recommended resources}
\begin{itemize}
\item {}\emph{Byrne's Euclid} \cite{byrne} --- a colored version of the first six books of Euclid's Elements edited by Oliver Byrne.
\item {}\emph{Kiselyov's geometry} \cite{kiselev} ---
a classical textbook for school students written by Andrey Kiselyov; it should help if you have trouble following this book.
\item {}\emph{Lessons in Geometry} by Jacques Hadamard \cite{hadamard} --- an encyclopedia of elementary geometry originally written for school teachers.
%\item Moise's book, \cite{moise} --- should be good for further study.
%\item Greenberg's book \cite{greenberg} --- a historical tour in the axiomatic systems of various geometries.
\item {}\emph{Problems in geometry} by Victor Prasolov\cite{prasolov} is perfect for mastering your problem-solving skills.
\item {}\emph{Geometry in figures} by Arseniy Akopyan \cite{akopyan} --- an encyclopedia of Euclidean geometry with barely any words.
\item {}\emph{Euclidea} \cite{euclidea} --- a fun and challenging way to learn Euclidean constructions.
\item {}\emph{Geometry} by Igor Sharygin \cite{sharygin} --- the greatest textbook in geometry for school students, I recommend it to anyone who can read Russian.
\item {}\emph{problems.ru} \cite{zadachi} --- an on-line collection of problems in Russian.
\end{itemize}
\section{Acknowledgments}
{\sloppy
Let me thank
Stephanie Alexander,
Thomas Barthelme,
Berk Ceylan,
Matthew Chao,
Quinn Culver,
Svetlana Katok,
Nina Lebedeva,
Alexander Lytchak,
Alexei Novikov,
and Lukeria Petrunina
for useful suggestions and for correcting misprints.
This work was partially supported by
NSF grants
DMS-0103957,
DMS-0406482,
DMS-0905138,
DMS-1309340,
DMS-2005279,
and Simons Foundation grants
245094, 584781.
}