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\title{Math 4520 Fall 2017}
\begin{document}
\title{Ancient Art and Geometry\\
\normalsize Math 4520
Fall 2017}
\section{Course Outline }
The following chapters include most of what I would like to cover:
\begin{itemize}
\item [Chapter 1:] A very brief discussion of Euclidean geometry and Euclid's Elements
\item [Chapter 2:] The Extended Euclidean plane and Projective Geometry.
\item [Chapter 3:] Desargues' Theorem. This is a model case where three-dimensional geometry can be applied to prove a non-trivial result in the plane.
\item [Chapter 4:] Three dimensional projective geometry, one, two, and three-point perspective.
\item [Chapter 5:] Drawing three-dimensional objects.
\item [Chapter 6:] Conic sections in projective geometry.
\item [Chapter 7:] Fields as they are used to define a projective geometry.
\item [Chapter 8:] Coordinates for a projective geometry.
\item [Chapter 9:] Finite projective planes and combinatorics
\item [Chapter 10:] Projections and collineations in projective geometry.
\item [Chapter 11:] Duality and Polarity, a common basic concept in mathematics.
\item [Chapter 12:] The cross ratio.
\item [Chapter 13:] Circle geometry.
\item [Chapter 14:] Hyperbolic geometry.
\item [Chapter 15:] Relativity Theory and causality in Physics.
\end{itemize}
Classical geometry usually includes Euclidean, Spherical and Hyperbolic geometry, but I feel that the principles and point of view of Projective geometry to be the skeleton on which the other geometries are attached. Accordingly we will discuss projective geometry both as a model geometry with a minimum of encumbering axioms as well as a concrete set of properties that we can use every day with what we see. It is both an abstract formalism and a practical insight to the geometry of everyday objects.
\section{Introduction}
In Plato's view, only that which turned the soul's eye from the
material world to objects of pure thought was worthy of a
philosopher's study. Music, astronomy, arithmetic, and especially
geometry were particularly recognized. The idea was to start with
certain unquestioned assumptions and precise definitions and proceed
logically and rigorously from there.
Euclid of Alexandria, a student of the Platonic school, lived from 323
BC to 283 BC. Around 300 BC he wrote the Elements, a collection of
thirteen books in which he compiled the geometry and number theory
developed in previous centuries and presented it in an axiomatic
way. His work has had a profound and lasting effect on almost all of
western mathematics. Included in this handout is a copy of the
beginning of Book I of the Elements, as well as a later ``revised"
edition by Playfair. It is ironic that Euclid's first Theorem, about
constructing an equilateral triangle, has a hidden assumption which is
just what he was apparently trying to avoid.
Mathematicians eventually got around to looking critically at Euclid's
work. One of the most detailed and thoughtful of these people was
David Hilbert, who is considered among the strongest and surely one of
the most influential mathematicians in the nineteenth and twentieth
centuries. Included is a copy of the beginning of Hilbert's
Foundations of Geometry, which is his attempt to achieve Euclid's goal
of presenting geometry as following from a system of coherent axioms,
with present day rigor. Hilbert has conveniently separated the axioms
into classes of incidence, order, congruence, parallels, and
continuity. But despite this heroic attempt at a natural and
reasonable presentation, we see that the system is quite complex and
involves some quite profound ideas.
Meanwhile, from a different perspective, we have a few pictures of
Egyptian Art and Asian Art. Plato referred to Art as the ``worthless
mistress of a worthless friend, and the parent of a worthless
progeny." [Perhaps some architecture students today feel the same
about Mathematics.] Looking at these few
examples of painting, possibly painting was at a
primitive level, inviting such attacks as Plato's. On the other hand,
sculpture was very ``realistic" and seems to have been much more
advanced. Some of the best ``drawings" seem to be obtained by
sticking a sculpture on a slab, blurring the distinction between
Drawing and Sculpture. Apparently the Greeks, both Mathematicians and
Artists, did not have a good grasp of what we today call
"perspective", although some people believe that there was some understanding, at least in some places.
It is my assertion that an understanding of perspective in one
area, Mathematics or Art, could have influenced the other; but this
did not happen until well over a thousand years after the prime of
Greek Mathematics, during the Renaissance. After the time of Euclid
and Archimedes, with a few individual twitches only, Greek Mathematics
died. It was preserved and copied with great reverence, but it did
not grow or develop until after the great Dark Ages. It took the
simple, straightforward curiosity of people such as Flippo
Brunellischi, the builder of the large dome in Florence, Leonardo da
Vinci, famous for the Mona Lisa, Alberti, who wrote a very influential
manual on the principles of perspective drawing, and D\"urer and others to see
beyond the strict rules laid down by Euclid and to do what was
necessary to draw a picture.
As Euclid's Elements was copied, it was studied intensely. Not only
were ``mistakes" found, but the question of the necessity of the fifth
Postulate arose, and this eventually became a major unsolved problem.
Without assuming something in its place, could Euclid's fifth
Postulate be proved from the previous four? Many people tried in
vain. In the nineteenth century, Gauss showed that the fifth
Postulate could be proved if one assumed that there was a triangle of
arbitrarily large area. Similarly, Legendre proved the fifth
Postulate, but he assumed that there was a triangle whose angles added
up to 180 degrees. There were many more attempts. It was not until
the nineteenth century that people realized that it may not be
possible to prove the fifth Postulate from the others at all. Indeed,
the belief in the ``absolute truth" of the axioms were gradually
replaced with the more formalist idea that meanings of the axioms
could be put in different contexts. Eventually it was shown that if
the system with the fifth Postulate is consistent, then it is
impossible to prove the fifth Postulate from the others. In other
words, if one has a consistent system with Euclid's fifth Postulate,
then one can prove the existence of a consistent system with an
alternative axiom that directly contradicts Euclid's.
The proof (of this impossibility of a proof) is very simple.
Playfair's version of the fifth Postulate says that ``Given any line
and a point not on the line, there is one and only one line through
the point parallel to the first line." In the Euclidean plane one
constructs a ``model" which is a system of points and lines (and
whatever else is needed) that satisfies all of Euclid's axioms, except
that the fifth is replaced by a property which states that ``Given any
line and a point not on the line, there is more than one line through
the point parallel to the first line." See the Figure 1.1. We will
see this in more detail later; this model is usually called the
``hyperbolic plane."
\begin{figure}
\begin{center}
\includegraphics{fig1}
\end{center}
\caption{}
\label{fig:parallel-lines}
\end{figure}
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%\midspace{.1in} \caption{Figure 1.1}
Closely related to the hyperbolic plane is what is now called
the ``projective plane" or the ``elliptic plane." It too can be
regarded as a model to show that Euclid's fifth Postulate does not
follow the others, if one has a very liberal interpretation of the
axioms (which is almost never done). Here the fifth Postulate is
replaced by the statement ``Any two distinct lines intersect in a
unique point." The model of a ``line" is taken to mean a great circle
(the intersection with the sphere of a plane through its center ) on a
sphere, and a ``point" is taken to mean a pair of opposite points on
the sphere. The notions of ``projection" and a convenient treatment
of the ideas developed by the Renaissance artists are most naturally
at home in the projective plane.
\vskip20pt
\section{Exercises:}
\begin{enumerate}
\item Find your own examples of perspective drawing of that does not follow the ``rules" of perspective. Bring them in to show to the class.
\item Explain what is wrong with the projective plane as a model
satisfying the first four Euclidean Postulates.
\item How would you prove Euclid's first Theorem, if you were
Euclid and had a second chance.
\item Discuss the apparent distortions in the photocopies of the
drawings shown. Is there a simple way to correct
the ``mistakes"?
\item Draw a ``square" in a horizontal plane as it is seen in a
vertical plane.
\item Draw a ``cube" as accurately as you can.
\item Draw a truncated pyramid with a triangular base.
\end{enumerate}
\section{Pictures}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.9\textwidth]{Hogarth}
\end{center}
\caption{Hogarth-satire-on-false-pespective-1753 \newline
\protect\url{ http://commons.wikimedia.org/wiki/File:Hogarth-satire-on-false-pespective-1753.jpg}
}
\label{fig:building}
\end{figure}
See also \protect\url{ http://commons.wikimedia.org/wiki/File:Hogarth-satire-on-false-pespective-1753.jpg} for a partial list of the ``mistakes" in the last picture.
\end{document}