Overview of Lessons on Models SP11

\textit{ $\C$ 2010, Prof. George K. Francis, Mathematics Department, University of Illinois} \begin{document} \maketitle \section{How to study this section of the course} \subsection{Lesson of Models} The Lesson on Models is an introduction to the two most important models of non-Euclidean geometry, the \textit{ Beltrami-Klein Model}, and the \textit{ Poincare-disk Model}. The former is generally just called the "Klein model", and the latter is also on of the two \textit{ conformal } models associated with Poincar\'e's name. By "conformal" we mean "angle preserving". Both are also referred to as "hyperbolic models" because \textit{ Hyperbolic geometry} is a synonym for \textit{ Non-Euclidean geometry}. There are better reasons for all this redundancy of nomenclature, but we won't give them here. Google them! The thing to memorize is that both models are \textbf{ inside} the unit disk. The Cartesian points on the boundary and outside of the disk are still there, the are just not part of the model; they a points, but not Points. In the Klein model, straight Lines are (segments) of straight lines (which is useful). In the Poincare model Angles have the same measure as Euclidean angles (which is useful for other purposes.) On the other hand, in the former it is hard to construct and impossible to recognize Right Angles. In the latter is hard to connect two Points by a Line, unless the Line is a diameter of the disk. So, even in the Poincare model some Lines are lines, but only if they're diameters. \subsection{Capitalization} At this point we abandon a strict adherence to the convention of capitalizing unless it is useful to make the distinctions. So the context determines in which geometry we're talking. \subsection{Circle Inversion} The first lesson introduces, and Bakshy's notes develop, the concept of \textit{ circle inversion }. This generalizes the concept of \textit{reflections} in a straight line you studied in analytic geometry. It is possible to study non-Euclidean geometry strictly axiomatically (which we do not do). For this you don't need models at all. The published (independent) discoverers of non-Euclidean geometry, Gauss, Bolyai and Lobachevsky did entirely without models. It was Beltrami who developed the first models, and Klein who knew how to adapt them to school geometry. It is also possible to treat the models srictly in terms of analytic geometry, for which you need the concept of circle inversion defined analytically. Hvidsten's book permits such an approach, but we do not take it. You should study Bakshy's condensation for your own edification. We will use a few excerpts from it. \section{The Complex Plane} Hvidsten introduces the essentials fo the complex plane at the end of his chapter on Cartesian geometry. They are developed and applied later in Chapter 8. This chapter was written at an almost graduate-level and needs to be explained. That is the approach we take here. But we do not complete Chapter 8 in this course. My intention is to get you to a plateau of comprehension so that you can profitably complete the chapter on your own at some later time when you get bored with the vapid math all around you. \subsection{Moebius Transformations} Felix Klein originated the modern concept that all geometry is based, ultimately, on groups of transformations. This is the properly developed 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. Indeed, he \textit{ used} this artefact as an example. It was known to the Greeks and used by medieval farmers, and engineers of the industrial age to make drive belts wear evenly on "both" sides (since they had only one side! to wear). For the record, and this belongs into an elementary topology course, M\"obius (as you would write it 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 another subject introduced in MA403. The last undergraduate course in Projective Geometry at the University of Illinois died in the '70s. [This introductory/guide will be continued.] \section{Unfinished Lessons} The 5 lessons in .pdf form are transcripts of interacive session in a class using a tablet computer. I expect we will replace these with Elluminate sessions. But they could still stand to be written up by somebody.