## Final Advice

6dec10

\begin{document} \maketitle \section{Introduction} Typically, the 3-hour final for MA403 has 3 kinds of questions: A \textit{theoretical} question asks for a complete statement of a theorem, and/or a short but complete proof. It is understood that you provide relevant definitions and helpful, properly labeled figures. For example: \begin{itemize} \item Discuss the meaning and uses of conjugation of isometries. State the most relevant definitions and theorems. \end{itemize} A \textit{practical} question involves calculations. For example, the classificaton of a composition of two given isometries as one of the five different kinds of isometries. It is understood here that you justify steps by naming the theorems you are applying. \begin{itemize} \item What isometry is the composition of two glide reflections whose mirrors are perpendicular? Provide an illustration. \end{itemize} An \textit{essay} question is designed to tell me how well you understand an entire subject. While the question "What is Euclidean Geometry" might be appropriate for a comprehensive final, for a midterm a more specific question, like "Compare Euclid's synthetic with Descarte's analytic approach to geometry" is more appropriate because it covers only the material up to date. Since there was no hourly on isometries (Part III), there will be a short, but specific essay question on that subject, as in the midterm. Your response is evaluated on the most pertinant points, not on length or breadth. Keep it short and simple. Above all, avoid wasting time on sentences which convey no information, or which repeat what you have already said. You will be given a list possible topics which you can prepare in your journals. \section{The Essay} Regarding an answer to "What is Euclidean Geometry?", the obvious answer "It is the geometry invented by Euclid" is correct but such an uninformed tautology would earn zero credit. A washlist of topics covered in the course would earn at best a C, because it is merely a memory dump and give no hint of comprehension. Besides, I don't want a transcript of the index in your journal. What would get a B or better is a sensible discussion of the different methods of doing Euclidian geometry, for example: \begin{itemize} \item Why don't we do geometry with postulates anymore, as Euclid did? \item What have numbers got to do with geometry. How do we use vectors? \item Compare projective or affine geometry with Euclidean geometry. Etc. \item Compare synthetic with analytic geometry? \item How to the three parts of the course fit into the the five threads. \end{itemize} \section{Drawing Figures} All figures on the final need to be drawn by hand, using your "gnomon" (transparent, rightangled ruler). You may also use a compass. In particular lines need to look straight and right angles need to look like right angles. But the illustrations are just that, there is no requirement for the accurace possible in KSEG. You should devinitely review how to hand-draw: \begin{itemize} \item Figures used in Ceva, Menelaus and Desargues. \item Euler's line and Feuerbach's (nine-point circle.) \item Cubing a given line segment in 3pt and 2pt perspective. \item Measuring the side ratios of a perspective box. \item Subdividing and multiplying perspective lengths. \item Figures in connection with factoring isometries. \end{itemize} \end{document}