640:435:02 Geometry

Class meets: TF1(8:40am -- 10:00am), ARC-207.
Instructor: Dr. Zheng-Chao Han
Office Hour: Wednesday 9:30am--11:30am in Hill 230.
Email: zchan at math dot rutgers dot edu (I prefer to answer math questions in person during office hours, not through emails; I try to process my emails once per day ).
Text: The following is the required text for this course: Additional supplementary material will be provided/recommended as the course progresses. Later in the semester, we will do some reading and discussion of Books I--III of the classic The Thirteen Books Of The Elements, by Euclid (translated with introduction and commentary by Sir Thomas L. Heath), which contains rich information about our subject from its birth more than two thousand years ago until the nineteenth century. Dover publishes an economical edition: The Thirteen Books Of The Elements, Vol. 1 (Books I and II) (second edition, ISBN 0-486-60088-2); 1956. There is also an online version of Euclid's Elements .

Another useful source is Geometry by Its History by Alexander Ostermann and Gerhard Wanner. A free electronic version can be obtained for free via the Rutgers Springer Mathematics E-books package---Rutgers affiliated users can also order each print copy for $24.95 (Shipping and handling are included). The specific Springer link for the book is here. This book will not be used systematically in this course, but it provides a lot of interesting geometric arguments which complement the more algebraic approach of the David A. Brannan, Matthew F. Esplen & Jeremy J. Gray text, and one can browse through and work on specific topics without too much difficulty.

Sakai The course will use Sakai for all material during the semester. All enrolled students should have automatic access to the site after logging in to Sakai. Current information about syllabus and homework will be found there.

Note: Do not forget to "reload" the assignments pages - if you visited them before, your browser may be showing you only the old cached page.

General Comments on the Course

This course uses the classical Euclidean plane geometry as an anchor point to study some of its natural outgrowth: affine, projective, spherical, and non-Euclidean geometries. The study of these geometries will, in turn, deepen our appreciation for the classical Euclidean geometry. One unifying theme in this course's approach to these geometries is Klein's transformation groups. Technically we will use a lot of analytic methods (setting up and analyzing equations in appropriate coordinates, matrix manipulations), so it is essential that students review the material in Math 250 (Linear Algebra)--- most of the material will be needed starting in the second week! You can find a list of review problems from our sakai course site. On the other hand, our approach will not be purely computational. We will emphasize the geometric flavor of the subject, and whenever possible and beneficial, will provide direct geometric argument. In particular we will blend in fair amount of deductive proofs (also called axiomatic or synthetic proofs).

Out hope is that, after the course, you will have an appreciation for the liveliness, diversity and connectedness of mathematics, and the excitement and pleasure of discovering mathematics, and that you would be comfortable to attack geometric problems using a combination of methods learned in this course.

Emphasis will be placed on geometric understanding and logical reasoning. As such, mere memorization of facts would be of little help. Nor can most regular assignments be completed by simply looking up a magic formula on a page from the texts. Instead, you should be prepared to fully participate in the discussions(in-class and out-class), do extra readings and research, develop and communicate your ideas. You are also encouraged to try to use a combination of geometric exploration, model making, and thought experiments to help you in the learning process. Group discussions and brainstorms will be strongly encouraged. An important aspect of the course is to help you sort out your ideas and present them in a logical way. So it is expected that you present your work in a coherent way, using complete English sentences. More guidelines are given below.

Course Material

This course uses a fair amount of linear algebra, especially in chapters 1, 2, 3 and 7. The attached file here contains a list of review problems for the relevant material. Problems 1, 2, 3, 6, and 7 will be needed starting in the first two weeks. Please review these topics on your own in the first week. You can focus your review to 2x2 or 3x3 matrices.

Structure of Assignments

Homework and Quizzes: You will have weekly regular assignments. The regular assignments are to help you work through the ideas discussed in class and gain a fuller understanding of the technical aspect of the ideas. Discussion and cooperation with each other is strongly encouraged at every stage of the course work, except at the writing-up. In your submitted work, ideas that come from other people should be given proper attribution. If your work has emerged from work with other people, write down whom you have worked with. If you have referred to some sources, cite them. Short quizzes may occasionally be given to test basic understanding on concepts.

Here is a link to the homework assignment site.

Assignment Grading Guidelines

The grading of the regular assignments will be based on correctness of concepts and content, soundness of logical structure in your arguments, and exposition. You should present your work just as you would do for a writing assignment in any other subject, giving the necessary background information, definition of terms you are about to explain, logical arguments, and your conclusions. More specifically, if you are presenting the solution to a problem, explain first what the problem is; if you are going to use some terms or concepts, try to give as clear a definition as possible(because mathematical concepts in a typical student's mind are often vague and may change from context to context, but in scientific discussions precision of concepts is needed); if you are giving an argument, try to explain the main points before you launch into the details. This may seem hard in the beginning. But you can improve quickly if you keep a journal to record what you have been thinking and doing in your work, and then try to organize the ideas in a coherent way as if you wanted to explain your ideas to a friend or to convince him/her of your arguments. It is important that you learn to explore geometry on your own, instead of limiting yourself to answering the questions raised by the instructor. It is good practise to raise further questions of your own at the end of each assignment (as simple as questions like 'what if we are in a different situation like...').

Attendance and Make-up Policy: Class attendance is expected. Poor attendance will be used to decide borderline grade situations. Any changes to the syllabus, homework assignment and any announcement for the midterms and final exam will be made in the lectures. No late work will be accepted. There will be no make-ups for quizzes. A make-up midterm will be given only if you have a valid reason such as serious illness (not a slight cold) or a family emergency, and provide an acceptable, written excuse (not an email message), or you will receive a grade of zero. If possible (particularly if you want to be sure that your excuse is an acceptable one), contact me before missing an exam.

Course Grading Policy

Your course grade will be determined on the following basis:

For comments regarding this page, please send email to zchan at math dot rutgers dot edu.