# 640:435:01 Geometry

Class meets: MTh2, RAB-204.
Instructor: Dr. Zheng-Chao Han
Office Hour: M11:15-12:15, Chem. Bldg-102; T3:00-4:00pm, Hill-230.
Email: zchan@math.rutgers.edu (generally not for answering math questions. I try to process my emails once per day ).
Text: The following is the required text for this course:
• David A. Brannan, Matthew F. Esplen & Jeremy J. Grayd; Geometry (first edition); Cambridge University Press, 1999 (510 pp.); (ISBN 0-521-59787-0)
Additional supplementary material will be provided/recommended as the course progresses. The classic The Thirteen Books Of The Elements, by Euclid (translated with introduction and commentary by Sir Thomas L. Heath) 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 .

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. The unifying theme in approaches to these geometries is Klein's transformation groups. Technically we will use a lot of analytic methods (setting up and analyzing equations in approriate coordinates, matrix manipulations), so it would be beneficial for the students to review the material in Math 250. However, 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).

The 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 you complete most regular assignments 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 may also 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 compelte English sentences. More guidelines are given below.

## Course Material

You may find a copy of our section's syllabus and homework assignment posted on line. Both are subject to adjustment. Any updated information should be posted on this web page. However, the most accurate information will be from the lectures.

Here are

• Solutions to Homework 1.
• Solutions to Homework 2.
• Notes on Key Concepts and Properties of Projective Geometry.
• Solutions to Homework 3.
• Here are some practice problems for our midterm, to be given in class on Oct. 28.
• Here are solutions to our midterm.
• The writing assignment--- First draft due Dec. 2.
• here is the revised writing assignment.
• here are some notes on non-Euclidean geometry, which may be helpful to your writing projects. You won't need all the technical discussions in the notes. For those interested in the subject, the notes can serve as an entry into the subject. Due to the time contraint, I was not able to supply any diagrams to help your reading. You should supply them when reading the technical discussions. We hope to cover some of these by the end of the semester.
• Here are solutions to the Homework of Chapter 7.
• Here are solutions to the first homework on Euclidean proof.
• Here are solutions to the second homework on Euclidean proof.
• Here are some notes on orthogonally intersecting circles and the interpretation of hyperbolic geometry.
• Here are review guides for the final exam ( and solutions to select problems), which is to be held on Dec. 23, from 8-11am, in RAB-204.

## Structure of Assignments

Homework and Quizzes: You will have weekly regular assignments(due each Thursday), and one or two writing assignments, of term paper