Last revised: December 9, 2004

Instructor: Dr. Paul Bamberg, Senior Lecturer on Mathematics (to be addressed as "Paul," please)
Email: bamberg@tiac.net
Best phone number (Monday noon - Thursday noon) 617-493-3100 (in Quincy House 102)
Other phone number (Thursday PM - Monday AM)  508-460-6569
Science Center office: Science Center 423.  Office hours there: Thursday 10:30-11:30 AM
Office Hours in Quincy 102: almost any time between 3PM Monday and 10AM Thursday, excluding Tu 10-1 and TuW 7-10PM.  The best way to arrange a meeting is by email.

Web site:
The offical site is www.courses.fas.harvard.edu/~math138

For the benefit of "shoppers" I have left last year's notes and assignments on the site.  These will be replaced by revised versions.

Most of the handouts from class are created in Mathcad so that I can use SmartSketch to produce diagrams.  I save these in .rtf  format and put them in the NoteRTF folder. If you lose the copy handed out at lecture, you can read and print these from Microsoft Word or from Wordpad.  If you are a computer purist who refuses to touch anything connected with Microsoft, you may be out of luck.
Beware- some of these files are enormous because of the graphics that they contain.

Homework problems are in the HWRTF folder. Beware: some problems may require diagrams that can only be done by hand, so you will probably need hard copies of problem sets, which will be handed out in class.

Windows executable software is in the WindowsExe folder, frequently zipped up with any necessary data files. You will see this demonstrated in class.

Things like instructions for the term project, review guides for exams, and this syllabus can be found in the documents folder.

For interactive Web pages go to the Interactive folder, where you will find a link to my own server.  These pages were tested with Netscape version 7, but they should display correctly in any standard browser.

Prerequisites:
A good knowledge of linear algebra before the end of the term, for example, Math 21b or E-21b (even this term) or Math 23a.
Calculus is not needed, although it's hard to take a linear algebra course without learning calculus first.
An elementary understanding of Newtonian mechanics and the special theory of relativity would make things easier but will not be assumed.
Group theory will be presented "from scratch."
Freshmen are encouraged to take the course -- those who took it in the past two years did extremely well!

Some Extension school students (especially high school seniors) have done extremely well in this course in the past three years, but others have found it a bit too challenging. Last year all Extension students did OK on the online placement test and also did well in the course.  Extension students who are interested in the material but who do not have plenty of time to spend on it should seriously consider enrolling for noncredit.  This lets you do everything but take the final exam. Undergraduates in a similar situation could register pass-fail (and will then take the final exam).

Placement tests:
Extension school students should take the online placement test. Here is a link. http://www.extension.harvard.edu/2004-05/courses/math.jsp#oltest
Notice that you can take the test twice.  If you are weak in one of the areas covered by the test (algebra and trig, geometry, vectors and matrices), brush up on the subject and try again.

Textbooks:
Ryan, Euclidean and Non-Euclidean Geometry, an Analytic Approach (in the Coop)
This book, although short, has a number of nice features:
1. It emphasizes the isometries of various plane geometries and their representation by 2x2 and 3x3 matrices.
2. Group theory is introduced in an appendix and used where appropriate.
3. Hyperbolic (Bolyai-Lobachevsky) geometry is presented using the terminology of the physics of space-time.
Like most books on "non-Euclidean geometry" it considers only two of the eight alternatives to Euclidean plane geometry.  It also touches only lightly on the axiomatic approach and on the rich history of geometry.

Yaglom, A Simple Non-Euclidean Geometry and its Physical Basis (as a Course Pack from www.hpps.harvard.edu/coursepackets)
This book, translated from the Russian, is a flawed masterpiece.  It presents in detail two alternatives to Euclidean geometry in which the "points" and "lines" of the geometry correspond to concepts of physics, and it then explains how these examples fit into the set of nine Cayley-Klein geometries, to which Ryan's Euclidean, elliptic, and hyperbolic geometries also belong.
The flaw is that much of the text is devoted to Yaglom's "Galilean" geometry, while the big picture is relegated to "supplements" at the end of the book.

Other useful books:
M.K. Bennett, Affine and projective geometry
This is a great reference on finite geometries, a topic that will appear mainly at the start of the course.  Those with a taste for the discrete may find it a good source of ideas for term projects. A copy is on reserve in Cabot Library, and there is also one in the math library on the third floor of the Science Center.

H. Meschkowski, Noneuclidean Geometry
This short book has a nice discussion of Euclid and Hilbert's axiomatic approach to geometry.  It is out of print.  I asked the publisher for permission to make copies, but the royalty they wanted was twice the original cost of the book!

David Hilbert, Foundations of Geometry(10th edition, Open Court Press, a bargain at $13.95 from amazon.com or elsewhere)
This is the book that straightened out Euclid's approach after 2000 years.  It looks elementary but is very subtle.  Read it only after you get into the subject.  Good inspiration for projects.

Euclid, Elements, online at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html
The copyright has expired on this classic textbook, but it is still useful.  From time to time there will be references to Euclid in the problem sets, and this is where you can look them up by book and proposition number.  This version has nice Java applets and good navigational features.  When you ask yourself, "How could Euclid ever have proved that?" this is the place to look. 
Lawyers are still trying to decide whether the rules that forbid you to download copyrighted music and movies also forbid you to download copyrighted translations of theorems of Euclid.

Software:
You should have access to a computer that will run Windows applications.  Many features of the geometries that are presented will by illustrated by an elaborate application program named CK, whose development was generously funded by the Dean for Undergraduate Education and the Extension School, that makes it easy to work interactively with five different plane geometries.  Students who enjoy programming will be able to "get under the hood" of this application and see how everything is implemented with 3x3 matrices. 
Thanks to the efforts of Luke Gustafson, who took the course two years ago, there is a PHP program on the Web site that does finite affine and projective geometry interactively.  You can use this to do calculations for your homework.  There is also an equivalent Windows application that does finite affine geometry.  Download affine.zip from the Web site to get the application and the three data files.  Creating more data files for this application makes a nice project.
For computational term projects the development environments of choice are Visual C++ (version 6 if you can find it), KDevelop under Linux,  and Mathematica (available for free download by Harvard undergraduates and not too expensive for Extension students).  I will be happy to provide expert assistance to anyone using one of these.  Use Java or old-fashioned C and you're on your own!
Making a thorough study of one of the four Cayley-Klein plane geometries not covered in this course and enhancing the existing software to support it would make a nice ALM project for an Extension student, one that I will be happy to supervise.

A couple of computationally messy proofs will be done in Mathematica. Harvard undergraduates can get their own copy from http://www.fas.harvard.edu/computing/download. Allow a couple of days to get a password. Extension students who are degee or certificate candidates may be eligible to get Mathematica, but it is not worth fussing about.

Tools of the trade:
You should own a nice long straightedge, a triangle, a protractor, and a good compass.  Staples on JFK St. is a good place to shop for these.  Don't put the compass in your carry-on luggage when you travel by air!!

Homework and sections:

Weekly problem sets will be due at class on Wednesdays. Optional sections will be held by the course assistants on Tuesday evenings.

Course requirements:
Weekly problem sets
A 1 1/2 hour midterm
A final exam (Extension students will have to be available 2:15-5:15 PM to take this)
A course project, which may be purely mathematical or computational or historical.  Students are strongly encouraged to work in pairs. Some or all of these projects will be presented during Reading Period. Extension students enrolled for graduate credit should do a project with a computational component in order for the course to count toward the CAS or the ALM in IT.

Grading policy:
Midterm exam: 60 points
Final exam: 100 points
Problems sets: 60 points
Term Project: 30 points (15 for written version, 15 for presentation) Up to 3 points extra credit for an early presentation
Point totals (out of 250) will be converted to letter grades according to the following scale:
Points                Grade
235                        A
225                        A-
210                        B+
190                        B
175                        B-
160                       C+
145                        C
135                        C-
130                        D+
125                        D
120                        D- = pass

Pass-fail status:
This should be a good fifth (or even sixth) course for concentrators in physics, appled mathematics, or computer science.  Such students are encouraged to register pass-fail and concentrate on the aspects of the course that especially appeal to them.  Any pass-fail student who attends class regularly, does a reasonable job on the homework, and shows up for both exams will pass the course. Pass-fail students who miss classes and/or neglect homework will be held to a higher standard on the exams.
The course project is not required for pass-fail students, though it could be an alternative to some of the homework.  If you register pass-fail and then do A work, I will be happy to write a letter to that effect for your file.
Mathematics concentrators should enroll for a letter grade.
Pass-fail is not available in the Extension School, so I have added a non-credit option for Extension students.  Non-credit students will be treated just like credit students, except that they will not receive course credit or an official grade.

Topics that will not be covered in any detail in lecture but that might be good for course projects:
Three-dimensional geometries, except to the extent that they are used to create models of plane geometry
Finite subgroups of the rotation group, regular polygons, and Platonic solids (Ryan, Chapter 3)
Co-Euclidean and co-Minkowski geometries (Yaglom, Supplements)
Differential geometry and Riemannian geometry
Hilbert's axiomatization of Euclidean geometry (Meschkowski, Chapter 2)
Analytic models of plane geometries based on complex numbers and their generalizations (Yaglom, Supplement C)
Lie groups in general, beyond the groups of 3x3 matrices that arise naturally as isometries of the geometries that we study.

Tentative schedule of lectures:

Wednesday, February 2       2500 years of  history of plane geometry: axioms for affine geometry (Ryan and Yaglom introductions)
Monday, February 7           Addition and multiplication in a finite affine plane (excerpts from Bennett)
Wednesday, February 9       Introducing coordinates into a finite affine plane (excerpts from Bennett)
Wednesday, February 14          Overview of models for the nine Cayley-Klein plane geometries (Yaglom, Supplement A)
Wednesday, February 16     Elliptic, parabolic, and hyperbolic measure of angle and length (Yaglom, Supplement A)
Monday, February 21          Holiday
Wednesday, February 23     Representing Euclidean reflections and rotations by matrices (Ryan, Chapter 2)
Monday, February 28         Building Euclidean isometries as products of reflections (Ryan, Chapter 2)
Wednesday, March 2     The connection between affine geometry and linear algebra (Ryan, Chapter 2 and Appendix E)
Monday, March 7                Galilean geometry and its isometry group. (Yaglom, Chapter I)
Wednesday, March 9           Duality in Galilean geometry (Yaglom, Chapter I)
Monday, March 14              Doublets and self-dual axioms for Galilean geometry Yaglom, Chapter I and supplement B)
Wednesday, March 16        Cycles and cyclic rotations in Galilean geometry (Yaglom, Chapter II)
Monday, March 21              Special relativity; Lorentz transformations as the isometry group of Minkowski geometry (Yaglom, Conclusion)
Wednesday, March 23        Midterm exam on material through March 12
Monday, April 4             Spherical geometry and trigonometry (Ryan, Chapter 4)
Wednesday, April 6            Matrix representation of the isometries of spherical geometry (Ryan, Chapter 4)
Monday, April 11                 Area and congruence in spherical geometry (Ryan, Chapter 4)
Wednesday, April 13            Finite projective geometry (excerpts from Bennett)
Monday, April 18               Transformations of a finite projective plane
Wednesday, April 20          Using homogeneous coordinates to prove theorems of real geometry (Ryan, Chapter 4)
Monday, April 25              Polarities and hyperbolic geometry (Ryan, Chapter 7)
Wednesday, April 27          Distance and angle in hyperbolic geometry (Ryan, Chapter 7)
Monday, May 2             Isometries of the hyperbolic plane and their representation by matrices (Ryan, Chapter 7)
Wednesday, May 4          Area in hyperbolic geometry; hyperbolic trigonometry; the parallel postulate revisited
Monday, May 9                  Doubly hyperbolic geometry
Wednesday, May 11             Student project presentations  (3 point bonus)
Monday, May 16                Student project presentations  (2 point bonus)
Wednesday, May 18           Student project presentations  (no bonus)