Mathematics 138
Guidelines for term projects - 2004
	Projects will be presented in class during Reading Period, on May 12, 17, and 19.  The final written version of the project is due at class on May 19, independent of when it is presented.
	You should team up with one or two classmates to do your project, but even after people organize themselves into groups, there will be about 12 projects to present in three 80-minute sessions. This leaves about 20 minutes for each project.  This means you must have a very tight presentation!  The ability to present the essence of something in less than 20 minutes is an extremely useful skill in the "real world" where lots of decision makers have short attention spans, and one of the goals of the project is to give you some practice with this.  
	You may hand out no more than eight pages about your project to everyone in the class.  Take it to the copy center at 77 Dunster St. I have made arrangements for this with Russell Coleman; ask to speak to him if there is any problem. In the order form, put Math 138 as the course, "Paul Bamberg" as "ordered by" with phone 3-3100.  The billing address is Science Center 325.  It saves money if you get it there a day or two in advance.  Make the copies double sided, collated and stapled, on 3-hole punched paper.  You may also make transparencies from your handouts.  77 Dunster St. will do this, I think, or you can do it on my copier in Quincy 102. You can also use a computer to display things. Either bring a laptop with a VGA output, or place the material in an accessible location on the Web, or put it on my laptop.  The third approach in the safest but takes a bit of advance planning.
	The overall scope of a project should be about the same as one to two lectures worth of course material, depending on how much you can build on what has been done in class.  If you choose to do something like co-Minkowski geometry, co-Euclidean geometry, co-Minkowski geometry, doubly hyperbolic geometry, larger affine planes, or inversive geometry, which is closely related to what was done in class, then the one- lecture guideline is appropriate.  If you choose to do something like Hilbert's non-Archimedean geometry that is unrelated to the approach in class, then the two-lecture guideline is appropriate. This means that all you can do in your 20 minutes is to hit the highlights.
	The final written version of your project should not exceed 10 pages in length.  This can include what you show in class, and in that case, if you used large fonts for better visibility of transparencies, you may count two pages as one. Hand-drawn diagrams and symbols are acceptable, though they make revision more difficult. If you do a computer project, give me the source code electronically; there is no need to print it.
	Cite all your sources, and make it clear what is completely original, what is an original reworking of published material, and what is a straightforward report on published material. Any reasonable style for doing this is OK.
Make it clear which member of your team did which aspect of the project.  Both members of the team should contribute both to the written version and to the presentation.
Almost any topic in geometry is appropriate, as long as it will be comprehensible to me and your classmates.  Here are some ideas for the general focus of a project.
1. Mathematical - textbook based
Find a topic from a textbook (even Ryan or Yaglom) and present it in a form appropriate to this course.  It will be a big plus if you prove a theorem that is not proved in the text (though perhaps stated there as a problem), an even bigger plus if you invent and prove a theorem.
2. Mathematical - literature based
Go to Mathematical Reviews, or check the bibliography of a textbook or the CRC Concise Encyclopedia of Mathematics (in Cabot).  Read at least two related articles and present the key results.

3. Mathematical - original 
Don't worry about possibly reinventing the wheel.  Choose some topic from the course and extend or generalize it in an interesting way.  Many of the Problems (with Roman numerals) from Yaglom
suggest good topics.  So does Supplement C on complex numbers.  Then there are all the Cayley-Klein models from early in the course, described in Supplement A.
If you state that you used no references, you will receive credit for the originality of your work even if you end up rediscovering something that is well known.  But if you browse through textbooks and articles, you must cite them!
4. Physical
Galilean, Minkowskian and co-Minkowskian, and hyperbolic and co-hyperbolic geometry can be presented as physics rather than as geometry, and the isometry groups then become the symmetry groups that physicists love.  In this case your project can look like physics, but your classmates should say to themselves, "Only someone who really knows geometry would have thought of that."
5. Instructional
Yaglom first presented the content of his book to high students in a mathematics club in Moscow. You might try your hand at reworking some material for people without Math 21.  In this case you might create a complete problem set and a number of exam questions.  If you can persuade someone, perhaps a former high-school teacher, to use your materials, so much the better.
You might combine 4 and 5 by creating materials that could be used in a high-school or Harvard physics course, perhaps for teaching special relativity from a geometric perspective.
6. Computational - data only
The file format used by affine.exe and by Luke Gustafson's PHP software is pretty obvious, and we need files for larger affine senates.  For n= 8 or n=9 there will be lots of collineations that hold three points fixed, and you can investigate them by computer.  The case n=7 was done two years ago, but n=8 or n=9 would be very interesting.
7. Computational - C++
I will happily make the source code for affine.exe and CK.exe available for modification,and I can explain (preferably to a small group of people all at once) how to make and test enhancements. You will need access to Visual C++ version 6. (I have one copy available for loan.)
8. Computational - PHP
If you know any object-oriented programming language, it will be easy to learn how to create Web pages with fancy diagrams in PHP. I can supply instructions for installing PHP and related tools, and I have some documentation for the geometry package. This sort of project is the best way to get your name on the course Web site next year. If four or more people want to learn PHP, I will
organize a tutorial session to get you started. 
9. Historical
Geometry has a long and rich history.  Investigate some of it using primary sources as well as textbooks.  (Textbooks sometimes get history wrong -- someone writes something that is incorrect, and then other authors copy it).  Hyperbolic geometry is the most obvious topic, but the development of new axioms for Euclidean geometry or the Klein "Erlanger Programm" are other possibilities.
10. Commercial
One hears that it is possible to purchase a term paper on any topic.  Try purchasing a term paper
that would be appropriate for this course, report on your experience, and critique the quality of the 
mathematics and of the writing. Unfortunately, no financial subsidies are available.