Math 381W
Non-Euclidean Geometry
Spring 2011
The course presents a rigorous treatment of the foundations of Euclidean and
non-Euclidean geometries. The discovery of non-Euclidean geometry in the first
half of the 19th century shattered the traditional conception of geometry and
ended a debate that lasted for more than two thousand years. This fundamental
challenge to the concept of space was instrumental in the development of the
theory of relativity by Einstein and has had far-reaching philosophical
implications.
We take an axiomatic and proof-oriented approach in introducing the Euclidean
and non-Euclidean geometries. Assignments will emphasize writing rigorous and
fully justified mathematical proofs. The final exam will consist in a project
on a topic in non-Euclidean geometry. Students will first submit a project
proposal and receive feedback on it. During the last week of classes students
will give short oral presentations of their projects.
Final projects will be due on a specified date during the exam period.
Textbook: ''Euclidean and Non-Euclidean Geometries: Development and
History'' by Marvin J. Greenberg, 4th edition, W. H. Freeman, ISBN:
0716799480
Office hours :
Friday, 11:30-12:30
Grading: based on
performance on assignments (50%) and final project/ essay (50%). There is no
midterm exam for this course.
Homework Assignments
General comments regarding the mini-project :
Overall we were quite impressed by the level of research and
thought put into these papers. Very frequently, the results made enjoyable and instructional
reading. For the most part, the mini-projects successfully conveyed an introductory summary
and some original results of Taxicab Geometry. The average was 7.5 out of 10. We made
extensive comments to provide as much feedback as possible. The main comment that applies to
most of you relates to proper citation. See the "Important notes" for the
final project listed below. There is an item that discusses citation. Read the information
provided by the library link and learn how to reference properly.
Information regarding the final project :
A draft version of the final project will be submitted in class on Friday, April 1,
2011. We
will read the project and return it to you with comments and feedback on Friday, April
8 (our last class). The revised (and final) version of the project is due by noon on
Thursday, April 21, 2011. You can either hand it to me in person or, if I am not in the
office, slide it under my office door.
Information regarding the oral presentation of
the final project: On Tuesday, April 5, 2011, students will give a short (5
minutes)
oral presentation in class to introduce their final project to me and the rest of the
students. I expect you to
summarize in a few sentences (possibly drawing a diagram on the board and referring to it)
the topic of your project and to explain why it is interesting and important to study it.
The presentation should not exceed 5 minutes, so everyone gets a chance to speak.
The presentation will count in the evaluation of the final project.
Important notes
Essay size and format : essays should be at least 8
and no more than 15 pages long. I strongly
encourage you to use a word processor to type the text. Diagrams and
drawings can be done by hand. Each page should have 200-250 words.
Also pay attention to grammar and spelling, crucial aspects for
successfully conveying ideas in essays (and proofs for that matter).
The choice of the topic is entirely yours. The requirement
is
that the topic has to be relevant to the material covered in the course,
that
is, non-Euclidean geometry. If you seek for advice, I
will only comment on your potential topics (which one I find more
suitable, etc), but I will not make any new suggestions myself.
I expect the essays to have a good balance between
historical
background, mathematical content, and applications, as in the mini-project
on taxicab geometry. I strongly suggest that
you choose a subject where topics covered in class are used in a
non-trivial
way. The textbook provides topics for projects at the end of each chapter.
This is a good source, but you are encouraged to consult others, as well
as to be creative about it.
Proper citation is key to a research paper. Please see
www.lib.sfu.ca/help/writing
for information on how and when to cite. There are a variety of different
style options. We wil accept any citation style, as long as it is properly
used.
Also remember that you have an audience. You have to convey the ideas to
the reader. Transitioning between narrative and
mathematical exposition can be tricky. The best advice on how to do this properly is
to read (good) published mathematical papers.
You are encouraged to consult books, articles, online material,
etc, as long as you reference the sources properly. Also, if you seek
inspiration from such
sources, you are not allowed to simply copy that work into your
essay. You should restate the ideas and the material that you choose to
present using your own words, including your own views and perspectives on
the respective topic/ issue.
Academic dishonesty (including plagiarisms) is very well described on
the university policies webpage:
http://www.sfu.ca/policies/files/Students/S10.01.pdf
Excellent tutorial for understanding plagiarism (on SFU library
webpage):
http://www.lib.sfu.ca/help/tutorials/plagiarism-tutorial
Check this webpage frequently for updates on assignments, etc