Math 360: Foundations of Geometry

Offered:     Spring 2008
Instructor:  Kevin J. Mitchell
Office: Lansing 305 
Phone:  (315) 781-3619
Fax:    (315) 781-3860

Office Hrs:  Mon and Wed 12:45 to 2:30, Thurs 9:30 to 11:00, Fri 1:30 to 2:30.
             Often available at other times by appointment.

Class:    M-W-F 10:10 to 11:05 in Lansing 301                   
Text:     Euclidean and Non-Euclidean Geometries, (Fourth Edition) 
          by Marvin Jay Greenberg
Optional: A Certain Ambiguity: A Mathematical Novel, by Gaurav Suri 
          and Hartosh Singh Bal. (Princeton University Press)

About the Course

This course is about geometry and, in particular, the discovery (creation) of non-Euclidean geometry about 200 years ago. At the same time, the course serves as an example of how the discipline of mathematics works, illuminating the roles of axioms, definitions, logic, and proof. In this sense, the course is about the process of doing mathematics.

The course provides a rare opportunity to see how and why mathematicians struggled with key ideas---sometimes getting things wrong, other times having great insights (though occasionally they did not recognize this fact). History is important to this subject; this course should convince you that mathematics is a very human endeavor.

The course focuses on Euclid's Parallel Axiom: "For any line l and any point P not on l, there is a unique line through P parallel to l." In particular, could this axiom be deduced as a consequence of the earlier and more intuitive axioms that Euclid had laid out for his geometry? Mathematicians struggled with this question for 2000 years before successfully answering it. The answer had a profound philosophical effect on all later mathematics, as we will see.

One of the goals of the course is to convince you that if you "believe in" Euclidean geometry (the ordinary, everyday geometry that you studied in high school), then you must also "believe in" hyperbolic geometry which is quite different from and contradictory to Euclidean geometry. For example, in hyperbolic geometry, triangles have less than 180 degrees (see the figure to the left) and there are no rectangles! An obvious question, then, is whether the universe is actually Euclidean or hyperbolic. Note that it is not possible to "prove" that the world is Euclidean by measuring a physical triangle and showing that it has 180 degrees. Why? However, if the universe were hyperbolic, it might be possible to show this by measuring triangles. How? In fact, the great mathematician Gauss tried to do exactly this! (The figure to the left shows part of a tiling of the hyperbolic plane by congruent triangles. Do they look congruent?)

The course rewards careful, attentive reading and regular review of previously covered material. Be sure to read with a pencil in your hand, especially to draw figures that relate to the material being discussed.

Rough Outline of Weekly Readings


There will be three take-home exams. The first will be assigned near mid-term (end of week 7, given about March 7, but this date may change depending on the pace at which we cover material). The second or Penultimate Assignment will be given out about 2 weeks before the end of term. It will review much of the material from the second half of the course. The third exam will be given out on the last day of class (or shortly thereafter) and will be due on the at the time scheduled for the final exam for this course (Monday, 12 May 2008 at 7:00 pm). There will be regular homewark assignments, a total of about 12 or so for the entire term. As well, you will be required to give class presenations on specific homework problems.

The take-home final will cover material primarily from the last few weeks of the course and will include a careful review of hyperbolic geometry that will require a number of hyperbolic constructions. (E.g., construct an equilateral triangle with three 45 degree angles!)

Assignments are due at the beginning of class. Late assignments will incur a substantial penalty and will be rejected entirely if more than two days late. Since the course builds on each assignment, it is extremely important for you to get them done on time.

I encourage you to discuss material with each other. However, each person should write up her or his work individually. On the take-home exams you should not even discuss the problems with your classmates.

Together the two take-home exams and final assignments count for 55% of your final grade. The presentations and homework assignments will make up the remainder (45%, split about 10% and 35%) of your grade. I reserve the right to consider class participation (including attendance) as a factor.

Office Hours

My office is located in Lansing 305. My extension is 3619. I have scheduled office hours on Mon and Wed 12:45 to 2:30, Thurs 9:30 to 11:00, and Fri 1:30 to 2:30. I am often in my office at other times of the day, and I encourage you to drop in to get hints or help with course assignments or just to chat.

More Reading

In some years I have assigned projects in the course. This time I expect to assign more class presentations of homework, in part, because the end of Spring Term always seems especially hectic. Nonetheless, I thought I would present the list of readings and topics that I provide when I do require projects for the course. They would be fun for you to read.

Let me begin by singling out a new novel that I just read. If the text for the course were less expensive I would have everyone buy this book and read it. It is wonderfully relevant for those who are taking this course. A Certain Ambiguity: A Mathematical Novel by Gaurav Suri and Hartosh Singh Bal. (Princeton University Press).

Additional Sources on Reserve

The following texts have been placed On Reserve in the Library. Readings will be assigned in some of these texts, others are for your reference.

A note about the Center for Teaching and Learning (CTL)

A Note about the Center for Teaching and Learning (CTL): Hobart and William Smith Colleges encourages students to seek the academic collaboration and resources that will enable them to demonstrate their best work. Students who would like to enhance their study skills, writing skills, or have other academic inquiries should contact the CTL. You may visit the CTL web site to learn more about the services and programs that are available.

If you are a student with a disability for which you may need accommodations, you are required to register with the Coordinator of Disability Services at the CTL and provide documentation of the disability. Services and accommodations will not be provided until this process is complete. The web site for information pertaining to registration with the CTL and documenting disabilities is:

Hobart and William Smith Colleges: Department of Mathematics and Computer Science