Offered: Spring 2008 Instructor: Kevin J. Mitchell Office: Lansing 305 Phone: (315) 781-3619 Fax: (315) 781-3860 E-mail: mitchell@hws.edu 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)

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.

- 1. Chapter 1, "Euclid's Geometry", pages 1--9 in
*Wolfe*, and Chapter 1 in*Meschkowski*. Introductory and historical materials. - 2-3. Chapter 2, "Logic" and Chapter 1 in
*Wylie*. A quick overview of logic that will be crucial in our development of geometry. - 4. Chapter 3, "Hilbert's Axioms". The basis of all the geometry we will study, as developed by Hilbert in the early part of this century.
- 5. Chapter 3, Completion of "Hilbert's Axioms." Chapter 4, "Neutral Geometry". Geometry with no parallel postulate, i.e., with no assumptions about the existence of parallel lines.
- 6. More on "Neutral Geometry".
- 7-8. Chapter 5, "History of the
Parallel Postulate". Attempts to "prove" the parallel axiom through the
ages.
**Take Home Mid-Term Exam**. - 9-10. Spring Break.
**Conduct Eratosthenes experiment**to measure the earth's circumference (ideally March 21). Chapter 6, "The Discovery of Non-Euclidean Geometry". See also Chapter 3 in Wolfe. - 11-13. Chapter 7, "The Independence of the Parallel Postulate" or why the
parallel postulate cannot be derived from the other axioms of Euclidean
geometry.
**Penultimate Assignment**at the end of Week 13. - 14. Chapter 7, More about "The Independence of the Parallel Postulate" using the Poincare model of hyperbolic geometry.
- 15. Chapter 8, "Philosophical Implications" of the existence
of more than one type of geometry. Course conclusion.
**Take Home Final Exam Assigned**. - 16.
**Final Exam Due**: Monday, 12 May 2008 at 7:00 pm in our usual classroom. Discussion and Answer session.

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.

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).

- "Polygons and Area". Chapter 13 (pages 184-197) in
**Moise**. Great for those interested in math education. - "Constructions with Ruler and Compass". Chapter 19 (Part I: pages 264-275) in
**Moise**. Great for those interested in math education. - "Constructions with Ruler and Compass". Chapter 19 (Part II: pages 278-294) in
**Moise**. For those with an interest in abstract algebra. - "Proportionality without Numbers". Chapter 20 (pages 295-313) in
**Moise**. For those with an interest in analysis. - "Double Elliptic Geometry". Chapter VII in
*An Introduction to Non-Euclidean Geometry*by David Gans. (Also see Appendix A of our text by Greenberg.) - "Single Elliptic Geometry". Chapter VIII in
*An Introduction to Non-Euclidean Geometry*by David Gans. (Also see Appendix A of our text by Greenberg.) - "Elliptic Geometry and Trigonometry". Chapter VII in
**Wolfe**. (Also see Appendix A of our text by Greenberg.) - "Symmetries and Groups". Chapter 1 of
*Groups and Geometry*by Roger Lyndon. - Material from Chapters 1 through 5 of
*Ruler and the Round*by Nicholas Kazarinoff. Great connections between abstarct algebra and geometry. - Chapters I and II of
*Projective Geometry*by John Wesley Young. - "Metric Postulates for Plane Geometry" by Saunders McLane in
*Selected Papers on Geometry*. On Reserve. - "Geometry and the Diamond Theory of Truth". Chapter 4 in
**Trudeau**. - "Historical Background of Mathematical Logic". Chapter 1 in
*A Profile of Mathematical Logic*by Howard DeLong. - "Buffon's Needle Experiment" by Brindell Horelick and Sinan Koont.
Module 242
*UMAP Modules Tools for Teaching 1977-79*. (Determining the value of pi using probability.)

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.

- Bonola,
*Non-Euclidean Geometry*. An amazing book which could be used as a source for projects or for an independent study. Contains many original sources. - Heath (trans.),
*Euclid: The Elements*. If you want to know what Euclid actually said, it's here. It can be rough going because of the style, but all mathematicians should look at this at least once in their lives. Now is your chance! Also available on line at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html - Hilbert,
*Foundations of Geometry*. Concise! From the master himself. - Kulczycki,
*Non-Euclidean Geometry*. Chapter 1 presents a nice historical treatment of the subject. - Meschkowski,
*Non-Euclidean Geometry*. You should read the introduction. - Moise,
*Elementary Geometry from and Advanced Standpoint*. Lots of nice topics and a resource for projects. - Trudeau,
*The Non-Euclidean Revolution*. It can serve as a companion for our text. Read it for pleasure, or suggest it to your friends when they ask you what this course is about. - Singer,
*Geometry: Plane and Fancy*. A nice source for material on the hyperbolic plane and especially geometry on the sphere. - Wolfe,
*Non-Euclidean Geometry*. A bit old-fashioned. It was used as the text for this course 30 years ago. Has some nice material. - Wylie,
*Foundations of Geometry*. Another possible choice as a text for this course, except it is out of print. *Selected Papers on Geometry*. Contains some articles that could be the basis of a final project.

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: http://www.hws.edu/studentlife/stuaffairs_disabilities.aspx

Hobart and William Smith Colleges

Department of Mathematics and Computer Science

Copyright © 2008

Author: Kevin Mitchell (mitchell@hws.edu)

Last updated: January 2008

Department of Mathematics and Computer Science

Copyright © 2008

Author: Kevin Mitchell (mitchell@hws.edu)

Last updated: January 2008