- About the course
- Outline of Weekly Readings
- Office Hours
- Project Guidlines
- Additional Sources on Reserve
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
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
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
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 a tiling of the hyperbolic plane
by congruent triangles.)
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
- 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. Fall Recess. Chapter 5, "History of the
Parallel Postulate". Attempts to "prove" the parallel axiom through the
ages. Take Home Mid-Term Exam.
- 9-10. Chapter 6, "The Discovery of Non-Euclidean Geometry". Mathematics is
a creative activity. See also Chapter 3 in Wolfe.
Project Topic and Outline Due: November 1.
- 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. Thanksgiving recess
- 14. Chapter 7, More about "The Independence of the Parallel Postulate" using
the Poincare model of hyperbolic geometry.
Chapter 8, "Philosophical Implications" of the existence
of more than one type of geometry.
- 15. Project Presentations: December 9 and 11. Course conclusion.
Take Home Final Exam Assigned on December 11.
- 16. Final Exam Due: Tuesday, 17 December 2002 at 7:00 pm in our usual
classroom. Discussion and Answer session.
There will be two take-home exams, one just after mid-term (during the week of October 14)
and the other will be due on the
at the time scheduled for the final exam for this course (Tuesday, 11 December 2002 at 7:00 pm).
As well, there is a required project for the course (see notes below). It will be due on December 6.
I will ask you to present the key ideas from your project to the class
during the last week of the term.
The take-home final will cover material primarily from the second half 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
Together the take-home mid-term and final assignments count for 45%
of your final grade. The project counts for 15%.
Homework assignments will make up the remainder (40%) of your
grade, though I reserve the right to consider class participation as a
My office is located in Lansing 305. My extension is 3619. I have scheduled
office hours on M & W 1:00 to 3:00 pm, T & T 9:30 to 11:00 am.
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.
Maturing as a mathematics student involves reading (engaging) sources
other than those texts covered in class. Equally important is being able to report
the results of your efforts in both written and oral forms. For these reasons, I am
requiring an independent project for the course. I don't expect you to create original
mathematics (that's what a Ph.D is for). What I would like you to do is to read a chapter
from one of the texts below and to report on and summarize the major results and arguments
in the reading.
Your project is due Monday,
9 December 2002 (the beginning of the last week of classes).
It should be the equivalent of about 10 pages typewritten. Feel
free to submit a handwritten copy if your writing is very neat and legible.
Your topic selection should be approved in advance. I will ask for a project proposal and a brief
outline on Friday, 1 November 2002 if you have not already submitted one. During the last two days of
class you will each have 25 minutes to give a brief presentation and answer questions related
to your topic.
The following are suggested readings/topics.
Other topics are possible, but please seek approval prior to 1 November 2002. Names in bold
refer to texts in the additional sources section below.
- "Polygons and Area". Chapter 13 (pages 184-197) in Moise. Great for those interested in
- "Constructions with Ruler and Compass". Chapter 19 (Part I: pages 264-275) in Moise. Great for those interested in
- "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.
- Chapters I and II of Projective
Geometry by John Wesley Young.
- "Metric Postulates for Plane Geometry" by Saunders McLane in Selected Papers on Geometry.
- "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
- With prior approval, any geometry related topic in these journals: American Mathematical Monthly,
Journal of Undergraduate Mathematics,
Mathematics Magazine, or
College Mathematics Journal.
- With prior approval, any geometry related topic in some other text.
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!
- 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
- 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
- Selected Papers on Geometry. Contains some articles that could be the basis of
a final project.