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| Quick links to: | Announcements | General information | Assignments | Notes | External resources |
01/07/08 Course web page now online! The course web page now exists! This web page will be your central resource (outside of classes) for information about the course; please come here often to look for assignments, announcements and other information.
Course description Around 2300 years ago the Greek mathematician Euclid laid out a set of axioms and definitions for plane geometry. His axioms included the famous `parallel postulate', which states:
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.
Reformulated in a more familiar form, this states:
Exactly one line can be drawn through a given point parallel to a given line.
Over the next two thousand years there were many failed attempts to show that this postulate followed from the other, more `self-evident' axioms that Euclid gave. Eventually, in the early 1800s, the reason for this failure was discovered: there are geometries which obey all of the rest of Euclid's definitions and axioms, but for which the parallel postulate fails! In this course we'll investigate these `non-Euclidean' geometries from a modern viewpoint. We'll study elliptic and hyperbolic geometries, as well as projective and Möbius geometries. We'll make use of Klein's Erlangen Program for studying and classifying geometries via group actions.
Prerequisites Math 0430: Introduction to abstract algebraic systems (groups, rings, etc.) is the main prerequisite. The course will use methods from many branches of mathematics, notably abstract algebra, calculus (single variable and multivariable), and complex analysis. I'll assume some knowledge of single-variable calculus; other concepts will be introduced or reviewed as necessary. You should also bring to the course a willingness to abandon familiar concepts and to embrace some strange new ideas.
Assessment The final grade will be composed of grades for coursework, two midterm exams, and a final exam, in the following proportions.
Coursework will consist of a series of weekly homework assignments.
There will be around 12 homework assignments in total. Homework will
be assigned every Friday, and completed assignments will be
collected at the beginning of class on the following Friday.
You're encouraged to discuss homework problems with each other, but
you should write up your solutions individually. Homeworks should be
presented neatly and carefully.
Late homeworks will not
be accepted without prior arrangement (genuine emergencies excepted):
if for some reason you think you won't be able to get your homework in
on time, please come and discuss this with me before the due
date.
The exams will be based primarily on the course material, so at exam
time it will be important to have a complete set of notes to review.
Exams will be closed book; calculators will not be permitted.
If you need to miss a class, make sure that you (1) get a copy of the
notes for the class you missed from a classmate, (2) read
those notes, together with the appropriate portion of the textbook,
and (3) come and see me if anything doesn't make sense, or if you want
to know more about what you missed.
Exam dates The final exam is on Tuesday, April 22 from 8am to 9:50am, in a room to be determined. There will be two midterm exams, provisionally scheduled for Wednesday, February 13 and Wednesday, March 26, which will be held during class time. The midterm exams will be held in the usual classroom for this course (Benedum 522). Please let me know as soon as possible if any of these dates causes serious problems for you, so that alternative arrangements can be made.
Textbook The textbook for the course is ‘Modern Geometries: Non-Euclidean, Projective and Discrete, 2nd edition&rsquo by Michael Henle. ISBN 0-13-032313-6. It's published by Prentice Hall, 2001.
Syllabus Here's a rough syllabus for the course. Please bear in mind that the information below isn't exact: some sections listed may be only partially covered, and we may cover additional material not listed below. When exam time comes around your course notes should serve as the primary reference.
| Week beginning | Textbook sections | Topics to be covered |
|---|---|---|
| 01/07 | Chapter 1 | Introduction and History |
| 01/14 | Chapter 2 | Geometry of the complex numbers |
| 01/21 (No class Monday) | Chapter 3 | Transformations; the Riemann sphere |
| 01/28 | Chapter 4 | Klein's Erlanger Program |
| 02/04 | Chapter 5 | Möbius transformations; clines |
| 02/11 | — | Review; Midterm I |
| 02/18 | Chapter 6 | Types of Möbius transformations |
| 02/25 | Chapter 7 | Hyperbolic geometry |
| 03/03 | Chapter 8 | More on hyperbolic geometry |
| 03/10 | — | Spring Break! |
| 03/17 | Chapters 9–10 | Hyperbolic length and area |
| 03/24 | — | Review; Midterm II |
| 03/31 | Chapter 11 | Elliptic Geometry |
| 04/07 | Chapter 13 | The projective plane |
| 04/14 | Chapter 14 | More projective geometry finite projective planes |
Assignments will be posted in this section. All assignments are due at the beginning of class on the indicated due date. Numbered problems refer to the textbook.
| Assignment | Problems | Due date |
|---|---|---|
| Assignment 0 | See detailed description. | Friday January 11 |
| Assignment 1 |
Chapter 2: Exercises 2, 3, 5, 9, 10, 12, 17, 19, 20 Bonus problem: using the series definition of the complex exponential function, explain why exp(w+z) = exp(w)exp(z) for any two complex numbers w and z. |
Friday January 18 |
| Assignment 2 |
Chapter 2: Exercise 16 Chapter 3: Exercises 1, 2, 3, 5, 6, 7, 15 |
Friday January 25 |
| Assignment 3 |
Chapter 2: Exercise 13 Chapter 3: Exercises 8, 10, 11, 12, 13 Bonus problem: is stereographic projection angle-preserving or angle-reversing? Justify your answer. (Hint: first you need to make sure that this question is well-defined.) |
Friday February 1 |
| Assignment 4 |
Chapter 2: Exercise 18 Chapter 3: Exercise 14 Chapter 4: Exercises 1, 2, 3, 4, 8, 9, 11 |
Friday February 8 |
| Assignment 5 |
Chapter 4: Exercises 6, 7, 14, 16 Chapter 5: Exercises 4, 6, 8, 11 |
Monday February 25 |
| Assignment 6 |
Chapter 5: Exercises 2, 3, 14, 17, 19, 20 |
Monday March 3 |
| Assignment 7 |
Chapter 5: Exercises 24, 25 Chapter 6: Exercises 1, 3, 7, 14 |
Friday March 7 |
| Assignment 8 |
Chapter 6: Exercise 15 Chapter 7: Exercises 1, 3, 5, 8, 11 |
Friday March 21 |
| Assignment 9 |
Chapter 8: Exercises 3, 6, 7, 9, 10 Chapter 9: Exercises 4, 7, 12, 16 |
Friday April 4 |
| Assignment 10 |
Chapter 10: Exercise 5 Chapter 11: Exercises 2, 4, 7, 9, 16 |
Friday April 11 |
I'll put digital copies of any handouts or overheads used during the course in this section, along with past exams and other course-related materials.
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| Spring 2006 | Midterm 1 | Midterm 2 | Final |
| Spring 2007 |
Midterm 1 Solutions |
Midterm 2 Solutions |
Final Solutions |
| Spring 2008 |
Midterm 1 Solutions |
Midterm 2 Solutions |
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| Review problems |
Midterm 1 review Solutions |
Midterm 2 review Solutions |
Final review Solutions |
Here's a Java applet that demonstrates parallelism in the
hyperbolic plane. You can move the blue line around by dragging its
two control points, and you can also move the third point
independently. The black lines are the two lines parallel to
the blue line and passing through the third point.
Note This applet needs Java version 1.4.2 or later to run.
Here are some external links of varying relevance to the course. Let me know of anything that you'd like to see added here! A reminder: treat all information coming from any of these web pages with caution; there's a lot of good information out there, but errors and inaccuracies abound.
Return to Mark Dickinson's home page