| Subsection | Pages | What to do |
|---|---|---|
| Consistency of Hyperbolic Geometry | 223-227 | Read all. |
| The Beltrami-Klein Model | 227-232 | Read all. |
| The Poincare Models | 232-238 | Read all (but just skim the paragraph on p. 237 describing the Poincare half-plane model). |
| Perpendicularity in the Beltrami-Klein Model | 238-241 | Read all. |
| A Model of the Hyperbolic Plane from Physics | 241-243 | Skip. |
| Inversion in Circles | 243-257 | Read definition of inverse on p. 243 and skim Propositions 7.1 to
7.5. Read the paragraph on p. 247 following the proof of Proposition 7.5 and relate this to the straightedge/compass construction we did in class on Monday. Read p. 248-249 on the definition of length in the Poincare disk model. Skip the rest of the section. |
| The Projective Nature of the Beltrami-Klein Model | 258-270 | Skip. |
| Section | Problems to do | Submit | Due date |
|---|---|---|---|
| Scorpling Flugs | Theorems 1-5 | Theorems 1,4 | Thursday, May 22 |
| Chapter 1 | RE: all; E: 1,5,8,12 | None | |
| Problems on logic | 1-7 | 2,4b | Friday, May 23 |
| Problems on sets | 1-2 | 2a,c,f | Tuesday, May 27 |
| More problems on logic | 1-4 | 4 | Tuesday, May 27 |
| Chapter 2 | E 3-9,11,12 | 6 (Prop 2.4 and 2.5),9,11 | Thursday, May 29 |
| Chapter 2 | ME 1,2,6,7,8 | 6,8 | Monday, June 2 |
| Chapter 3 | E 1,2,6,9,12,16 | all | Friday, June 6 |
| Chapter 3 | E 24,25,26,29,32,36 | all | Tuesday, June 10 |
| Chapter 4 | E 9-14 | all | Friday, June 13 |
| Chapter 4 | E 4-7 | 5 or 6 | Wednesday, June 18 |
| Chapter 5 | E 1,8,9,11 | all | Thursday, June 19 |
| Chapter 6 | E 2,3,4,5,7(a,b),14,15 | all | Tuesday, June 24 |
| Chapter 7 | K-E 2,3,4,5 |
Course Overview and Text
This course examines three themes: the axiomatic method, specific axiom systems for geometries (both Euclidean and non-Euclidean), and the history of Euclidean and non-Euclidean geometry. A major goal for the course is learning to construct valid proofs within the specific axiom systems we study. Upon successfully completing this course, a student should be able to
The prerequisite for this course is Math 122. The main rationale for this prerequisite is to ensure that you have a certain level of mathematical experience rather than understanding of specific mathematical concepts.
Grading, Coursework, and Policies
In class, we will discuss new material, respond to questions from reading the text, and work through assigned problems on which there are difficulties. When we discuss new material, the focus will be on ``the big picture.'' That is, we will look at new ideas in their simplest form and how these ideas fit together. Often, we will not consider details and variations in depth during a first pass through new material. Your mastery of the details will begin outside of class with a careful reading of the text and work on the assigned problems. We will address the details by responding to questions on the reading and problems that you bring to class. You are expected to participate in class by being present (and alert), by responding to questions I pose, and by asking the questions that you have.
Outside of class, you should read the relevant sections of the text carefully. This will generally include working through the reasoning of arguments and filling in steps that are omitted in calculations. You should keep a list of specific questions from the reading and find answers to those questions either in class, with me outside of class, or with study partners.
The text is also a source of exercises that are essential in building understanding and skill. I will assign homework sets from the textbook on which you will need to spend considerable time and effort. I will also designate problems to be collected and evaluated. For these problems, you should write up careful solutions using the standards of proper technical writing. You should not get in the habit of focusing only on the problems designated to be turned in. You will need to do many more problems in order to become facile with the concepts, techniques, and applications.
We will have three exams during the course. The exams are scheduled for every other Friday: May 30, June 13, and June 27.
To determine course grades, I calculate a total course score with homework problems weighted at 40% and exams weighted at 60%. I assign a preliminary course grade based on an objective standard (ususally 93.0-100% for an A, 90.0-92.9% for an A-, 87.0-89.9% for a B+, 83.0-86.9% for a B, etc.). I then look at each student's performance subjectively. Occasionally I will assign a course grade that is higher than the objective standard. For example, if a student has a grade of B according to the objective standard but has shown steady improvement, I might assign a course grade of B+.
Course Web Pages
Web pages for this course are located
at
www.math.ups.edu/~martinj/courses/summer2003/m300/m300.html
You can get to this page by following links at www.math.ups.edu/~martinj. Assignments will be listed and class handouts will be available to download as PDF files.