Assignment #1 (25 points) - due
Wed. Sept. 3 at the beginning of class.
Read Sections 1.1, 1.2, 1.4
Read the Wikipedia articles
on Euclidean geometry and non-Euclidean geometry at http://en.wikipedia.org/wiki/Euclidean_geometry
and http://en.wikipedia.org/wiki/Non-Euclidean_geometry
Turn in Exercises 1.4.1, 1.4.3, 1.4.4, 1.4.5, 1.4.13
Assignment #2 (30 points) - due Wed. Sept. 10 at the beginning of class
Read Sections 1.5, 1.6
Turn in Exercises 1.5.4, 1.5.6, 1.5.7, 1.5.8, 1.5.9, 1.5.10
Assignment #3 (30 points) - due Wed. Sept. 17 at the beginning of class. Solutions will be handed out in class that day, since the following class is the first test.
Read Sections 2.1 and 2.2
Turn in Exercises 1.6.8, 1.6.9, 1.6.10, 1.6.11, 1.7.1 (you are not asked for proofs. Give experimental evidence from working with Geometry Explorer. You can print out diagrams from Geometry Explorer, or sketch by hand what you see on Geometry Explorer).
Test #1 will be Fri., Sept. 19, in class, covering all of Chapter 1 (except 1.3, which we are skipping), and Assignments #1-3.
Assignment #4 (25 points) – due Fri. Sept. 26 at the beginning of class.
Turn in Exercises 2.1.6, 2.1.7, 2.1.8, 2.1.9, 2.1.10
Assignment #5 (30
points) – due Fri. Oct. 3 at the beginning of class.
Read Sections 2.4 (through p. 77), 2.5, 5.1
Turn in Exercises 2.5.1, 2.5.4, 2.5.5, 5.1.4, 5.1.5, 5.1.9
Assignment #6 (30 points) – due Fri. Oct. 10 at the beginning of class.
Read Sections 5.2, 5.3. 5.4 We’ll be going through these somewhat quickly, but you need a good understanding of the main points.
Turn in Exercises 5.1.6, 5.2.4, 5.2.11, 5.2.12, 5.3.4, 5.3.5
Test #2 will be Wed., Oct. 15, in class, covering Sections 2.1, 2.5, 5.1, 5.2, 5.3 and Assignments #4-6.
Assignment #7 – due Wed. Oct. 22 at the beginning of class.
Read Sections 5.6, 7.1, 7.2. Review the Wikipedia article on non-Euclidean geometry. Take a glance at 7.3 just to see what sorts of theorems are proved (we’ll be going into much more detail in class).
Turn in the following exercises:
1. Exercise 5.4.7
2. Exercise 5.6.5
For 3.-6., decide whether each of the following is a subgroup of the group of all isometries. You don’t have to prove everything, but at least give some reasoning for each of the properties. We’ll be discussing in class what degree of rigor is required, so come to class!
3. all translations (and the identity)
4. all reflections (and the identity)
5. all rotations (and the identity). Do not assume the rotations all have the same center!
6. all glide reflections (and the identity)
7. Give another example of a subgroup of the group of all isometries.
8. Answer the following True/False questions concerning the development of non-Euclidean (hyperbolic) geometry. Explanation not required, but be sure you understand it.
a. The system consisting of
b. The system consisting of
c.
d. Postulate V can be proved from the first four postulates.
e. The system consisting of
f. Saccheri and Lambert were seeking to develop a new geometry.
g. Although Saccheri and Lambert discovered important theorems in hyperbolic geometry, they did not realize that hyperbolic geometry was a new, consistent geometry.
h. Gauss, Bolyai and Lobachevsky all came to believe that hyperbolic geometry was consistent.
i. Gauss used the axioms of hyperbolic geometry to prove that hyperbolic geometry is consistent.
j. There is exactly one model of hyperbolic geometry.
k. Euclidean geometry is the actual geometry of the physical world. Hyperbolic geometry and other geometries, while they are interesting axiomatic systems, are unrelated to the physical world.
Assignment #8 – due Fri. Oct. 31 at the beginning of class. (If you are going to put homework in my mailbox, you need to do this by 11am on the due date!)
Time to starting thinking about your project for this class (http://www.math.uiuc.edu/~kmortens/402-Fa08/project.htm) Project topics are due Nov. 5.
Read Section 7.3 and turn in the following exercises:
7.3.1, 7.3.2, 7.3.3, 7.3.11, 7.3.12, 7.3.13