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Math 403, Spring 2007

Assignment #1 (25 points) - due Friday Feb. 2 at the beginning of class.
Read Sections 1.1-1.7
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 the following 5 exercises:

1.1

1.6

1.7

*What is the definition of parallel used in this book? Give the definition and write an explanation of what it means, as if you were explaining it to a fellow student in the class.

**Use Theorem 1.1 to prove that parallel lines lAB and lCD with C not on lAB have no common point. (This exercise is at the bottom of page 7 and it tells us that the definition of parallel used in this book corresponds to the commonly used definition of parallel as meaning not intersecting.)

Solutions to Assignment #1

Assignment #2 (25 points) - due Friday Feb. 9 at the beginning of class.

Reminder: Test #1 is Wed. February 14 (postponed to Feb. 16 due to snow)

Read Sections 1.11 and 1.12 (We are skipping 1.8, 1.9, 1.10 for the most part).

Turn in the following 5 exercises:

1.10

1.11

1.17

1.20

*What is the name for the point with barycentric coordinates a=1/3, b=1/3, c=1/3?

Solutions to Assignment #2


Assignment #3 (20 points) – due Monday Feb. 19 at the beginning of class

Read Section 1.13 (we will omit the material on pages 31-32)

Turn in 2 exercises:

Exercise 1.22 This problem is asking you to write X as a combination of C and D (in the form of Theorem 1.1) and also as a combination of E and B.

Exercise 1.23 For this one, prove only one direction of Ceva’s theorem: the direction in which you assume the identity and prove that the lines are concurrent.
Solutions to Assignment #3

Assignment #4 (25 points) – due Friday March 2 at the beginning of class
Read Sections 2.1, 2.2, 2.3.

Read the section of http://www.cut-the-knot.org/triangle/pythpar/Geometries.shtml on Affine Geometries (scroll about 1/3 of the way down the page to find this short section).
Turn in the following 5 exercises:

1.24 on page 30 (10 points)

Exercises 1-4 on the HW4 handout

Assignment #5 – (35 points) due Friday March 9 at the beginning of class

Do the Geometer’s Sketchpad lab from March 2.

Turn in Exercises 1 and 3-7 from the lab.  Free printable graph paper

Also turn in a proof of Theorem 2.16 from the book. Note: earlier I had listed exercise 2.16 by mistake.  You can choose which of these you want to turn in, Theorem 2.16 or exercise 2.16.  But please do look at Theorem 2.16 before the test on Monday.

Note: Test #2 is on Monday, March 12. It will cover Sections 1.12, 1.13, the portions of 2.1-2.5 which we have covered, and the material on affine transformations.

Assignment #6 (20-25  points) due Friday March 30 at the beginning of class

Read all of Chapter 3.  We’ll be going through this chapter very quickly since much of it is familiar from Multivariable Calculus.

Turn in Exercises 3.2, 3.4, 3.6, 3.13, 3.15, 3.16

The project assignment is now available. The first due date, which is just to choose your project topic and partner (if you wish) is Fri. March 30.

Assignment #7 due Monday April 9 at the beginning of class

Read Sections 4.1, 4.2 and 4.3.

Turn in the following exercises:

  1. Exercise 4.2 from the textbook
  2. Show that isometries maps a circle of radius r onto a circle of radius r. Be sure to show the “onto” part!
  3. Give an example of an isometry having no fixed point, an isometry having exactly one fixed point, and an isometry having (at least) two fixed points.
  4. Recall that a function from the plane to itself is called a similarity if there is a number k such that all distances are multiplied by the factor k by this function. Prove that the composition of two similarities is a similarity.  (the numbers k will not necessarily be the same).
  5. Prove that any given similarity can be written as the composition of a translation and a similarity which takes the origin to the origin.
  6. Theorem 4.3, with the work “isometry” replaced by “similarity”, is true, and the proof is nearly identical to the proof given in the book for isometries. But you can accept this theorem without proof. Use it to show that a similarity maps lines to lines.

 

Reminder: Test #3 on Friday April 13, covering material through Section 4.2.

Assignment #8 due Friday April 20 at the beginning of class.

This is the last homework assignment of the semester.

Read Sections 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10.  Since the semester is drawing to a close, we will not be able to cover all of this material, but I will cover at least the major points of it in lecture, leaving out details of proofs as necessary.  Lecture notes will be handed out. We’ll be talking about the assigned problems in class Mon. and Wed; they are mostly a reading test!

Turn in:

1. Exercise 4.8

2. What is the main result of this chapter? (just read the book until you find “The main result of this Chapter is …”)

3. List all of the orientation-reversing isometries.  List all of the orientation-preserving isometries.

4. List all isometries which have exactly one fixed point.

5. For each of the following types of isometries, is it orientation-reversing or preserving? What are the fixed points? It can be written as the composition of how many reflections?

 a. identity

 b. translation

 c. rotation

 d. glide reflection

 e. reflection

 f. central reflection (fits into which of the categories (a)-(f)?)

 


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