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Math 403, Spring 2008

Assignment #1 due Friday Jan. 25 at the beginning of class.
Get a copy of the textbook if you haven’t already and 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 written answers to the following questions:

  1. What is an axiomatic system? (mention undefined terms, axioms, definitions, theorems)
  2. Give at least two examples of axiomatic systems.
  3. State Euclid’s five postulates. Explain a little of the history of the 5th postulate.
  4. What are the axioms for hyperbolic (non-Euclidean) geometry? Approximately when was hyperbolic geometry accepted as a new, valid geometry?
  5. What does it mean for an axiomatic system to be consistent? Is Euclidean geometry consistent? Is hyperbolic geometry consistent?
  6. Prove Euclid’s first proposition: given a line segment, there exists an equilateral triangle having the given segment as base.
  7. What is synthetic geometry? What is analytic geometry?
  8. Who was Euclid? When and where did he live? What are Euclid’s Elements? What is in the Elements besides plane geometry?
  9. Consider the following axiomatic system, having “building”, “sidewalk” and “between” as undefined terms:
    Axiom 1. There are at least two buildings on campus.
    Axiom 2. There is exactly one sidewalk between any two buildings.
    Axiom 3. Not all the buildings have the same sidewalk between them.

    Deduce the following theorems in the above axiomatic system:
         Theorem 1. There are at least three buildings on campus.
         Theorem 2. There are at least two sidewalks on campus Consider the following axiom set. 

 

Note: this assignment concerns important background information, but is rather different from future assignments, which will primarily ask you to do proofs and solve problems.

Assignment #2 due Friday, Feb. 1 at the beginning of class

Read Sections 1.7, 1.8, 1.10, 1.11 (up to the middle of page 20. We will not be covering the material involving complex numbers)

Write the definitions of the following terms. Use the definition which is given in the textbook:

  1. line determined by A and B (Section 1.5)
  2. midpoint of a segment
  3. parallel lines
  4. parallelogram
  5. median of a triangle
  6. centroid of a triangle
  7. centroid of the points A, B, C, D, E
  8. mass-point
  9. centroid of mass-points

 

In addition, do the following exercises. Be sure to include some sentences to explain your reasoning.

1.1, 1.2, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11

For 1.2, be sure to read the note in the paragraph immediately proceeding it.

Assignment #3 due Wednesday, Feb. 6 at the beginning of class

Test #1 on Fri., Feb. 8 will cover all the material through 1.11 (but not Section 1.9, which we skipped) Since you won’t get this assignment back before the test, solutions will be distributed when you turn it in.

Do the following exercises (10 points each):

1.14, 1.15, 1.17

Assignment #4 due Fri. Feb. 15 at the beginning of class

Read Sections 1.12 and 1.13

Exercises 1.20, 1.21, 1.22, 1.23

Assignment #5 due Fri. Feb. 22 at the beginning of class

Exercise 1.24 (10 points)

Also turn in the two problems on http://www.math.uiuc.edu/~kmortens/403-Sp08/hw5.doc

Assignment #6 due Fri. Feb. 29 at the beginning of class

Test #2 on Monday, Mar. 3 will cover Sections 1.12, 1.13 (not Pappus’ Theorem), 2.1, 2.2 (at least part of 2.2). Since the material from this homework assignment is on the test, solutions will be handed out when you turn it in.

 

Read Sections 2.1 and 2.2

Exercise 1: Give an example of a map (function) from the plane to itself which is not one-to-one.

Exercise 2: Give an example of a map (function) from the plane to itself which is not onto.

Exercise 3: Prove that a translation preserves midpoint. In other words, prove that if M is the midpoint of X and Y, then τA(M) is the midpoint of τA(X) and τA(Y) for every translation τA.

Exercise 4: Prove that a translation preserves the centroid of mass points (a,A), (b,B), (c,C).

Exercise 5: Prove that a central dilatation preserves parallel lines.

Exercise 6: Prove that a central dilatation preserves the ratios of lengths of vectors. In other words, if X, Y, Z are collinear and if X-Z=q(Y-Z), then δC,r(X)- δC,r(Z)= q[δC,r(Y)- δC,r(Z)] for every central dilatation δC,r.

Exercise 7: What is the definition of “collineation”?

Exercise 8: What is the inverse of τA? What is the inverse of δC,r?

Also, from the book,

Exercise 2.2 (you can use any of the propositions, theorems, exercises which appear in the book before this exercise)

 

Assignment #7 due Fri. Mar. 14 at the beginning of class (nothing due Mar. 7, but start early as this assignment is really two weeks’ worth).

Read Sections 2.3, 2.4, 2.5, 3.1, 3.2

Turn in the problems from the March 5 Geometer’s Sketchpad lab

Also turn in Exercises 2.8, 2.10, 2.11, 2.12 from the book.

Assignment #8 due Fri. Mar. 28 at the beginning of class.
Note: the project assignment is now available (www.math.uiuc.edu/~kmortens/403-Sp08/project.html) First due date is Fri. April 4 to submit the topic.

Read Sections 3.3, 3.4, 3.5

Exercises: (A) list six different groups of transformations (note: the groups can have a finite number of transformations if you wish).  And from the book: Exercises 3.4, 3.7, 3.11, 3.14

Assignment #9 due Mon. April 14 at the beginning of class.

Read Sections 4.1, 4.2, 4.3, 4.5, 4.6

Exercises: 4.2, 4.3, 4.4, 4.7

Assignment #10 (last one) due Mon. April 21 at the beginning of class.

Read Sections 4.7, 4.8, 4.9, 4.10

Exercises: 4.8, 4.11, 4.12, 4.14

Be sure to be aware of due dates for project (see project page)