Math 323S: Geometry Homework Assignments

Day 01: 26 August

  1. Read: Chapter 1
  2. Exercises: 1,2,5,6,7

Day 02: 28 August

  1. Read: Chapter 2, up to the section "Incidence Geometry" (pg. 69)
  2. Exercises: 16 and Major Exercises 1, 4, and 5 (all from Chapter 1).

Day 03: 04 September

  1. Read: Chapter 2: Sections: "Incidence Geometry" (pg. 69) through the end of Chapter 2
  2. Exercises: 2, 3, 6, 9(a,c), 13 (from Chapter 2). (Note that there is a misprint in 3(a): Where the book has 'CD', it should be 'CB'. There is no point D in the problem.)

Day 04: 09 September

  1. Read: Chapter 3: Through page 111. We'll begin to discuss the betweenness axioms in class on Wednesday, so I want you to be prepared.
  2. Exercises: 14 (from Chapter 2)
  3. Major Exercises: 2, 3 (from Chapter 2)

Day 05: 11 September

  1. Read: Continue Chapter 3: Up to "Axioms of Congruence" on page 119.
  2. Exercises: 1, 2, 3 of Chapter 3. (Look at these exercises before Wednesday so that we can discuss them in class.)

Day 06: 16 September

  1. Re-read: The entire section on betweenness in Chapter 3 (up to page 119). The discussion about ordered fields at the end is not critical; you already understand the real number and the rationals, and that's enough (right now) for what we will do. However, Example 2, where 'points' are the points in the Cartesian plane inside a circle and 'lines' are chords of the circle, will be very important going forward.
  2. Exercises: 4, 5, 10 (Chapter 3)
  3. Read and think about (but don't turn in unless you want to) Exercises: 9 and 13 (Chapter 3)

Day 07: 18 September

  1. Read: "Congruence" in Chapter 3 (pages 119 to 129).
  2. Exercises: 11, 12, 19 in Chapter 3. Look at these before class; we'll discuss them before you start actually writing them up.

Day 08: 20 September

  1. Re-Read: "Congruence" in Chapter 3 (pages 119 to 129).
  2. Exercises: 20, 21, 22 of Chapter 3. When you write up 21, ignore the last sentence about Example 4.

Day 09: 25 September

  1. Read: "Continuity" in Chapter 3 (pages 129 to 138). This is a first pass through, though, so just read it for the general ideas. We'll go into more depth on this next week.
  2. Problems for this material have been moved to next week. You should read through all of the exercises 20–35 and convince yourself that you can do them, even though you don't need to turn the others in. They are all important, so be sure that you understand them all.

Day 10: 27 September

  1. Preparation: Look through your homeworks and the answers that I have provided in email Read the Conclusion section, but don't worry too much about the final comment on existence of sets.
  2. Homework: Study for next week's test, especially reviewing your homework and comparing your solutions with the comments and solutions that I have provided.

Remember: You will receive a take-home test on September 30 to be completed and turned in by midnight on Tuesday (01 October).

No class on Monday, September 30!

Day 11: 02 October

  1. Re-Read: "Continuity" in Chapter 3 (pages 129 to 138), and finish reading Chapter 3, especially the section Hilbert's Euclidean Axiom of Parallelism (pp. 138–142). We'll discuss this in depth this week
  2. Exercises: 23, 24, 26, and 33 in Chapter 3.

Day 12: 07 October

  1. Read: The beginning of Chapter 4, (pp. 161–168). Don't worry about the remarks on Elliptic Geometry, though.
  2. Alternate Interior Angle Theorem
  3. Existence of Parallels

Day 13: 09 October

  1. Exterior Angle Theorem
  2. Exercises: (Chapter 4) 1(b), 3, 4, 8, 9
  3. Read: "Measure of Angles and Segments" in Chapter 4 (pages 169 to 173).
  4. Have a good Fall Break

No class on Monday, October 14! (Fall Break)

No class on Wednesday, October 16! (postponed to October 25)

Day 14: 21 October

  1. Read: Equivalence of Euclidean Parallel Postulates and Saccheri and Lambert Quadrilaterals (pg 173–183)
  2. Exercises: 10, 11 from Chapter 4

Day 15: 23 October

  1. Read: Finish Chapter 4 (pp. 183–191)
  2. Exercises: Major Exercise 3.2 (pg 153–154)

Day 16: 25 October

  1. Read: Read Chapter 5 (pp. 209–227)
  2. Exercises: Exercises 4.17, 4.18, 4.24. (You should look at Exercises 4.22 and 4.23 before starting 4.24. You don't need to turn those in, but you may find it helpful to use their results in Exercise 4.24.),

Day 17: 28 October

  1. Read: The statement of the Parallel Projection Theorem (which assumes Hilbert's Euclidean Parallel Property), pg. 232.
  2. Exercises: 5.10, 5.11, 5.13, 5.14. (You may enjoy looking at the other exercises in this section 5.12 and 5.15-18, and you may use 5.12 if you need it, but you don't need to turn it in.) (Not due until November 11)

Day 18: 30 October

Notes on Hyperbolic Geometry
  1. Read: Notes on Hyperbolic Geometry, Sections 1 & 2 (The Klein Model)
  2. Exercises: Exercise 2.1 and 2.2 (Not due until November 11)

Day 19: 04 November

  1. Read: Notes on Hyperbolic Geometry, Sections 3.1 through 3.4 (The Sheet Model)
  2. Exercises: Exercises 3.1, 3.2 (Not due until November 11)
You will receive a take-home test on Monday, November 04 to be completed and turned in by midnight on Tuesday, November 05. This test will cover material up to Chapter 4 of our text.

Day 20: 06 November

No class on Wednesday, November 06!

Day 21: 11 November

  1. More about the Hyperbolic Sheet Model. Re-read Sections 3.1 and 3.4.
  2. Exercises: Exercises 3.4, 3.5, and 3.6. (For Exercise 3.6, you can assume that the First Hyperbolic Law of Cosines is known. See Equation 3.11 or 2.3 in the notes.) You may enjoy doing Exercise 3.7 as well (the Second Hyperbolic Law of Cosines), which has no analog in Euclidean geometry.

Day 22: 13 November

  1. Read: Notes on Hyperbolic Geometry, Sections 4.1 and 4.2 (The Poincaré Disk Model)
  2. Exercises: Exercises 4.1, 4.2

Day 23: 18 November

  1. More about the Poincaré Disk. Re-read Sections 4.1 and 4.2.
  2. Exercises: 4.3, 4.4

Day 24: 20 November

  1. Constructions in the Poincaré Disk.
  2. Internet based resources:
    1. a javascript program for ruler and compass constructions in the hyperbolic plane,
    2. a Wikipedia article on tilings of the hyperbolic plane, and
    3. an interactive webpage for viewing hyperbolic tilings of low degree.

Day 25: 25 November

  1. The Poincaré Upper Half-Plane. Read Section 5 of the Hyperbolic Notes.
  2. Exercises: 5.1, 5.2

Day 26: 02 December

TBA

Day 27: 04 December

TBA