Modern Geometries
(MATH 444)
Offered Fall only
Fall 2004
Time and Place: MW 3:30-4:45 pm, Room 307 Haag Hall
Professor: Richard Delaware
Office Hours: See Current Semester Office Hours
Office: Room 306 A, Manheim Hall
Mailbox: Room 206, Haag Hall, Dept. of Mathematics & Statistics
Phone: 816.235.2850
Email: delawarer@umkc.edu
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Syllabus, Texts, Grading
PREREQUISITES:
- The prerequisites for this course are Math 300 (Linear Algebra I), and
Math 301 (Sets and Algebraic Structures).
TEXTS:
- Required: EUCLIDEAN AND NON-EUCLIDEAN GEOMETRIES: DEVELOPMENT AND HISTORY, 3rd Ed., by Marvin Jay Greenberg.
- Recommended: JOURNEY INTO GEOMETRIES, by Marta Sved (Paperback)
- In Greenberg, we will cover Chapters 1-6, and five sections of Chapter 7.
EXAMS:
- There will be two 75 min. Exams (100 points each) during the semester, each given a letter grade for your convenience.
- There will be at least 3 announced 20 min. Quizzes (30 points each).
- A photocopied answer sheet will be available in class immediately after each Quiz or Exam.
- The FINAL EXAM (150 points) will be held in our usual classroom, 1:00-3:00 pm, on Thursday Dec. 16.
HOMEWORK:
- A list of suggested homework problems will be posted here. Selected problems will be collected and graded.
- Homework should be done on standard 8½ x 11 paper, folded lengthwise with your name at the top. When due, put it on my desk at the start of class.
- Instructions for the 10 page expository paper are on separate web pages.
- Late assignments of any kind are NOT accepted, except by prior arrangement.
GRADES:
- 40% -- 2 Exams and Final Exam
- 15% -- 3 Quizzes
- 30% -- Homework
- 15% -- 1 Paper
- I use grading software to provide grade reports. At the end of the semester before the Final Exam is recorded, I will use the same software to drop your "most damaging" score. The score for your paper and the Final Exam cannot be dropped.
NOTES:
- You will be proving many things in this course, and there are no answers in the back of the Greenberg book. So, you must discuss proofs with your classmates and me if you are to do well.
- You will not be able to do every suggested problem. But to succeed, you must try as many as possible.
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The Sosland Journal - An Opportunity for Student Publication
The Sosland Journal publishes a section of the journal devoted exclusively to student writing in UMKC writing intensive courses. To give you sense of the range of work the journal publishes, take a look at the 2003-2004 winning essays. These essays from philosophy, economics, and astronomy (by a Math 464 WI student!) illustrate the quality of work students are doing and the range of work the Sosland Journal publishes.
The next edition of the Journal will publish the best work submitted from the latest Fall and Winter semesters. Any student who has done writing in a UMKC writing intensive class this year is eligible to submit work from that class. All such submitted work is entered in the Ilus W. David Writing Competition, offering cash awards of $25-$200, and publication in the Sosland Journal.
The entry form includes the deadline for submissions which closely follows the last day of classes in the Winter semester. Print off the form yourself and then discuss it with me. Submissions from Math 444 or Math 464 WI students must be recommended and signed by me, Richard Delaware. I will try to keep on hand some copies of the submission forms, and will discuss this possibly with those of you who write excellent papers or essays as the semester progresses.
A student from Math 464 WI has won a prize in each of Fall 2002 and Fall 2003, and two students were published in Fall 2004. (See the department's What's New web page.)
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Suggested Homework Problems [Subject to change]
Chapter 1
Review Exercise: p. 26.
Exercises, p. 27: 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13.
Major Exercises, p. 31: 1, 3, 7.
Projects, p. 35: 1, 3. [I have references if you choose 1.]
Cartoon, p. 37: Take Note!
Chapter 2
Review Exercise: p. 62.
Exercises, p. 63: 1, 2, 3, 4, 5, 6, 7, 9, 12.
Major Exercises, p. 65: 2, 5.
Projects, p. 68: SKIP.
Chapter 3
Review Exercise: p. 103.
Exercises on Betweeness, p. 104: 1, 2, 3, 6, 9, 10, 12, 14, 18.
Exercises on Congruence, p. 107: 20, 24, 25, 26, 27, 28, 34.
Major Exercises, p. 111: 1, 2 (reflection).
Projects, p. 114: 1, 2, 3, 4.
Chapter 4
Review Exercise: p. 134.
Exercises, p. 136: 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,32,33(use 27).
Major Exercises, p. 143: 1, 4, 6 (you may use Major Exercise 1 here), 8.
Projects, p. 146: 2, 3.
Chapter 5
Review Exercise: p. 163.
Exercises, p. 164: 1,2,3,4, 8,9,10,12,18-26 (For 21, read mid-p.170 first.)
Major Exercises, p. 174: 1, 2, 3, 4, 5.
Projects, p. 176: 3, 4.
NOTE: Exercises 18-26 and Major Exercises 1-7 could become a paper.
Chapter 6
Review Exercise: p. 201.
Exercises, p. 203: 1,2,3,4,5,6,7a,8,9,11,12,13,14,15. [The author says 15 is the most important exercise in the course.]
Major Exercises, p. 209: 4, 9, 10. [NOTE: Actually read ALL Major Exercises 1-13.]Projects, p. 221: 5.
Chapter 7
Review Exercise: p. 270 [SKIP questions 11 and 15.]
P-Exercises, p. 279: P-1, P-4, P-6, P-10, P-19. These are all "major" exercises.
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Weekly Schedule - Major Events [Subject to change]
---------------------------------------------------------------------Week 15
Day 27 M Nov. 29 - Homework 6 DUE
Day 28 W Dec. 1 - LAST DAY OF CLASS, Student Evaluations
---------------------------------------------------------------------Week 16
Day 29 M Dec. 6 - NO CLASS, paper drafts continue
Day 30 W Dec. 8 – NO CLASS, paper drafts continue, last day for regular office hours
---------- F Dec. 10 – Papers DUE if possible, by 4:30 pm in my mailbox Haag Hall 206
---------------------------------------------------------------------
M Dec. 13 - LAST DAY to turn in Paper. No extensions.
R Dec. 16 - Final Exam = Exam 3, 1:00 - 3:00 pm
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Quiz and Exam Outlines
Quiz 1 (pp. 72-92) - Possibilities [Subject to change]
Exam 1 (Chapters 1-3) - Possibilities [Subject to change]
Quiz 2 (Chapter 4) - Possibilities [Subject to change]
Exam 2 (Chapters 4-6) - Possibilities [Subject to change]
Quiz 3 (Chapter 7, pp.225-249) - Possibilities [Subject to change]
To State, Use, or Explain
- MetaMathematical Theorem 1 (MMT) (Consistency of Hyperbolic Geometry, p.225)
- Poincare Model, undefined terms, pictures (pp.232-235)
- Definition of Inverse P' (p.243)
- Definition of Power (p.246)
- Prop 7.5 (Only circles through inverses are orthogonal to the given circle, p.246)
- Definition of Cross-Ratio: (AB,PQ) (p.248)
- Definition of Poincare Length: d(AB) (p.248)
- Definition of "Poincare Congruence" (for Poincare Segments) (p.249)
To Prove, wholly or in part, and Use
- Corollary to MMT (The Parallel Postulate is independent, p.225)
- Prop. 7.1 (Basic Inverse Properties, p.243)
- Prop. 7.2 (P inside circle, Construct P', p.244)
- Prop. 7.3 (P outside circle, Construct P', p.244)
- Prop. 7.4 (Construct Poincare Line through two ideal points, p.245)
- Lemma 7.1 (Power exists, p.246)
- Verification of I-1 (Poincare) (Construct Poincare Line through two ordinary points, p.247)
- Verification of C-4 (Poincare) (p.248)
- Facts about d(AB) (p.249)
- Verification of C-1 (Poincare) (p.249)
- Verification of C-3 (Poincare) (p.249)
Sample Quiz 3 - Possibilities [Subject to change]
Sample Quiz 3
Final Exam/Exam 3 (Chapter 7; some Chapter 6) - Possibilities [Subject to change]
To State, Use, or Explain
- Ideas of Hyperbolic Geometry found in Chapter 6 (pp.187-200).
- Poincare Model, undefined terms, pictures (pp.232-235)
- Prop. 7.5 (d orthogonal to g iff d passes through points and their inverses in g, p.246)
- Definition of Cross-Ratio: (AB,PQ) (p.248)
- Definition of Poincare Length: d(AB) (p.248)
- Definition of "Poincare Congruence" (for Poincare Segments) (p.249)
- Corollary to Prop. 7.6 (p.251) (d orthogonal to g iff d preserved by inversion in g, p.251)
- Prop. 7.11 (p.253)
- Lemma 7.4 (p.255) (Relation between Euclidean Distance & Poincare Model Hyperbolic Distance)
- Verification of C-6 (Poincare) (pp.254-256)
- Theorem 7.1 (p.256) (Congruence of hyperbolic triangles)
- Review the ideas listed under Quiz 3 above
To Prove, wholly or in part, and Use
- Verification of I-1 (Poincare) (Construct Poincare Line through two ordinary points, p.247)
- Verification of C-4 (Poincare) (p.248)
- Facts about d(AB) (p.249)
- Verification of C-1 (Poincare) (p.249)
- Lemma 7.3 (p.251)
- Prop. 7.10 (p.253) (Cross-ratio is preserved by inversion in g)
- Theorem 7.2 (Bolyai-Lobachevsky Formula giving the relation between the Poincare Length of a Poincare segment and its associated Angle of Parallelism) (pp.256-257)
- Appropriate Chapter 7 P-exercises (pp.279-286)such as those assigned for homework
- Review the ideas listed under Quiz 3 above
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Handout
Undefined Terms
point
line
incident
between
congruent
Informal Logic Rules
Rule 0: No unstated assumptions may be used in a proof.
Rule 1: The following are the six types of justification allowed for statements in proofs:
- "By hypothesis..."
- "By axiom..."
- "By previously proved theorem..."
- "By definition..."
- "By a previous step in the argument..."
- "By a logic rule..."
Rule 2: To prove a statement H Þ C, assume the negation of the statement C (RAA hypothesis) and deduce an absurd statement, using the hypothesis H if needed in your deduction.
Rule 3: The statement "~(~S)" means the same as "S".
Rule 4: The statement "~[H Þ C]" means the same as "H & ~C".
Rule 5: The statement "~[S1 & S2] means the same as "[~S1 or ~S2]".
Rule 6: The statement "~["xS(x)]" means the same as "$x~S(x)".
Rule 7: The statement "~[$xS(x)]" means the same as ""x~S(x)".
Rule 8: If P Þ Q and P are steps in a proof, then Q is a justifiable step.
Rule 9:
- [[P Þ Q] & [Q Þ R]] Þ [P Þ R].
- [P & Q] Þ P, [P & Q] Þ Q.
- [~Q Þ ~P] Û [P Þ Q]. [Contrapositive.]
Rule 10: For every statement P, "P or ~P" is a valid step in a proof. [Law of Excluded Middle.]
Rule 11: Suppose the disjunction of statements S1 or S2 or ... or Sn is already a valid step in a proof. Suppose that proofs of C are carried out from each of the case assumptions S1, S2,..., Sn. Then C can be concluded as a valid step in the proof. [Proof by Cases.]
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