Foundations of Geometry -- Mathematics 531

Frank Thorne - Fall 2012

University of South Carolina

Instructor: Frank Thorne, LeConte 400G, thorne [at] math [dot] sc [dot] edu

Office Hours: Monday 3:30-4:30, Tuesday 3:30-4:30, Friday 2:30-3:30.

  • Learning outcomes:

    Successful students will master a variety of concepts in Euclidean geometry. In particular:

    In addition, students will also:

    Warning. You should expect 5-8 hours of homework a week in this class, which is more than most other instructors assign; in my experience there is no other way to learn the material. Your consistent effort will certainly lead to improved understanding, and it will almost certainly lead to you earning high grades.

  • Texts : We will read selections from four textbooks. The Isaacs book is required; please plan on reading parts of it closely. We will only read a little bit from Stillwell -- skimp on this if you must -- but he is very good at making complicated ideas easy. The books are each approximately $40 on Amazon. The bookstore charges much more so I recommend you order the books online.

  • Lectures : LeConte 303B, MWF, 9:05-9:55. Too early if you ask me. Bring coffee; I certainly will.

  • Exam schedule : In-class midterms on Wednesday, September 28 and Wednesday, October 31. The final exam will be given on Saturday, December 15 at 9:00 a.m. (I did not pick the time...)

  • Homework :

    Warning. I assign a lot of homework.

    The homework is intended to take 5-8 hours a week. That is a lot. Please count on making a consistent effort to do well in this class! Starting the night before is a bad idea.

    If homework takes you more than 10 hours on any given week, then that is more than I intended; please let me know.

    There will be at least one bonus problem on each homework, each worth one or two points, up to a maximum score of 11/10 on each week's homework. This is the only way to earn extra credit; please note that bonus problems will not be accepted late.

    Graduate credit: If you want graduate credit, you are required to do at least one bonus problem from every other homework (i.e., do one from HW1 or HW2, etc.) Bonus problems on top of that count for extra credit.

  • Grading :

    Please note. You will be graded both on correctness and on quality of exposition. Indeed, a major focus of Math 531 is the ability to communicate mathematical ideas clearly. The standard is that someone who doesn't know the answer should be able to easily follow your work. In particular, please write in complete English sentences and draw clear diagrams where appropriate. Any work that is confusing, ambiguous, or poorly explained will not receive full credit.

    Grading scale: A = 90+, B+ = 85+, B = 78+, C+ = 73+, C = 66+, D = 52+.

    Please note that my grading scale is more generous than the usual 10-point scale. However, I am a (slightly) stricter grader than most. This is intended to balance out.

      Grade component     % of grade  
      Two in-class exams     20% x 2  
      Final exam:     35%  
      Homework:     25%  

  • Contacting me : Please contact me if you have any questions about the course, about my expectations, about my lectures, about the homeworks, about the reading, or about anything else. The syllabus is demanding and it is my job to help you succeed. The best ways to get help are to come to office hours (no appointment necessary) or to e-mail me (I will almost always reply within 24 hours). If neither of these work for you then please e-mail me to set up an appointment.

  • Make-up policy :

    If you have a legitimate conflict with any of the exams it is your responsibility to inform me at least a week before the exam. Otherwise makeup exams will be given only in case of documented emergency.

    Late homework will be accepted once per student, up to a week late; after that, no late homework please except in case of emergency.

    Academic honesty and attendance are expected of all students.

    Calculators will not be needed or allowed. You will eventually require a compass and straightedge, including on exams.

  • Rough outline (times are very approximate)

  • Schedule and homeworks :

    (Note: Daily topics are listed only for some days, and when in the future are subject to change.)

  • (F) August 24: The power and methods of geometry.

    Homework 0, optional, due Wednesday, August 29.

    If you turn this homework in, I will give feedback but the grading will be very lenient. This homework asks you to write proofs, an essential skill to review for this course.

  • (M) August 27: Congruent triangles.

    Homework 1, due Monday, September 3. Isaacs, Chapter 1B, 2, 3, 5, 7, 8; bonus: 9. Solutions (due to Matt Boylan).

    If you also hand in Homework 0 then only three problems are required from Homework 1 (your choice).

  • (W) Sep 12: Circles and arcs (Isaacs, 1F)

  • (F) Sep 14: Circles and arcs (cont.); polygons in circles (Isaacs 1G, but for this section we will follow the book less closely)

    Homework 2, due Monday, September 17 (by 5:00): Isaacs, Chapter 1D, 4, 6, 7, 8, 9, 10; 1E, 1, 3. Bonus: 1D, 12, 13.

    Solutions by Matt Boylan: Part 1, Part 2.

  • (M-F) Sep 17-21: Similarity (Isaacs, 1H)

  • (M) Sep 24: Similarity continuted (Isaacs, 1H, covered by John Willis)

    Homework 3, due Wednesday, September 26. Isaacs, Chapter 1F, 1, 2, 6, 7, 10; 1G, 2; 1H, 3, 5, 6, 8. Bonus: 1F, 12, 14; 1H, 11.

    Solutions by Matt Boylan: Part 1, Part 2, Part 3.

  • (W) Sep 26: Exam review/problem session (covered by Matt Boylan).

    Practice Exam 1 (solutions to be posted later).

    Also, Matt Boylan's exam review sheet from 2011 (Part 1, Part 2) might be helpful. (On Part 2, only through Chapter 1G.)

  • (F) Sep 28: First midterm exam, on Chapter 1 of Isaacs.

    Exam 1 (with solutions).

  • (W) Oct 3: Euclid, Propositions 1 and 4.

    Homework 4, due Friday, October 12

  • (F, M) Oct 5, 8: Euclid, Propositions 1-8, 13-20, 26.

    Homework 5, due Wednesday, October 17. Solutions.

    Homework 6, due Monday, October 29.

    Exam 2, Wednesday, October 31. Study guide, practice exam with solutions.

    Homework 7, due Monday, November 19: Isaacs: Ch. 2A, 1, 2, 3, 4 or 5 (bonus: do both); 2B, 1; 2C, 2 or 3 (bonus: do both); 2E, 1, 2, 3, 4 (bonus: 5).

    See Matt Boylan's website for solutions.

    Homework 7 bonus (one point each): For each of the Axioms of Order appearing in Group II of Hilbert, imagine that Hilbert had left this axiom out. Then, either (1) prove that this axiom follows from the other axioms, or (2) construct a ``model of geometry'' (i.e., draw a bunch of points and say which are between others) in which the other axioms are true and the omitted axiom is false.

    Homework 8 (now complete), due Friday, December 7.

    Solutions, except for a few problems which Matt Boylan wrote solutions to, which you can find on Matt Boylan's website.

    Final exam information:

    Here is a final study guide. The final will be roughly half computations and constructions, and half proofs. Here is a partial practice final, for the proofs section only. Here is the diagram. I wrote two versions of this document, and selected this one at random. You will be required to choose six out of the other seven proofs and do them, and there will be one more proof. The remainder the exam will be computations (no proofs required) and constructions.

    The world's hardest easy geometry problem.