The University of Colorado

Boulder, Colorado

Mathematics 3210, Spring 2007

Euclidean and non-Euclidean Geometry

• Location: none -- semester over.
• Schedule: none -- semester over.
• Professor: Walter Taylor
  • Email address: walter.taylor@colorado.edu
  • Office Phone: (303) 492-8344 (Please use email if possible.)
  • Office : Mathematics Building 255
• Office hours:
  • Regular hours: none -- semester over.
  • If you need to contact me, please use email. If necessary, I can arrange to meet you in person.
• Attendance: To be considered properly enrolled in the course, you must have this hour free on all MWF-days. At no time will I accommodate anyone who has registered for the class with a schedule conflict, academic or otherwise, or anyone who attempts to do so.

  You are expected to be in class every day. I can tell you more or less the page numbers that we covered in a class you missed, but not much more. I cannot begin to recreate a class situation for you, since much of the material is spontaneous, and depends on student participation. This is especially true in Geometry. Any serious or systematic unexcused absence will result in a serious penalty in your grade; if a problem arises, please let me know at once.

• Early-semester attendance: Please take note of this attendance policy from the CU course-planning website:
  • "You must attend regularly to guarantee your place in a course during the first two weeks of the semester. If you fail to do so, you may be administratively dropped at the discretion of the department offering the course. Check with your instructor regarding their specific policy regarding being dropped for nonattendance. ....... It is your responsibility to know whether or not you are still registered in each of your classes at the end of the drop/add period."
  • Since Math 3210 is somewhat in demand, and since it is not practical to extend enrollment beyond the specified class size, this policy will be strictly adhered to. In particular, attendance will be taken at every meeting of the class during the first two weeks. This policy applies whether you are waitlisted or already enrolled.
• Class decorum: No eating. No drinking. No reading of newspapers, etc. No conversation or talking, except with the prof. No headphones. No wireless in-ear devices. No use of laptops or any other electronic gadgetry. Cell phones and pagers off. No exceptions. If you need to do any of these things, please leave the room.

This page pertains only to Professor Taylor's section of Mathematics 3210, for the spring semester of 2007. As far as I know, this is the only section occurring this semester. For other sections or other semesters, other details and regulations will no doubt apply.

An attempt will be made to keep this page up-to-date, but this is not guaranteed. Students are responsible for every assignment made in class, whether or not it ultimately appears on this page.

Material

In this course we step aside from the fast track of calculus, and examine the assumptions that underlie the subject. We look again at Euclid and his axioms, and especially modern rephrasing of Euclid's axioms. It turns out that there are alternative developments of geometry, in which things work out a little differently. (For example triangles have angle-sum less than 180 degrees.) Calculations as such (e.g. of areas) are mostly irrelevant to this understanding of geometry. (They figure in some theoretical contexts, but calculational prowess is not at issue here.) It's more about understanding why things are true ... in other words, about proofs.

So the two important features of the course are: (1) working with geometric proofs, and (2) the understanding of unfamiliar geometric worlds, such as mentioned above.

Textbook

ISBN: 0-387-98650-2

To see the cover of the text, click here .

Be advised that this is a serious mathematics book, written by a mathematician for mathematicians. There is little repetition, no fluff, and no blue highlighting of the important parts. (All parts here are important.) It may be your first encounter with a text of this quality. If so, rejoice: you are getting the genuine article here. We will read parts of it together, slowly enough that it can be comprehended.

The text contains much more than any class could cover in a semester. I would like us to cover at least Chapters 1, 2 and 7.

Grades

For average grades on exams see below. The average hw score was 5.98. This average includes all recorded scores, including zero for failure to submit the assignment. If we average only those papers that were actually turned in, the average is 6.79.

About homework

Homework must come the day it is due. Late homework will either be rejected or, in rare cases, accepted with a steep late-penalty. In any case, none whatever will be accepted after the prof's version has been made available.

Until further notice, homework will be scored on a basis of a genuine effort to solve each problem. It is expected that some lapses of logic will occur in first attempts. Thus a favorable score should not be taken as an endorsement of every aspect of the proofs.

Collaboration A modest amount of collaboration is permitted on homework, even encouraged, subject to the following rules. Your submission of homework includes the implicit assertion that you have followed these rules.

A group of two or three may work together, provided they begin as equals. It is not permitted to form a group where one person already knows how to do the problems. If you are in one group of two or three, you may not be in any other group for that week. "Working together" means talking and some scratch work. It never includes copying of finished work or complete sentences. Final writeup to be done away from the group. And, of course, as is always true in mathematics (or any other subject), at the start of your paper you write, "I discussed these problems with XXXX and YYYY." (You write this every week, even if it's always the same people.)

Exams

• You must be present at all exams.
• No books, calculators, cards or other aids.
• Exams will cover proofs that are, in most cases, available in detail for your study prior to exams. Therefore, the proofs may be marked at a standard that is higher than the standard that was used for the homework. It is expected that all statements will be correct and supported by correct reasons.
• The final examination will be Monday, May 7, 10:30-1:00 You must keep this time free for the exam. Unless announced otherwise, the final is to be in the regular classroom.
• Hour exams are in the usual classroom, at the usual class time, and are limited to the usual fifty-minute hour.
• Exam dates:
  • First Hour Exam: Wednesday, February 21
    • Average 68.1
    • Quartiles 87 70 43
  • Second Hour Exam: Wednesday, March 14
    • Average 90.4
    • Quartiles 97 93.5 92
  • Third Hour Exam: Wednesday, April 11
    • Average 85.8
    • Quartiles 100 93 74
  • Final Exam: Monday, May 7, 10:30am -- 1:00pm
    • Average 163.1 out of 200
    • Quartiles 183 172 144

Homework.

• HW 1, Due Friday, January 26
  • pages 16-17: 1.9, 1.10, 1.13, 1.14;
  • pages 23-26: 2.5, 2.7, 2.9, 2.12, 2.15, 2.16.
• HW 2, Due Friday, February 2
  • pages 43-45: 3.1, 3.3 (OK to assume ABC is a right triangle);
  • pages 50-51: 4.8;
  • pages 71-72: 6.1, 6.3(a), 6.5(a,b).
  • The prof's comments on the first two homework sets may be found here , and the prof's proof of the regular pentagon may be found here .
• HW 3, Due Friday, February 9
  • pages 71-72: 6.6;
  • pages 79-81: 7.1(a) (use Prop. 7.2), 7.2 (use Ex. 7.1(a)), 7.4, 7.6, 7.9 (use Ex. 7.6), 7.10.
  • The prof's comments on the third homework set may be found here .
• HW 4, Due Friday, February 16
  • pages 88-90: 8.1(a), 8.2 (use Prop 8.4(b)(ii)), 8.3, 8.4, 8.5 (for 8.5(b) you could use 7.4).
  • The prof's comments on the fourth homework set may be found here .
• HW 5, Due Friday, March 2
  • pages 88-90: 8.6 (skip I1-I3);
  • page 96: 9.2 (Exercise 7.1(b) may be helpful);
  • page 103: 10.1, 10.2, 10.4, 10.6, 10.11.
  • The prof's comments on the fifth homework set may be found here .
• HW 6, Due Friday, March 9
  • pages 103: 10.9;
  • pages 111-112: 11.5;
  • pages 301-302: 33.1 (should be easy for us!), 33.7(abc);
  • pages 316-319: 34.1, 34.3.
  • Note: it's best to do 34.1 before 33.7, and then use 34.1 in doing 33.7.
  • The prof's comments on the sixth homework set may be found here .
• HW 7, Due Friday, March 23
  • pp. 316-319: 34.4 (semi-hyberbolic case), 34.5 (semi-hyberbolic case) and your choice of 34.9 or 34.10.
  • pp. 325-326: 35.1, 35.3, 35.8 (semi-hyperbolic case).
  • The prof's comments on the seventh homework set may be found here .
• HW 8, Due Friday, April 6
  • pp. 316-319: 34.11.
  • pp. 325-326: 35.6, 35.9.
  • pp. 342-346: 37.2, 37.3, 37.14.
  • The prof's comments on the eighth homework set may be found here.
  • This set will be returned on Monday, April 16.
• HW 9, Due Friday, April 20
  • pp. 366-373: 39.2, 39.5, 39.6, 39.8, 39.14(a), 39.14(b).
  • The prof's comments on the ninth homework set may be found here.
• HW 10, Due Friday, April 27
  • pp. 384-387: 40.2, 40.3, 40.5, 40.6, 40.7, 40.8
  • The prof's comments on the tenth homework set may be found here.