Math 460: Geometry - Summer 2012

Math 460: Geometry - Summer 2012



Important Notes


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Instructor Contact and General Information


Instructor: Luís Finotti

Office: Ayres Hall 243

Phone: 974-1321 (please do not ask me to call back -- leave your e-mail)


Office Hours: M, Th 11am-12pm or by appointment (subject to change!!)


Textbook: M. J. Greenberg, ``Euclidean and Non-Euclidean Geometries'', 4th Edition. W. H. Freeman, 2008.

Prerequisite: Math 300 or Math 307.

Class: MTuWThF 8am to 9:30am at Ayres B004. (Section 301.)

Exams: Midterm: 06/25 (Mon), in regular classroom and time; Final: 07/03 (Tue), also in our regular classroom and time.

Grade: 25% for quizzes/HW + 20% for each Midterm + 35% for the Final. Note the weight of the Quizzes/HWs!


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Course Information

Firstly, this is a summer course, in which 16 weeks are squeezed into 5. So, as you can imagine, the pace is quite fast. You cannot just ``catch up on the weekends'' in a course like this, as by then we will have covered way too much material. You should catch up immediately if you fall behind, as you will not be able to follow classes and things just start to accumulate in a faster pace than you will likely be able to catch up. I strongly recommend that you review, do problems, and study every day.

This is a course on Euclidean and Non-Euclidean Geometries. We will take the axiomatic approach to geometry, starting from scratch, and develop Euclidean geometry and then deviate to non-Euclidean geometry.

Therefore, proofs will be a core of the course, although concrete examples will be given when possible. In any event, you should be able to write and read proofs, and master all the essential material from Math 300/307. You should not expect to see simply High School geometry!

Course Structure

The goal is to cover Chapters 1 to 7 from the text, and perhaps Chapter 8 if time allows it. I will not push it, though, as I'd rather you have a good grasp of the first seven chapters then covering Chapter 8.

I will likely skip more technical details, or just mention them briefly, as I think they are not necessarily fit for most taking the course. On the other hand, I'd gladly discuss these technicalities with interested students outside the lectures.


Homeworks will be assigned after every class and the selected problems and due dates will be posted at the section Homework of this page. No paper copy of the HW assignments will be distributed in class. It is your responsibility to check this page often! Besides HW assignments, other important information will be posted here. (Check the section Important Notes often!)

The HWs will be due Tuesdays and Fridays. I will likely grade just a couple of the problems you will turn in, but you will not know in advance which.

Points will be taken from messy solutions in all assignments, and you need to show work in all questions (unless stated otherwise)! (The same applies to exams!)

Calculators will not be allowed! In fact, they would be completely useless for most (if not all) of this course.

I will do my best to post solutions to the most difficult problems (on demand). If I do, they will be posted in this page.

I will try to select problems likely to be assigned early, so that if you like to do your HW early, you can do problems in the list of Problems Likely To Be Assigned below, even before I assigned them. I might change a problem or two for the actual assignment, but that is not very likely.

Also, some times I might get too ambitious in posting HW problems, i.e., I might think we will cover a section during the week, put exercises from it in the next assignment, and then end up not being able to finish it. In this case I might have to take a few problems off the assignment. The bottom line is the following: the HW assignment is not final until I remove the "More to come" from it. (If you've done problems which were removed, just saved them for the following assignment.)

Finally, if there is still a "More to come" in an assignment on a Tuesday or Friday, please write me right away so that I can update it. (If I delay in replying, you can proceed with the Problems Likely To Be Assigned.) If I only realize on the due date (Monday or Thursday) that an assignment was not complete (with the "More to come" still there) and nobody tried to contact me, I may add new problems just then, giving you little time to finish the assignment.

In my opinion, doing the HW is one of the most important parts of the learning process, so the weight for them is greater than the weight of a single midterm, and I will assume that you will work very hard on them.

Also, you should try to come to my office hours if you are having difficulties with the course. I will do my best to help you. Please try to come during my scheduled office hours, but feel free to make an appointment if that would be impossible.

Finally, it is your responsibility to keep all your graded Quizzes, HW, and Midterms! It is very important to have them in case there is any problem with your grade. You can check all your scores at Blackboard. (Blackboard will be used only for scores. This is the official site for the course.)

Missed Work

There will be no make-up quizzes or exams. If you miss a quiz or exam and have a properly documented reason, your final will be used to make-up your score.


You will have to check your e-mail at least once a week, preferably daily. I will use your e-mail (given to me by the registrar's office) to make announcements. (If that is not your preferred address, please make sure to forward your university e-mail to it!) I will assume that any message that I sent via e-mail will be read in a week or less, and it will be considered an official communication.


I have an On-line Feedback Form where you can anonymously send me your comments and suggestions. I will consider your comments and try to do whatever I can to resolve possible problems before it is too late. So, please, feel free to use it whenever you have any constructive comment or suggestion. (In fact, I would greatly appreciate it.) If you don't want you comments to be anonymous, just send me an e-mail or come by my office and we can discuss the problem.


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Legal Issues


All students should be familiar and maintain their Academic Integrity: from
Hilltopics 2011/2012, pg. 41:

Academic Integrity

Study, preparation and presentation should involve at all times the student’s own work, unless it has been clearly specified that work is to be a team effort. Academic hon- esty requires that the student present his or her own work in all academic projects, including tests, papers, homework, and class presentation. When incorporating the work of other scholars and writers into a project, the student must accurately cite the source of that work. (See Academic Standards of Conduct, pg. 12.)

All students should follow the Honor Statement: from Hilltopics 2011/2012, pg. 12:

Honor Statement

``An essential feature of The University of Tennessee is a commitment to maintaining an atmosphere of intellectual integrity and academic honesty. As a student of the University, I pledge that I will neither knowingly give nor receive any inappropriate assistance in academic work, thus affirming my own personal commitment to honor and integrity.''

You should also be familiar with the Classroom Behavior Expectations.


Students with disabilities that need special accommodations should contact the Office of Disability Services and bring me the appropriate letter/forms.

Sexual Harassment and Discrimination

For Sexual Harassment and Discrimination information, please visit the Office of Equity and Diversity.  

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Important dates:


Thursday, May 31 - Classes begin.

Monday, June 4 - Last day to add, change grade options, or drop without a "W".

Monday, June 11 - Midterm I.

Monday, June 25 - Last day to drop with a "W".

Monday, June 25 - Midterm II.

Tuesday, July 3 - Last Class Day and Final.


More dates.


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Problems Likely To Be Assigned

This list is subject to change without prior notice. The official assignments will be posted below.


Chapter 1 (pg. 43): 1, 2, 6, 14. (Please look at all others.).

Chapter 2 (pg. 91): 1(c), 3, 5, 6 (Proposition 2.3 only), 9, 10(b), 14(a), 17.

Chapter 3 (pg. 146): 3, 5, 13, 20, 22, 27, 34. (Again, look at all others and try to do a few more if possible.)

Chapter 4 (pg. 192): 1(a), 6, 7, 12, 17.


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You must form groups of at most 3, in principle. (You can do it by yourself, if you prefer too.) If you'd like a group, but don't know people in the class, let me know and I can try to help (by collecting those together).

E-mail me your group and choice of problem ASAP. (If too many groups make the same choice, I might have to reassign a different problem, likely by a random choice.)

Each group will write the solutions carefully (one for the whole group) and likely present it in class (all members, likely).

This project will have the same weight as the midterm.

The projects are due on July 2nd (the day before the final/last day). You will turn in the written solution and the whole group will present it.


Here are the possible projects:

  1. Problem 35 on pg. 152. (Describe also circles in this model.)
  2. Problem 1 on pg. 158.
  3. Experiment with concrete examples of triangles in the Upper Half Plane model to see if the sum of the inner angles of triangles still add up to 180 degrees. Is the sum even constant? Can you see something that influences the value of them? (Can you make it get bigger and smaller with concrete examples?)
  4. Major Exercises 1 and 2 from Chapter 4 (on pg. 200).

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Solutions to Selected HW Problems

Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the ``answer''. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the answers. Be careful that just because our ``answers'' were the same, it doesn't mean that you solved the problem correctly (it might have been a ``fortunate'' coincidence), and in the exams what matters is the solution itself. I will do my best to post somewhat detailed solutions to the harder problems, though.

Disclaimer: I will have to put these solutions together rather quickly, so they are subject to typos and conceptual mistakes. (I expect you to be a lot more careful when doing your HW than I when preparing these.) You can contact me if you think that there is something wrong and I will fix the file if you are correct.

Solutions to Selected HW Problems (Click on ``Refresh'' or ``Reload'' if you don't see the changes!)


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HW1 - Due on Tuesday 06/05:

Chapter 1 (pg. 43): 1, 2, 6, 14. (Please look at all others.).


HW2 - Due on Tuesday 06/12:

Chapter 2 (pg. 91): 1(c), 3, 5, 6 (Proposition 2.3 only), 9, 10(b), 14(a), 17.


HW3 - Due on Tuesday 06/19 (if no exam on Monday):

Chapter 3 (pg. 146): 3, 5, 13, 20, 22, 27, 34. (Again, look at all others and try to do a few more if possible.)


HW4 - Due on Friday 06/29:

Chapter 4 (pg. 192): 1(a), 6, 7, 12, 17.


PLEASE, HIT ``REFRESH'' (OR ``RELOAD'') IN YOUR BROWSER WHEN VISITING THIS PAGE!!!!!!! I usually get messages asking for the update in the HW when it has already been updated. Since I change this page often, some times the browser don't see the changes. But, if you hit refresh and there is still problems missing, feel free to write me.


If it is already Monday or Thursday afternoon and there still is a ``More to come'' after the HW assignment due on the coming Thursday or Monday, write me an e-mail at, and I'll update it and let you know.


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