Math 316 - Euclidean and Non-Euclidean Geometry - Fall 2019

General Information:

Meeting Time:MWF 11 - 11:50
Location: Jones 306
Instructor:Ryan Vinroot
Office: Jones 100D
Office Hours: Mon 2-3, Wed 3-4, Thurs 10-11 and 2-3:30 (also by appointment).
Textbook:Euclidean and Non-Euclidean Geometries (Fourth Edition) by Marvin Jay Greenberg
Class Participation - 10%, Homework & Quizzes - 60%, Final Project - 30%. The grading scale will be the standard 10 percentage point scale, so that a final score of 93 or higher is an A, 90-92 is an A-, 87-89 is a B+, 83-86 is a B, 80-82 is a B-, 77-79 is a C+, 73-76 is a C, 70-72 is a C-, etc.
Attendance & Lecture Policy:It is expected that you attend all lectures, with exceptions minimized. It is greatly appreciated when you are on time. Please do your best to stay awake and attentive during lecture, please do not email or text during lecture, and keep all cell phones/hand held devices/tablets/laptops put away during lecture (unless you are specifically writing notes on a tablet). While it is understandable that you may miss a lecture here and there, or be sleepy in class once in awhile, repeated absences, late arrivals, naps, or general non-attentiveness will negatively affect your class participation score. Your participation score will be based on attendance as just described, along with your participation or attention during problem discussion.
Prerequisite: Math 214 - Foundations of Mathematics
Course Summary: The main goal of this course is to understand Euclidean and non-Euclidean geometry from the axiomatic standpoint, and in the context of the historical development of mathematical thought. The textbook is quite expansive, and has much more material than we can cover, but we will plan to cover topics from Chapters 1 through 8. This begins with an introduction to Euclid's original axioms and some classical geometry, followed by incremental changes or additions to these axioms. This all leads to the concept of non-Euclidean geometry, which is essentially based on the altering of Euclid's parallel postulate. By the end of the course, the goal is for you to all understand the meaning and significance of non-Euclidean geometries, both mathematically and historically, and to be able to prove statements within these systems.

Dates & Course Announcements:

  • All relevant announcements will be listed here. Check back frequently (don't forget to refresh your browser) for updates.
  • Important Dates and Class Holidays:
    • Mon, Sep 6: ADD/DROP DEADLINE
    • Sat, Oct 12 - Sun, Oct 15: NO CLASS (Fall Break)
    • Fri, Oct 25: WITHDRAW DEADLINE
    • Wed, Nov 27 - Sun, Dec 1: NO CLASS (Thanksgiving Break)
    • Fri, Dec 6 - LAST DAY OF CLASS
    • Mon, Dec 16, 9:00 AM - 12 Noon - FINAL EXAM (Final project due)
  • (8/28) I will determine regular office hours after the add/drop period. For this short week, my office hours will be: Wed Aug 28 2-3:30, and Thurs Aug 29 10:30-12 and 2:30-4.
  • (8/30) I have posted the first HW assignment below, which is due in class, next Fri, Sep 6. Please read the HW policy carefully, and I will also talk more about HW in class.
  • (9/2) My office hours this week will be: Mon 2-3, Wed 3-4, Thurs 10:30-11:30 and 2:30-3:30.
  • (9/5) Because of the closure of William & Mary, HW 1 will now be due on Mon, Sept 9. If you've already finished with it, and want something to read, I found the following paper on Pierre Wantzel and why his work on the impossibility of the trisection of angles and duplication of the cube was overlooked for so long: Why was Wantzel overlooked for a century? The changing importance of an impossibility result. This is a great example of a project-like topic for this course.
  • (9/9) My office hours this week, which likely will become my weekly office hours, are: Mon 2-3, Wed 3-4, Thurs 10-11 and 2-3:30.
  • (9/9) There will be Quiz 1 next Mon, Sept 16, in class. Other than notes or examples from class, the following are problems you should do on your own in preparation for the quiz, some of which will be problems on the quiz: pgs. 91-92 #3, 6, 7, 8.
  • (9/16) I have set my office hours for the semester to be the same as they were last week. So, unless otherwise specified, I will be in my office the following times during the week: Mon 2-3, Wed 3-4, Thurs 10-11 and 2-3:30.
  • (9/18) As promised in class, I am giving you some links having to do with the projective plane. The first is a video which gives some animation for the map in Example 7 on pgs. 84-85 of our text corresponding the real projective plane to the top half of a sphere, so you should watch it before, after, and/or while you read that example: The Real Projective Plane.
    Here is the video we watched in class which shows why the real projective plane with a disk removed is a Möbius strip: Real projective plane and Moebius strip.
    Finally, here is a video with a short (8 minute) lecture on homogeneous coordinates for projective planes, which you could watch in parallel with reading Example 8 on pgs. 86-87 of the text: Projective geometry and homogeneous coordinates.
  • (10/7) We will have Quiz 2 in class on Fri, Oct 18 (the Friday following Fall Break). The quiz will have a few "complete the proof" problems similar to those from HW #4, and will come from: pgs. 150-152 #21 (except for finishing Example 4 part), 22, 23, 27. You should also read back through Chapter 3, pgs. 103-129, to get a better view of the main purpose and history of the Betweenness and Congruence Axioms.
  • (10/9) I have to cancel my morning office hours tomorrow, Thurs Oct 10, 10-11 am. I will still have my afternoon office hours tomorrow 2-3:30 pm.
  • (10/25) Please see below, at the bottom of the Projects section, there is now a Latex file that you can download which contains explicit guidelines (along with Latex tips) for the Final project. Please read it carefully, especially the first section, and let me know if you have any questions.
  • (11/1) We will have Quiz 3 in class on Fri, Nov 8. You should read the material on Saccheri and Lambert quadrilaterals in the book, pgs. 176-188, and know the definitions of these quadrilaterals, but you are not responsible for the proofs in that section. You should be able to prove that the diagonals of a Saccheri quadrilateral are congruent. You should also do the following problems similar to those from HW #6: pg. 195 #10, 13. Finally, you should read Chapter 5, pgs. 209-227, but you are not responsible for the proofs in that chapter.
  • (11/1) My office hours next week are different next week. They will be: Mon Nov 4 2-3:30, Tues Nov 5 1:30-3, and Wed Nov 6 2:30-4.
  • (11/18) My office hours this week will be as follows: Mon (Nov 18) 2-3:30, Wed 12:30-2, Fri 3-4:30.
  • (11/25) The Final quiz, Quiz 4 will be in class on Fri, Dec 6. The quiz will contain True/False questions, many of which will come from the Review Exercise True/False questions in the book on pgs. 267-269. The quiz will also have two qualitative questions coming from our discussion and reading of Chapter 8, which we will discuss in class on Mon, Dec 2 and Wed, Dec 4.
  • (12/4) At the end of Chapter 8 of the textbook, there is some discussion of the use of hyperbolic geometry in the work of the artist M. C. Escher. Here is an article on this topic: Escher and Geometry.
  • (12/6) Your final project is due by 12 Noon on Mon, Dec 16. Any written portion, or portion which can be sent as an attachment to email, can be sent to me in that way by that due time. I will be available in my office during exams from 10 AM-12 Noon and 2-3:30 PM on Mon Dec 9, Wed Dec 11, Thurs Dec 12, Fri Dec 13. We can talk about your project at any of these times, and you can also pick up your graded last quiz.


There will be no midterm, and no timed final exam. Instead of a final exam, there is a final project due by the end of our final exam time slot, which is Mon, Dec. 16, 9 AM-12 Noon (so the project will be due at noon that day). This final project is worth 30% of your final grade.

There are several options for the final project. First, if you need to do your writing requirement for the Math major, this may be done through the final project. This means you would need to sign up for my section of Math 300 for this semester, and your final project will be an individual paper and treated slightly differently. You should only do this if both of the following hold: (1) you are not doing an honors thesis in Mathematics, and (2) you are not doing your COLL 400 requirement in Mathematics. If you opt for the Math 300 project, you will need to tell me and register before the end of the Add/Drop period.

If you are not using the final project as your Math 300 writing requirement, then you will have the option of doing an individual project, or a group project (with 2-3 people in a group of your own choosing) with an individual contribution. I would like to know by Fall Break whether you are doing a group or individual project, and who is in your group.

There are many project ideas in the textbook, but you are not limited to these. While an expository paper is one possibility for your project (even for a group project), I am open to ideas. For example, you might create a lesson plan for a high school geometry class and make a video of the lecture. I will give more specific criteria for the project later in the semester, but the following are the basic guidelines:
(1) Your project must have significant mathematical content in the form of geometry.
(2) Your project must have some other element related to geometry (history, pedagogy, philosophy, art,...).
(3) If you are doing a group project, you must let me know who is in your group by Fall Break.
(4) If you are doing a group project, other than the main component of the project, each group member must turn in an individual written component explaining what the main idea is from their perspective, and their contributions.
(5) If your project is something other than a paper, it must be approved by me, and also have a written component (whether it is a group or individual project).
(6) All written components of the project must be in LaTex (which I can help you with if needed).
(7) All components of the final project are due by the end of our Final Exam block, and so due by 12 Noon on Mon, Dec. 16.

More information on the project guidelines, and the latex file you should use for all written components, is here: Latex file with Project guidelines.

Homework & Quizzes:

There will be a total of 12 graded homework sets and quizzes (with 4-5 quizzes and the rest homework, depending on how things progress). A total of 60% of your final grade will be based on these scores, with your top 10 scores of these making up your grade. Quizzes will be in class, and will cover reading along with lectures and examples.

Homework problems will be a very important part of the course, and there will be homework assigned each week when there is no quiz. Proofs and computations should be written carefully and neatly, with attention paid to the completeness of your argument and clarity of your steps. Individual homework assignments should be written up by yourself, although some collaboration while working on the homework is fine, and encouraged as long as the work you turn in is your own formulation of a solution. You should not, under any circumstances, attempt to copy solutions to problems online (although I know this is very tempting), as this will have to be treated as plagiarism. Instead, email me for a hint, or discuss problems in a group of classmates.

Each quiz and each homework will be scored out of 50 points. There are no make-up quizzes, except for official travel related to W&M, specific accommodations, religious holidays, or some other valid reason which is discussed with me prior to the quiz.

Homework problems are due on the posted due date at the start of class, either turned in as a hard copy, or emailed to me as a pdf of a latex file (scanned homework is not accepted). Homework is considered on time if it is turned in, or in my inbox, at most 10 minutes after the start of class. Everyone will get one free pass for a one-weekday late HW without penalty (by 5 pm the weekday after it is due). Late penalties for homework are as follows, which are strictly enforced unless a student has accommodations:
10% off if it is turned in after the beginning of class, but it is in my hands (on my door), or in my email inbox as a pdf by 5 pm on the day it is due.
20% off if it is turned in by 5 pm the next weekday after the due date.
20% more off for each (week)day late, turned in by 5 pm, thereafter.

Assignment Problems Due Date
1pg. 43 #1, pgs. 47-48 #16
pgs. 48-49 ("Major Exercises") #1, 2, 5
Mon, Sep 9
2 Give any proof of the Pythagorean Theorem
we did not do in class or is done in the book.
Write it up clearly and give your resource,
and who the proof is due to (if available).
pgs. 93-94 #12(a), 14(a), 14(b), 15(a)
Mon, Sep 23
3pg. 146 #1(a), 1(b), 2(a), 2(b), pg. 149 #15 Mon, Sep 30
4pg. 148 #9, pgs. 150-152 #20, 24, 26
Discuss, in a few paragraphs, one idea
for your final project, and give one source
(which can be website, article, or a
project from the book).
Mon, Oct 7
5 pgs. 193-194 #3, 4, 5(only the first part), 6a, 6b Fri, Oct 25
6 pg. 152 #32, pgs. 194-196 #8, 11, 12, 16 Fri, Nov 1
7 pg. 229 #3(a), 3(b), pg. 271 #4 (you can apply pg. 271 #3)
pgs. 271-272 #5, 6
Fri, Nov 18
8 pg. 272 #8, pg. 275 #12
pg. 348 #K-1 (other than Axioms I-1 and B-4)
Fri, Nov 25

Student Accessibility Services:

William & Mary accommodates students with disabilities in accordance with federal laws and university policy. Any student who feels they may need an accommodation based on the impact of a learning, psychiatric, physical, or chronic health diagnosis should contact Student Accessibility Services staff at 757-221-2512 or at to determine if accommodations are warranted and to obtain an official letter of accommodation. For more information, please visit the SAS webpage.