MATH 366: Geometry A tiling of the hyperbolic plane (Poincaré disc model) by congruent regular pentagons

Instructor: Dr. Evan Bullock
Office: Herman Brown 408
Office Hours: W 4-6, Th 1:15-2:15, or by appointment
Email:
Time: 02:00PM - 02:50PM MWF
Location: Herman Brown 427
Textbook: Euclidean and Non-Euclidean Geometries: Development and History (4th Edition) (Homeworks may refer to exercises or pages in the book by number, and the numbering for both is different in the 3rd edition.)

TA: Natalie Durgin

Description

One cornerstone of Euclidean geometry is the parallel line postulate: For each line L and each point p that does not lie on L, there exists a unique line M through p parallel to L.  For thousands of years, it was expected (but not proven!) that this follows from the other axioms of geometry. In fact it does not, and there are non-Euclidean geometries with radically different notions of parallel lines. Two examples are spherical geometry (in which no lines at all are parallel) and hyperbolic geometry (in which a line has many parallels through the same point). We will develop plane geometry using various sets of axioms, keeping careful track of which properties follow from which axioms. In particular, we will isolate the results that do not require the parallel axiom. Concrete models of non-Euclidean geometry will be constructed.

Prerequisites

While there are no formal prerequisites at all for the course, students without much previous experience writing mathematical proofs should expect it to be very challenging.

Homework

Assignment 11, due Friday, April 20.
Assignment 10, due Friday, April 13.
Extra credit problems on the "unwound circle group"
Pledged Homework 2, due Friday, April 6.
Assignment 9, due Friday, March 30.
Assignment 8, due Friday, March 16.
Assignment 7, due Friday, March 9.
Pledged Homework 1, due Friday, February 24.
Assignment 6, due Friday, February 17.
Assignment 5, due Friday, February 10. (Exercises 1 and 8 have been moved to Assignment 6.)
Assignment 4, due Monday, February 6.  (See this colored figure for Major Exercise 8.)
Assignment 3, due Friday, January 27.
Assignment 2, due Friday, January 20.  (See the sample proofs of Propositions 2.1 and 2.2.  See solution to problem 2.)
Assignment 1, due Friday, January 13.

Homework is very important in this class.  Homework will be assigned weekly, due at the start of class on Fridays unless announced otherwise.  It is very important that you work on every assigned problem: in some cases, results developed in the homeworks may be used in class and likewise some results may be stated without proof in class with the proof left to you on the homework. 

I encourage you to talk to other students about the homework problems, but you must write up your own solutions and they should reflect your own understanding.

Late homework will generally not be accepted, but come talk to me (or email) if there are special circumstances.  Your two lowest homework scores will be dropped, but I strongly encourage you to complete every assignment.

If you'd like to typeset your homeworks on a computer, I would strongly suggest learning to use LaTeχ; even if  you do typeset your homeworks, you may find it quicker and easier to leave space to draw figures by hand than to draw figures on a computer.  Please do include figures though: even if your proofs would be logically correct and complete without them, they make it much easier for the grader to understand your arguments.

Problems designated "extra credit" are generally either much trickier than typical homework problems or require background knowledge from other courses.  These problems should be regarded as entirely optional; they are typically only loosely related to the main ideas of the course, and I would encourage you to look at them if and only if you're confident that your solutions to the other problems are correct.

Exams

There will be a closed-book, closed-notes scheduled three-hour final exam.  There will also be two open-book, open-notes take-home midterms (which will replace the usual homework assignment for the week).

Pledged Homework 1, due Friday, February 24.
Pledged Homework 2, due Friday, April 6.

It is the policy of the mathematics department that no final exam may be given early to accommodate student travel plans.

Grades

Homework counts as 45% of your grade.  The final exam counts for 35% and each midterm counts for 10%. 

Some useful links

Non-Euclid, a Java application for doing hyperbolic geometry in the Poincaré disk and upper half-plane models
Wikipedia's list of dynamic geometry software (I've been using C.a.R. myself since it supports some nice macros for doing hyperbolic geometry in the Poincaré disk model; it also has a web version.)
John Polking's web resources on Spherical Geometry
A Java applet for generating tilings of the hyperbolic plane (in the Poincaré disk model).

Disability Support

If you have a documented disability that will impact your work in this class, please contact me to discuss your needs. Additionally, you will need to register with the Disability Support Services Office in the Allen Center.

Return to Evan Bullock's web site.


Creative Commons LicenseAll content on this website is licensed under a Creative Commons Attribution-ShareAlike 3.0 License.