Math 123: Non-Euclidean
Geometries
Spring 2008
Lecture Meeting Time: MWF
9:00a-9:50a
Lecture Location: MS 5127
Professor:
Talia Fernós
Office:
MS 6228
Regular Office Hours:
Monday 10-11am and 1-2pm
or by
appointment.
Email: I will only
answer emails requesting additional office hours. Please sign all your
emails with your full name and remember that you are
communicating with your professor, not sending a text message to your
friend.
talia@math.ucla.edu
Teaching Assistant:
Tye Lidman
Office:
MS 3931
Regular Office Hours:
Tuesday: 12-2pm
Email: tlid@math.ucla.edu
Discussion Session:
1a Boelter 5272 Tuesday 9:00A-9:50A
Text: "Experiencing
Geometry" by Henderson and Taimina
Website: http://www.math.cornell.edu/~henderson/ExpGeom/
*Links to some
resources in geometry
*Links to
resources in history of mathematics
*Supplements
to some of the chapters in the book
*Errata for
3rd edition
Other Online Resources:
Plots figures in the hyperbolic plane (Poincaré Disk Model)
http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html
Virtual Office Hours:
http://www.math.ucla.edu/classes/math/08s/123.1.08s/voh/
I will not
check the VOH. You may, and are encouraged, to use it as a forum to
discuss questions.
Please take advantage of this resource. You can even make anonymous
postings if you like.
Mentoring: A very
effective tool for learning is to engage in dialogue. It is through
this that you will be able to determine the scope of your
understanding. You are all encouraged to go to the Student Math Center
(SMC) to work and build your math network.
*SMC is at MS 3974
Hours: Monday through
Thursday, 09:00 a.m. to 3:00 p.m.
Course Content and Goals:
The goal of this course is
for you to learn the foundational principles of basic non-Euclidean
geometries. We will spend as much time as possible studying hyperbolic
geometry. The reason is not only that it is a fascinating world to
explore, but also related to many vibrant areas of current research.
We hope to cover:
Chapter 0: Historical Strands of Geometry
Chapter 1: What is Straight?
Chapter 2: Straightness on Spheres
Chapter 3: What is an Angle?
Chapter 4: What is straight on cylenders and cones?
Chapter 6: Triangles and
Congruences
Chapter 7: Area and holonomy
Chapter 14: Projections of a Sphere onto a plane
Chapter 16: Inversions in Circles
Chapter 17: Projections (Models) of Hyperbolic Planes
Plus: Excerpts from Chapters 4, 5, and 6 from "College Geometry" which
is on reserve in the libary.
Grading Scheme:
*Activities/Reflections: (25%) We will sometimes
engage in group activities. You will be required to be an active
participant (see participation below). Also, you are required to come
to class with a question or observation for possible discussion. At the
end of most classes you will have a few minutes to compose a well
written reflection on the day's class: this will usually take form in a
confusing topic or a point of illumination.
*Homework: (25%) There will
be several homework sets assigned throughout
this course. Your solutions should be clearly written with concise
logical arguments.
*Mini-Tests
and Quizzes: (30%) There will be quizzes and tests throughout the
quarter. You will be given at least 1 week advance notice.
*Final Project: (20%) Students will work in groups of 3-4
students on a
major project during the semester. A separate document will
outline
the details of the project, including topic selection, format, and
group selection process. The project will culminate in a
significant
written report, as well as a 20-30 minute oral presentation to the
class.
Workload: As with all
university courses, a great deal of the learning and understanding will
take place outside of class. You are expected to read the
textbook, to do all of the assigned problems (and more if you feel that
you need the extra practice), and to meet with the professor, the TA,
the people at the SMC, or your peers as necessary when you are
stuck. Do not wait to get help if you find yourself confused.
Participation:
This will be an interactive class. You should expect to
participate in small groups as well as in whole class discussions, and
to present work at the board. It is also important that you
listen to your peers, so that everyone feels comfortable talking in
class.
Requirements for Graded Work:
Your best effort is expected on all assignments, both in terms of
mathematics and presentation.
*Include enough of the problem in your write-up so that the reader can
follow your work. State your conclusions clearly. Use the
philosophy
that a reader who does not have the assignment sheet nearby
should
still be able to understand what you are working on.
*Your work needs to be legible. Write neatly and show your work
in a
logical, orderly fashion, written from top to bottom. Only one
line of
text should be written on each line of the page.
*If an assignment takes more than one page then the pages need to be
stapled together before you turn them in. I do not bring a
stapler to
class.
*Explanation and/or justification for all your answers is expected,
whether or not it is explicitly asked for in the problem.
Explanations
must be written in complete sentences.
Academic Integrity:
Mathematicians work collaboratively, and you are strongly encouraged to
discuss class material and homework problems with your peers.
Study
groups are encouraged. However, you must write your own solutions
with your own words and never copy somebody else's work. If working in
a group and somebody comes up with a particularly important or useful
idea, you should give them credit on your homework. Similarly, if you
find another resource useful, you should give credit to that source.
Homework Assignments:
Assigment 1 Due Wednesday, April 9th:
*Read Chapter 1.
*For each symmetry of the line find an object in the plane that does
not have that symmetry but does have all the other ones.
*Consider two reflections about non-parallel lines (so they intersect).
Show that the composition of these two reflections is a rotation.
*Read Chapter 2.
*Convince yourself that the great circles on the sphere are geodesics.
Use the book's suggestions on pages 31-32.
*Write a brief but convincing argument that explains that great circles
are geodesics.
*Assuming that great circles are geodesics, prove that all geodesics
are great circles. You will need Theorem 2.1.
*(This will be assigned on Monday:) What is the intrinsic curvature of
a geodesic on a sphere? Can you come up with something that has
infinite intrinsic curvature?
Assignment 2 Due Wednesday, April 16th:
*Read Chapters 3 and 4.
*In class we saw 3 types of geodesics on a cylinders: vertical lines,
horizontal circles, and "helixes". Assuming these are geodesics and
that the cylinder is a smooth surface, prove that every geodesic on a
cylinder is one of these 3 types.
* Problem 4.2
* This problem on Similarity
Assignment 3 Not Due:
From "College Geometry": Section 5.1 ex 8, 12, 13, 15; Section 5.2 ex
4, 5, 10b, 13a, 16
Assignment 4: Due April 30
Make all corrections on Homework 2 and the first Test.
Assignment 5: Due May 7
*Show that the identity always belongs to a group.
*Prove that the collection of isometries of the plane (which are affine
with orthogonal linear part) is a group.
*Show that affine transformations (not just affine isometries) preserve
convex combinations.
*Read Sections 5.1, 6.1, 6.2 (only what was covered in class), and 6.3
from "College Geometry"
*Do problems Section 5.6 ex 4, 5, 9, and more to come on friday.
Assignment 6: Due May 16
A)Consider the 5 axioms of Euclidean Geometry (taken say from the
appendix in Experiencing Geometry) and the cone C(a) of angle
0<a<infinity. For each axiom and each a determine whether each of
the 5 axioms hold. Explain.
B)Show that the hyperbolic plane satisfies the first 2 axioms
(this is worked out in the College Geometry book for the first axiom)
and the hyperbolic version of the 5th axiom. (We take as definition of
parallel that the two lines don't intersect.)
C)Consider the points A= (0,1) and B_t= (t, 2) the upper half plane.
a) Find the hyperbolic geodesic which passes through A and B_t (so
unless t=0 this is a semi-circle; you must find it's center and
radius)
b) Make a table of values with t going to 0 and the equation of the
geodesic from (a). Can you conclude anything about the limiting
behaviour of the geodesics as B_t approaches B_0?
D) Pick 3 points in the hyperbolic plane. Find the geodesics between
those points (explicitly) and draw the triangle that they determine.
Estimate the sum of the angles. Can you make triangles with different
angle sums? If so what range can you get? What's the biggest?
Smallest?
E) Show that the hyperbolic metric on the upper half plane is
symmetric. Symmetric means the following: Let A,B be two points in the
upper half plane, and d(A,B) the hyperbolic distance between them
(which is computed with the cross ratio). You must show that d(A,B) =
d(B,A).
F) Let C be the circle (in the hyperbolic metric of course) of radius 2
about the point (0,1). Let L be the line with equation x=0. Find the
intersection of L and C. (It has two points.)
Assigment 7 has 2 parts, due May 21st, 23rd.
*By May 21st:
*Hand in outline for group projects. Here is a pdf with a description of the instructions.
*The rest is May 23rd:
*From "College Geometry" Section 6.3 Ex's 4,6a-6c,
8a
* Recall
the lemma on page 4 from
this handout. Show that the Fact at the
bottom of page 4 is true.
*Will do work from "Experiencing Geometry" as a group Monday19th-20th.
Hand in your individual solutions.
Midterm May 30th.