MATH
3355 NONEUCLIDEAN GEOMETRY 
Fall 2015 
Instructor:
Roberto Hasfura 
SCHEDULE: TR 9:5511:10 a.m.in
room MMH 130
PREREQUISITES:
MATH 3326 or equivalent
TEXT: Euclidean and NonEuclidean
Geometries, M. J. Greenberg, Third or Fourth Edition, (Fourth Edition is
about $160; Third Edition can still be found online for far less money)
OFFICE HOURS: MW 45 p.m.,
TR 3:304:30 p.m.; and by appointment. In Room MMS 115K
INSTRUCTOR’S COORDINATESOffice: MMS 115K;
Phone Extension: X8240; Email: jhasfura@trinity.edu
COURSE URL: www.trinity.edu/jhasfura/Courses/Math3355
COURSE
DESCRIPTION:
The discovery of noneuclidean geometries in the early part of the
nineteenth century is an event not only of great importance in mathematics and
physics, but also one with significant epistemological import.
In this course, after a
brief discussion of the essentials of axiomatic systems we will examine the
merits and deficiencies of the geometry in Euclid's Elements (c.
300 B.C.) and then we will present the system postulated by Hilbert, which
redresses those shortcomings. We will also study the role of the parallel postulate
in that geometry as well as the history of the controversy surrounding it.
Finally, the geometry that arises by postulating the negation of the parallel
postulate (a noneuclidean geometry) will be
considered along with some of its models.
Hopefully, by the end of
the semester, the first paragraph in this course description will have been
justified.
HOMEWORK AND GRADING:
The homework will consist
of eight to ten takehome problem sets throughout the semester. Your final
grade in the course will depend mostly on the grades you get on these
assignments so you shouldn't treat them lightly (as you might treat a regular,
less weighty homework assignment.) The papers you will turn in must be
thoughtful, clear, and as complete as possible. (I will not give credit to
illegible or disorganized or otherwise messy work. I suggest that you work out
the problems in their entirety and carefully review them before you attempt to
produce the final draft. The solutions that you will hand in should be neatly
presented with all work shown.)
Your final grade in this
course will be computed in the following manner:
Final
Grade = .90(Average Grade on Assignments) + .10(Grade on Attendance).
With regard to
attendance, each unjustified absence will cost 20 (out of 100) points in your
Attendance Grade so that if, for example, you have five unjustified absences
your Attendance Grade will be 0 and you will not be able to get an A in the
course.
ACADEMIC INTEGRITY:
“All students
are covered by a policy that prohibits dishonesty in academic work. Under
Trinity’s Honor Code, a faculty member will (or a student may) report an
alleged violation to the Academic Honor Council. Students who are under the
Honor Code are required to pledge all written work that is submitted for a
grade: `On my honor, I have neither given nor received any unauthorized
assistance on this work’ and their signature. The pledge may be
abbreviated `pledged’ with a signature. As your instructor, I support
and adhere to the principles of the honor code. You will be assumed trustworthy
in all your dealings with me and with your fellow classmates. However, should a
violation of this trust be discovered, it will be reported to the Council which
will investigate and adjudicate. The goal is not vengeance against those who
violate the honor code, but fairness for those who adhere to it.”
NOTES:
1. Please do not
use cell phones in the classroom. If you need do it during class, please
discreetly leave the classroom and return when you are finished.
GRADE EQUIVALENCES:
94<#<100
A 
90<#<94
A^{} 
87<#<90
B^{+} 
83<#<87
B 
80<#<83
B^{} 
77<#<80
C^{+} 
73<#<77
C 
70<#<73
C^{} 
65<#<70
D^{+} 
60<#<65
D 
#<60 F 

HOMEWORK:
Hwk 1. Due in my office (please
use the accordion envelope labeled Geometry) before 5 p.m. on Friday 9/11. Read Chapter 1. Hand in solutions to all the Review
Exercises and Exercises 1, 2, 4, 5, 10, 11, 13 in Chapter 1.
Hwk 2. Due in my office
(please use the accordion envelope labeled Geometry) before 5 p.m. on Friday
9/18. Read Chapter 2. Hand in
solutions to Exercises 1, 2, 3, 4, 6, 7, 9, 11 in Chapter 2.
Hwk 3. Due in my office
(please use the accordion envelope labeled Geometry) before 5 p.m. on Friday
9/25. Read Betweenness
on Chapter 3. Hand in solutions to Exercises 1, 2, 3, 4, 5, 6, 9, 12 and 15 in
Chapter 3. These are Exercises 1, 2, 3, 4, 5, 6, 9, 13 and 16 in the third edition of the
book.
Hwk 4. Due in my office
(please use the accordion envelope labeled Geometry) before 5 p.m. on Friday
10/2. Read Congruence on Chapter 3.
Hand in solutions to Exercises 22, 23, 25, 26, 27, 29, 30
in that chapter. They are the same Exercises in the third edition of the text.
Hwk 5. Due in my office
(please use the accordion envelope labeled Geometry) before 5 p.m. on Friday
10/9. Read Chapter 3. Hand in
solutions to Review Exercises 117, Exercises 7ab, 35, and Major Ex. 1 in
Chapter 3.
(Problem 35 is not in the
third edition: I will send an email with a picture of the problem. The other
exercises are in your text (with the same numbering.)
Hwk 6. Due
in my office (please use the accordion envelope labeled Geometry) before 5 p.m.
on Monday 10/19. Read Chapter 4. Hand
in solutions to Review Exercises 114, 16, 1820 and 22; and Exercises 1ab, 2,
16, 17, 19, 20a, 22, 23 and 24 in Ch. 4.
(In the third edition of
the text these are all the Review Exercises; and Exercises 3, 9, 16, 17ab, 18,
19, 23, 27, 28 and 29 in Ch. 4.)
Hwk 7. Due in my office
(please use the accordion envelope labeled Geometry) before 5 p.m. on Wednesday
10/28. I
understand that everyone is working out of the Third Edition of the text. The
following exercises are from that edition. Read
Chapter5. Hand in solutions to the Review Exercises; and to Exercises 2, 3, 4,
5, 6, 10 and 11 in Ch. 5.)
Hwk 8. Due in my office
(please use the accordion envelope labeled Geometry) before 5 p.m. on Friday
11/6. The
following exercises are from the Third Edition of the text. Read Chapter 6. Hand in solutions to Exercises 6, 8,
10 and 14; and to Major Exercise 10 in that chapter.
Hwk 9. Due in my office
(please use the accordion envelope labeled Geometry) before 5 p.m. on Tuesday 11/24.
The
following exercises are from the third edition of the text (in parenthesis I
give the corresponding exercise in the fourth edition). Read Chapters 6 and 7 (inversion with respect to
circles, and the Klein and Poincare disk models only). Hand in solutions to
Exercise 11 (2) and Major Exercise 11 (11) in chapter 6; and exercises K11 and
K12 (K11 and K12) and P6, P7 and P9 (P6, P7 and P9) in chapter 7. .