Introduction to Modern Geometry

Fall 2011

**Instructor:** Dr. Nathaniel Miller
**Email:** nathaniel.miller@unco.edu

**Office: ** Ross 2210D
**Office Phone: **351-2297

**Course Meeting Times:** TR 3:30-4:45 in 0275 Ross Hall

**Office Hours:** T 9:15-11:00; R 2:30-3:30; and by appointment

**Course Web Page: ** http://www.unco.edu/NHS/mathsci/ClassSites/hoppercourse/math341/

**TA:** Lee Roberson

**Email:** Lee.Roberson@unco.edu

**Office: **Ross 2210E

**Office Phone:** 351-2254

**Office Hours: ** TR 12:30-1:30, W 11-1

"The royal road to knowledge,

it is easy to express:

to err, and err, and err again,

but less, and less, and less."

This course is about geometry, but it is also a course about learning to develop and express your own mathematical ideas. It will emphasize ideas and imagination in addition to techniques and calculations. We will be investigating not only the planar Euclidean geometry that you probably studied in high school, but also geometry on spheres, cylinders, cones, pool tables, and other surfaces. We will try to imagine what geometry would be like for someone living on each of these surfaces. We will be studying these geometries not only because they are useful and can tell us surprising things about the world in which we live, but also because they are beautiful and fascinating subjects.

This course may be very different from most other math courses that you have taken. High school and introductory college math courses usually focus on teaching methods of doing computations: mathematics as finding the correct answer. In this course, we are going to focus on another kind of mathematics: mathematics as a way of thinking about and trying to understand the world. We'll try to understand how people decide what is true, and how they reason about mathematics and geometry in particular. We are going to focus on the process as much as the results of mathematical thought.

Writing is an important part of this process. Contrary to what some other math courses may have led you to believe, it is virtually impossible to do mathematics without writing about it. Writing is a tool for communicating ideas to other people, but it can also be used as a tool for clarifying one's own ideas. It can be very hard to spot a flaw in a line of reasoning if you haven't written it down; conversely, writing down a line of reasoning is often the best way to expose any problems that it might have. For these reasons, practicing mathematicians invariably use writing as a vital part of their work, and writing will be an integral part of this course.

This course will require a willingness to invest significant amounts of effort grappling with developing your own ideas. In this course, as in the real world, you will be the ultimate arbiter of what you believe to be true. Deciding for yourself what is true is not easy, but it can be very rewarding.

**Writing Assignments:** Over the course of the semester, I
will be assigning a series of problems for you to write about. Some
of these will be informal writing assignments that will be graded only
on how complete they are; others will be formal writing assignments, for
which you will be expected to turn in a typewritten paper that will be
graded not only on the completeness and correctness of your answer, but
also on the clarity of your explanations. All of these problems will
go through a process of revision: I will make comments on them, and
then you will have an opportunity to revise them. You should explore
each question and write out your thinking in a way that can be shared with
others. Focus on your own ideas and understandings, and turn in whatever
your thinking is on a question, even if only to say, "I do not understand
such and such" or "I am stuck here." Be as specific as possible.
Conjecture. Use pictures. Respond to my comments and
questions. Only the final draft of the formal writing assignments
will be graded. There will also be at least one major group
project assigned during the semester that will make up a significant part
of your final grade. There will also be a take home midterm and a final project that will be presented during our scheduled final exam meeting time, on December 7th at 1:30 p.m.

At the end of the semester, I will ask you to turn in your portfolio containing all the work that you have done for this class, including all drafts of all papers, so please save everything. On average, you should expect to spend at least nine hours per week outside of class on this course. If you are concerned about the time that you are spending on this class, come see me.

**Grades:** Your final grade will be computed from your final formal and informal assignment averages. The informal assignment average will be weighted to be 40% of your final grade, and the formal average will be 60% of your final grade. Letter grades will be given following a traditional grade breakdown, including +/- grading.

**Due dates:** All assignments will have assigned initial due
dates. Revisions of formal writing assignments will be due sixteen
days after the previous draft, and only the final draft will be graded.
Informal assignments can be revised as many times as you like; however,
they should be substantially done within sixteen days of the original due
date. Progress made after this date will count half as much as progress
made before this date, unless the problem is at least 3/4 complete by then.

**Hints for Success:** The best approach is to strive for a solid
understanding of the course topics and to accept at the start that this
necessarily entails some struggling with ideas and feelings of frustration.
The course problems take time, especially time to explore and think about
the ideas. Often your will need to walk away for a while or for a
day, and return to a problem for a second or third look before writing
up your response. Expect this. However, do not get behind on
the problems. Try to cultivate an approach that is a nice balance
between "just getting it done" and avoiding it altogether. Stay connected,
and come see me if you are having difficulties.

**Texts: **The readings for this course will come from photocopied
handouts. They will come mainly from
*Experiencing Geometry In
Euclidean, Spherical, and Hyperbolic Spaces
*by David W. Henderson and
from Euclid's
*Elements*.

**Course Materials:** You will need a folder or binder in which
to keep your written work and a disk on which to keep your work done on
the computers. You will also want to have a sphere of some kind to
look at when we study spherical geometry. Tennis balls work well,
as do the plastic spheres often available at craft shops.

**Group Work:** We will often work in groups in this course.
Whenever a group hands in a written assignment, they are required to put
on the paper the names of those who participated fully, and only those
names. Each person must sign the final copy. Your signature
certifies that you participated equally in the project. It is dishonest
to turn in work that is not solely and equitably the creation of the team
members. You are not required to include on the report the name of
someone who started but did not finish, or who did not contribute their
share.

**Outside Sources:** A central aim of this course is to help
you learn to develop your own ideas about mathematical questions.
You therefore should **NEVER** consult any reference materials outside
of the course texts in answering questions for this course. This
includes materials found on the internet. The ideas that you present
should be your own.

**Office Hours: **My office hours are listed above, and will be held in 2210D Ross Hall. Please come see me! The best
way to make an appointment or to get in touch with me for any other reason
is to send me email.

**Attendance: ** It is absolutely vital for an interactive class
like this that you come to class and participate. Your attendance
will count as part of your class participation grade, and excessive late
arrivals will count as an absence.

**Licensure: **This course provides content necessary to enable
secondary licensure standards to address the K-12 Colorado Model Content
Standards in Mathematics.

**Disability Support Services:** Any student requesting disability accommodation for this class must inform the instructor giving appropriate notice. Students are encouraged to contact Disability Support Services at (970) 351-2289 to certify documentation of disability and to ensure appropriate accommodations are implemented in a timely manner.