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.)
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 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).
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.
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