TA: Ben Waters Background lectures: Mondays
5:00-6:00PM in HB 453 Homework session: Thursdays
5:00-6:00PM in HB 427
One cornerstone of Euclidean geometry is the parallel line
postulate: For each line L and each point p that does not lie on L,
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
Prerequisites and background lectures
While there are no formal prerequisites at all for the course, students
without much previous experience writing mathematical proofs (e.g. from
221, 300, 354, 356, 365, or 368) are likely to find it very challenging.
We will need to use bits and pieces of various mathematical topics
covered in depth by other 300-level course (e.g. complex numbers,
linear transformations, groups, fields, real numbers, proofs by
induction, etc.); since many students will already know about these
topics from other courses, we won't introduce them again in class.
For students have not seen these topics before (or would like a
review), they will be covered in background lectures by the TA (on
Mondays from 5:30-6:30 in HB 427). While students who already
know a particular topic well from another course are not required to
attend the background lectures on that topic, everyone is responsible
for learning the material covered by these background lectures, and
this material may appear on the homeworks and exams.
Assignment 1, due Friday, January 15. Assignment 2, due Friday, January
22. (Here is a proof of
as an example for how to write up these proofs.) Assignment 3, due Friday, January 29. (The
black and white figure 2.11 from the book may not be so clear about
lines are (or even what the points are) so here is a color version in which every black arc, line, or
circle is itself a whole line in P2(F).) Assignment 4,
due Friday, February 5. (You may want to play around with this hyperbolic
tiling java applet; it draws the tilings and allows you to
to apply translation-like transformations to the hyperbolic plane in
the Poincaré disk model.) Assignment 5,
due Friday, February 12. (There are some typos in exercise 3.6 in
the book: everywhere a ray
symbol appears in the problem should be a line symbol.) Assignment 6,
due Friday, February 19. Assignment 7,
due Friday, February 26. Assignment 8,
due Friday, March 12. Assignment 9,
due Friday, March 19. Assignment 10,
due Friday, March 26. Assignment 11,
due Friday, April 9. Assignment 12,
due Friday, April 16. (In 4e, "sinh 2z = 2sinh x cosh y" should
read "sinh 2z = 2sinh z cosh z".) Assignment 13,
due Friday, April 23. (Error corrected in 4e.)
Homework is especially 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
Late homework will generally not be accepted, but come talk to me (or
email) if there are special circumstances. Your two lowest
scores will be dropped, but I strongly encourage you to complete every
If you'd like to typeset your homeworks on a computer, I would strongly
suggest learning to use LaTeχ.
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 only if you're
confident your solutions to the other problems are correct.
There will be two take-home midterms and a scheduled final exam.
All exams are closed-book and closed notes and are subject to the
Rice University Honor Code.
The final exam is on Friday, April 30, from 2pm-5pm in HB227. The
final exam is closed book and closed notes, but the Hilbert plane
axioms will be provided as on the midterms, as will the trigonometric
formulas on pages 490, 492, and 495 in the textbook. It is the
policy of the mathematics
department that no final may be
given early to accommodate student travel plans.
The final exam will be comprehensive, but with an emphasis on topics
covered since the second midterm. Any topic that we covered in
class or on the homework may appear on the exam, but here is a reminder
of some of the most important topics we've covered:
Compass and straightedge constructions
You needn't remember the
details of how to carry out the constructions from the first few
homeworks, but should know what
sorts of things are possible and what things aren't.
recently described some compass-and-straightedge constructions related
to hyperbolic geometry in the Poincaré disk model (e.g. construction of
the Euclidean center of a hyperbolic circle).
Chapter 2 - Incidence Geometry
Elliptic geometry and other examples
Projective completion (compare with the hyperbolic version on
p. 286 and its interpretation in the Klein model on pp. 310-311)
Chapter 3 - Hilbert's Axioms
Betweenness arguments and
Archimedes's and Aristotle's axioms have already been well-represented
on the midterms and will not be emphasized on the final.
Chapter 4 - Neutral Geometry
Alternate interior angle theorem, exterior angle theorem, etc.
Saccheri and Lambert quadrilaterals
Uniformity theorem, Saccheri-Legendre theorem, etc.
Chapter 5 - History of the Parallel Postulate
Attempted proofs of the parallel postulate
Equivalent formulations of the parallel postulate (See also:
Exercise 1 on p. 269)
Chapter 6 - The Discovery of Non-Euclidean Geometry
Limiting parallel rays
Angle of parallelism
Chapter 7 - Independence of the Parallel Postulate
The Beltrami-Klein model, perpendicularity in the Klein model
The Poincaré disk and upper half-plane models
Isomorphism between the Klein and Poincaré models,
Inversion in circles, power
Constructions Poincaré lines through a given pair of points,
making a given angle, etc.
Distance and congruence in the Poincaré models
The Bolyai-Lobachevsky formula
Chapter 8 - not covered (but once you're done with your exams,
you should definitely read it: it's fairly light reading and explains,
among other things, why the geometry of the actual universe is probably
not as Euclidean as you might think.)
Chapter 9 - Geometric Transformations
I've written a short review
of Möbius transformations
so that you would have the statements of most of the important facts
about Möbius transformations (which were spread out throught the
homeworks and lectures) in one place.
Our treatment of complex numbers and Möbius transformations
did not follow the book, but the section on "Motions in the Poincaré
Model" (pp. 437-444) may be helpful.
Chapter 10 - Further Results in Real Hyperbolic Geometry
Area and defect
Horocycles and equidistant curves
Lambert quadrilaterals, complementary segments
Homework counts as 40% of your grade. The final exam counts for
30% and each midterm counts for 15%.
If you have a documented disability that will impact your work in
this class, please contact me to discuss your needs. Additionally, you
need to register with the Disability
Support Services Office in the