**Instructor:** Evan
Bullock

**Office: **Herman Brown 408

**Office Hours:** W 4-6, Th 2-3:30, or by appointment

**Tel:** (713) 348-2372

**Email:**

**Time:** 02:00PM - 02:50PM MWF

**Location:** Herman Brown 227

**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.)

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,
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.

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 2, due Friday, January 22. (Here is a proof of Proposition 2.2 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 what the 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 P

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 drag-and-drop 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 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 L

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.

The first take-home midterm is due Wednesday, February 24. The second take-home midterm is due Wednesday, March 31.

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.
- We've
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)

- Euclidean geometry exercises (similar triangles, Pythagorean
theorem, chords of circles, etc.)

- Chapter 6 - The Discovery of Non-Euclidean Geometry
- Limiting parallel rays

- Angle of parallelism

- Limiting parallel rays
- 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, stereographic projection
- 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
- Poincaré circles
- The Bolyai-Lobachevsky formula

- The Beltrami-Klein model, perpendicularity in the Klein model
- 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.

- 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.
- Chapter 10 - Further Results in Real Hyperbolic Geometry
- Area and defect
- Horocycles and equidistant curves
- Hyperbolic trigonometry

- Spherical trigonometry
- Lambert quadrilaterals, complementary segments

Wikipedia's list of dynamic geometry software (I've been using C.a.R. myself since it supports some nice macros for doing hyperbolic geometry in the Poincaré disk model.)

John Polking's web resources on Spherical Geometry

A Java applet for generating tilings of the hyperbolic plane (in the Poincaré disk model).

A short but pretty video about Möbius transformations and stereographic projections.

Return to Evan Bullock's web site.

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