Math 128A Classical Geometry Fall 2011

Updated 12/4/11

The hyperbolic tape measure (what was going to be the last lecture of the course).

Moebius Transformations Revealed is a very pretty video of Moebius transformations (of the entire complex line), with a very nifty way of thinking about reflections across circles. A geometry video with 1,870,863 views is a really amazing accomplishment.

Symmetry Artist: a simple, user-friendly drawing tool for symmetric figures. You draw something with the mouse and it creates the specified reflections or rotations of your curve as you draw.

The linear algebra and geometry sections from the California math standards for K-12 teachers (officially "Mathematics Teacher Preparation in California: Standards of Quality and Effectiveness for Subject Matter Programs"), with particularly course-relevant topics highlighted.

Evaluations are now submitted through eCommons and clicking on the Evaluation System link in the sidebar. Math's evaluation window closes Monday, 4:00 AM (before the final, of course)

Practice final exam problems
Midterm solutions (in case you want them for review)
Cayley exercises solutions

INSTRUCTOR
Instructor: Debra Lewis
Office: 4122 McHenry
Phone: 459-2718
E-mail: lewis [add the usual address info]

TA: Wei Yuan
Office:
E-mail: wyuan2 [add the usual address info]

TIMES AND PLACES
Lecture: MWF 12:30PM-01:40 PM, Engineering 2 194
Sections: Monday 4:00-5:10 PM, 1279 McHenry, and Friday 11:00-12:10 AM, 1257 McHenry
D.L.'s office hours: Monday, 11:00-12:00, Tuesday 12:00-1:30, Wednesday 2:00-3:00.
W.Y.'s office hours: Tuesday 2:00-3:00, Thursday 12:00-2:00

TEXT
The Four Pillars of Geometry, by John Stillwell. Springer, 2005.

FREE ONLINE GEOMETRY AND LINEAR ALGEBRA BOOKS

The UBC Euclid's Elements of Geometry website has an integrated version of three different English translations of Euclid's Elements: the 'classic' translation by Thomas Heath, a vintage (1847) visuals-based translation by Oliver Byrne, and a modern, Java-based translation by David Joyce; they also have a link to an online version of the original Greek. (A very reasonably priced hardcopy version of the Heath translation is available.)
Mathematical Illustrations, a manual of geometry and PostScript, by Bill Casselman.
From Wikipedia: "PostScript is best known for its use as a page description language in the electronic and desktop publishing areas. Adobe PostScript 3 is also the worldwide printing and imaging standard. ... everything, even text, is specified in terms of straight lines and cubic Bezier curves (previously found only in CAD applications), which allows arbitrary scaling, rotating and other transformations." Chapters 2, 4, and 10 are very relevant to our course; chapters 8, 12, and 14 would be equally relevant if we had time to go further. You can ignore the programming aspect and just read the geometry.
Linear Algebra, by Jim Hefferon.
This looks like a nice starter linear algebra book. Neat examples; watch for typos.
A list of free math books on a variety of topics (some very advanced)

PDFS FROM LECTURE, OR SUPPLEMENTS TO LECTURE
Areas of parallelograms (10/3/11)
Euclid's proof of the Pythagorean Theorem (10/3/11)
Euclid's proof of Thales Theorem (10/5/11)
The straight line in the intersections of circles problem (supplement to the 10/10/11 lecture)
Three points, three reflections (supplement to the 10/14/11 lecture - date 10/19 is from 2009)
Centroids for triangles (supplement to the 10/18/11 lecture)
Perspective drawing of tiled floors (10/21/11 lecture, improved version)
Practice midterm problems and solution suggestions .
Projections of parametrized lines, and linear fractional transformations (10/31 and 11/2 lectures)
Recognizing rotations (11/18)
Parallel lines in hyperbolic space (11/23)
Invariant lines under reflection in hyperbolic space (11/28)

POSSIBLY USEFUL WEB SITES
The Mathematica Demonstrations project has some interesting geometry demos.

Mathematica notebook for rectangle/parallelogram conversions and the Pythagorean theorem. You will need to download and install Mathematica Player (it's free) to run this if you don't have Mathematica on your machine.
I haven't used Player before; let me know if you have problems with it.

MIT OpenCourseWare course on linear algebra, taught by Professor Gil Strang.

Perspective Geometry, a website/online journal edited by Tomás García-Salgado

SUPPLEMENTAL TEXTS AND RESEARCH PAPERS

Secret Knowledge, by David Hockney (the use of optics in Renaissance art). On reserve in the Science and Engineering library.
The Science of Art, by Martin Kemp (perspective drawing etc.). On reserve in the Science and Engineering library.

OPTIONAL EXTRA CREDIT PROJECT

Projects should emphasize effective communication of mathematics. The point is not for you to learn specific new mathematical information, but for you to demonstrate that you can transmit your mathematical knowledge and insight to others. Also, hopefully, to have fun. Effective use of your medium of choice, e.g. a short paper, a brief oral presentation, a poster, a web page, or a set of PowerPoint slides, is important.
Quality, not quantity, is crucial. Outside of a classroom situation in which you're the student, you're much more likely to encounter upper bounds on length of documents, time for presentations, attention spans, etc., than lower bounds. Your audience should feel that they've made a good investment of their time and effort.
Your project should, of course, be relevant to the course, but it can/should build on your knowledge of other topics. 'Double-dipping' and repurposing work you've done for other courses is fine; please indicate that clearly if it's the case - establishing relationships between your work and the rest of science is an important part of research.
Practical consequences: the project will improve your grade by at most half a grade (e.g., B+ to A-); if you're solidly in the middle of a grade span and your project is so-so or very small, it might not change your course grade at all. It will not hurt your grade.

RELATED RECREATIONAL READING
Mason & Dixon, by Thomas Pynchon. 1997. A novel about 'pure' science, technology, politics, and some very complex geometry.

TENTATIVE LECTURE SCHEDULE

Monday	Wednesday	Friday
		September 23: 1.1-2. Introduction, constructions
September 26: 1.3-4. More constructions, multiplication and division	September 28: 1.5. Similar triangles	September 30: 2.1-2. The parallel and congruence axioms
October 3: 2.3-4. Area	October 5: 2.5 Pythagorean theorem	October 7: 3.1-3. The line and plane, distance
October 10: 3.4-5. Intersections of lines and circles, angles	October 12: 3.6-7. Isometries	October 14: 4.1-2. Vectors and linear independence
October 17: 4.3-4. Midpoints and centroids, the inner product	October 19: 4.4-5. The inner product and cosine	October 21: 5.1-2. Perspective drawing
October 24: 5.3. Projective plane models and axioms	October 26: MIDTERM	October 28: 5.4. Homogeneous coordinates
October 31: 5.5. Projection	November 2: 5.6. Linear fractional functions	November 4: 5.6-7. The cross-ratio
November 7: 7.1. Isometries of the plane	November 9: 7.2. Vector transformations	November 11: HOLIDAY
November 14: 7.3. Transformations of the projective line	November 16: 7.4-5. Spherical geometry and rotations of the sphere	November 18: 8.1-2. Extending the projective line to a plane, complex conjugation
November 21: 8.2-3. Reflections and Mobius transformations	November 23: 8.4. Preserving non-Euclidean lines	November 25: HOLIDAY
November 28: 8.5. Preserving angle	November 30: 8.6. Non-Euclidean distance	December 2: 8.7. Non-Euclidean translations and rotations

FINAL EXAM: Monday, December 5, 8:00–11:00 AM.

GRADING

There will be weekly homework assignments, given in class on Wednesdays and due at the beginning of class the following Wednesday. Your overall score in the course will be the best of at least three weighted averages of your average homework score, your midterm score, and your final exam score. Your two lowest homework scores will be dropped.

On the exams you'll be asked to do K out of N problems. You can do one or more of the remaining N—K problems for extra credit. Any points you earn on these problems will not be included in your exam score; at most, they will nudge a borderline grade or influence some of the wording in your narrative evaluation. You will be much better off doing K problems well than doing all N problems less impressively.

HOMEWORK POLICIES

Homework assignments will be posted online, but assignments are not 'locked in' until they've been given in lecture. Please let me know ASAP if you notice a discrepancy between an online assignment and the one given in class.

Most exercises will be from the text, but some may be taken from other sources.

Late homework will be discounted and may not be accepted. I do not need to convince you that we are consistent with such decisions; don't count on the "well, you let him/her turn in last week's assignment late, so you have to let me turn this week's assignment in late" argument.

Your homework should be neatly written and well-organized, with the pages securely fastened together and your name on every page. Many of the exercises involve several nontrivial steps; make it clear to your readers (and yourself!) what it is you're doing at each step.

Clearly number the exercises and try to submit them in numerical order; if any problems are out of sequence, indicate that at the beginning of the assignment. (You don't need to solve them in order, just submit them in order.) We should not have to hunt through several pages to find a particular problem.