WEEK 1: 1/15 and 1/17
Read Chapter 1 and Chapter 2.
(Problems 1.1 and 2.1)
1/15 Summary: "Straight" vs. "turning". Straight implies symmetry. Angle sum in high school geometry is 180 degrees. But there is a 90-90-90 triangle on the earth! What gives? Straight depends on who you ask. The polar bear joke. Symmetric vs. unsymmetric letters.
1/17 Summary: Exploration of triangles on the sphere. Found examples with angles from 180.01 to ??? Angle sum of "small" triangle on the earth is nearly 180 (but actually larger). A plane flying along a latitude line (other than the equator, say due west) is actually turning!
WEEK 2: 1/22 and 1/24
Read Chapter 3, problems 3.1-3.3.
1/22 Summary: Philosophy: calling "facts" from high school geometry into question. Is geometry just an approximation of "reality"? Computing areas of triangles on the earth. The Gauss-Bonnet theorem relates the area of a triangle to its angle sum -- a completely new phenomenon!
1/24 Summary: Introduction to wallpaper patterns. "Border" patterns as a warmup problem. How do you describe your design to the wallpaper printer?
WEEK 3: 1/29 and 1/31
Look for examples of "wallpaper" type symmetry.
1/29 Summary: More border patterns. Finding all symmetries of a given border pattern. Translational and reflection symmetries. What happens when you "combine" two symmetries?
1/31 Summary: When does a mathematician view two wallpaper patterns as the same? Enumerating all symmetries of a given pattern.
WEEK 4: 2/5 and 2/7
Homework: compute the symmetries for the two border patterns given in class.
2/5 Summary: Tilings vs. wallpaper patterns. Constructing examples. Building new examples from old ones by "subdividing". Proof that you can't tile with regular pentagons.
2/7 NO CLASS.
WEEK 5: 2/12 and 2/14
Homework (art project): Design your own wallpaper pattern a la Escher.
2/12 Summary: The "cut-and-paste" method of constructing new tilings of the plane from old ones. Experimentation.
2/14 Summary: Discussion of the mathematician's catalog of tilings/wallpaper in terms of symmetries, as for the border patterns. Handout containing pictures of the seventeen possible patterns.
WEEK 6: 2/19 and 2/21
2/19 Summary: Constructed a "Decision tree" for cataloging a given wallpaper pattern.
2/21 Summary: Finished up discussion of the decision tree with handout of "my version". Defined "glide reflections". Began discussion of good vs. bad axioms. Returned a bit to a discussion of whether latitude lines are "straight lines" on the earth.
WEEK 7: 2/26 and 2/28
Turn in your Escher art project 2/26.
2/26 Summary: Continued discussing good vs. bad axioms. Focused on the "parallel postulate" and whether it meets our criteria of (1) obviousness (2) not excluding interesting kinds of geometry.
2/28 Summary: Last lecture I mentioned "not excluding interesting kinds of geometry", but what other kinds are there? We've talked about "high school" geometry, spherical geometry, "punctured" high school geometry, and "Manhattan" geometry. Evaluated each according to whether they satisfy (1) existence of straight lines (2) uniqueness of straight lines (3) shortest distance is a straight line (4) any straight line is the shortest distance. We computed the circles in spherical and Manhattan geometry and found they could be weird-looking.
WEEK 8: 3/5 and 3/7
Midterm exam is Thursday, March 7th.
3/5 Summary: Review session for the midterm.
WEEK 9: 3/19 and 3/21
Withdrawal deadline is Friday, March 22nd
Homework: Read Problems 6.4 and 6.5 (SAS and ASA discussion). 3/19 Summary: Gave our first proofs of the semester. From the first axiom alone, showed that two distinct lines can meet in at most one point. Discussed SAS as an axiom vs. something to be proved. Proved that the base angles of an isosceles triangle are congruent. Proved the "alternate interior angles theorem". Discussed the notion of "criticizing" a proof by using a counterexample (say from spherical geometry) to see where the proof goes astray for the counterexample. In particular, at key places in the above proofs we used the uniqueness of a straight line connecting two points.
3/21 Summary: Continued our discussion of proofs, and worked in groups on a proof that the sum of any two angles in a triangle is less than 180 degrees.
WEEK 10: 3/26
No class Tuesday 3/26. See you next week.
WEEK 11: 4/2 and 4/4
4/2 Summary: Reviewed development from the axioms through our proof that the sum of any two angles in a triangle is less than 180 degrees. Then proved that the sum of all three angles is less than or equal to 180 degrees. Posed one of the fundamental questions in geometry for over 2000 years: can you prove that the angle sum is precisely 180 degrees only using the axioms we've given so far?
4/4 Summary: (Emotional high point of the semester) Discussed how one might attempt to prove something like the angle sum theorem and mentioned some famous attempts (Sacchieri, e.g.) Book after book filled with mathematics which the authors hoped would all be "false" (proof by contradiction). I ended lecture by answering the main question: you can't prove the angle sum is 180 under our current assumptions. The explanation for this is that you can construct a geometry which satisfies our axioms and in which there are triangles for which the angle sum is less than 180. I proceeded to do this (hyperbolic geometry).
WEEK 12: 4/9 and 4/11
4/9 Summary: Exploration of hyperbolic geometry.
4/11 Summary: No class (I'll be out of until next Monday).
WEEK 13: 4/16 and 4/18
4/16 Summary: Criticize Ptolemy's proof that the angle sum of a triangle is 180 degrees.
4/18 Summary: Criticize Nasiraddin at-Tusi's proof that a Sacchieri quadrilateral is a rectangle.
WEEK 14: 4/23 and 4/25
4/23 Summary: Pac-Man, Asteroids, tilings, and donuts. Discussed similar constructions in hyperbolic geometry, the last of which yielded a donut with two holes.
4/25 Summary: Semester presentations by (1) Kirsti/Amie/Amber on Escher and (2) Sara/Allie on tesselations and their applications in the elementary classroom.
WEEK 15: 4/30 and 5/2
Thursday, May 2nd is the last class meeting
4/30 Summary: Semester presentations: (1) Chris/Ray on spherical geometry, (2) Laura/Tim/Kylene on Manhattan geometry, (3) Kristen/Mary on astronomy, (4) James^2/Tiffany/Lindsay on Euclid.
5/2 Summary: Semester presentations: (1) Merin/Anna/Stephanie on Euclid, (2) Carrie/Candace on Manhattan geometry, (3) Melissa/Hillary on spherical geometry, (4) Liz/Mandi/Louise on spherical geometry in the elementary classroom.