Math 330 — Introduction to Higher Geometry; Spring 2008

Homework Assignments & Course Schedule

Here is a link back to the course syllabus/policy page.

In the following, "**[H]**" refers to the required course textbook,
* Geometry with Geometry Explorer*, by Michael Hvidsten.

(at jonathan.poritz@gmail.com) telling me:**Send me e-mail**- Your name.
- Your e-mail address. (Please give me one that you actually check fairly frequently, since I may use it to contact you during the term.)
- Your year/program/major at CSUP.
- What you intend to do after CSUP, in so far as you have an idea.
- Past math classes you've had.
- Have you taken a course on Euclidean geometry (with proofs) at some point in your education?
- The reason you are taking this course.
- Your favorite mathematical subject.
- Your favorite mathematical result/theorem/technique/example/problem.
- Have you had classes in which you were required to write "proofs"?
- What computer background/experience/knowledge do you have?
- Anything else you think I should know (disabilities, employment
or other things that take a lot of time,
*etc.*) - [Optional:] The best book you have read recently.
- [Optional:] When you are studying, do you like to listen to music? If so, what kind of music?

*Read in***[H]**:*Preface*, pp.*ix-xii*- §§1.1, 1.2, 1.4, 1.5

*Exercises:*- start the exercises due next Monday.

*NOTE:*Friday is the last day to add classes- Content this week:
- Overview and bureaucracy. Some history. Axiomatic geometry, general axiom systems. Models, consistency, completeness.

*Read in***[H]**:- §§1.6. 1.3, 1.7

*Exercises due Monday:*- 1.4.3, 1.4.4, 1.4.5, 1.4.9, 1.4.10, 1.4.11
- look over 1.4.1, 1.4.13, and 1.5.2 and be ready to say something out loud in class about them (no written work necessary)

- Content this week:
- More on general axiom systems. Various axiom systems for Euclidean geometry. Pasch's Axiom. Various constructions -- constructable numbers, the Pythagorean Theorem.

*NOTE:*Monday is the last day to drop classes*Read in***[H]**:- §§2.1, 2.2

*Exercises due Wednesday:*- 2.1.5-2.1.9, 2.2.7

- Content this week:
- Angles. Congruence of triangles: SAS ⇒ SSA, ASA and SSS. Parallel lines — some lemmata and theorem. Seeking clarity in proof writing, examples from the HW. The Golden Mean.

*Read in***[H]**:- §§2.3, 2.4, 2.6

- Content this week:
*M:*Existence of the circumcenter. Lemma characterizing the perpendicular bisector of a line segment.*W:*Finishing with circumcenter. Arcs, central and inscribed angles in circles.*F:*Finished proving cental angle of an arc is twice any inscribed angle with that arc. Started the concept of area, which should satisfy the criteria:- areas of rectangles are
*base*×*height* - areas of congruent figures are the same
- areas of figures cut up into finitely many, non-overlapping pieces are the sums of the areas of the pieces

- areas of rectangles are

- Handed out the first midterm, which is due next Wednesday.

*Read in***[H]**:- §§2.5, 3.1, 3.2, 3.4

- Content this week:
*M:*More on area (issues towards areas of polygons, such as convexity, interior/exterior of closed curves [The Jordan Curve Theorem], exhausting figures with infinitely many triangles,*e.g.,*to compute π). Started similar triangles.*W:*More similar triangles (making them with a line parallel to the side of a given triangle; the AAA theorem for similar triangles,*etc.*).*F:*Starting analytic geometry.

**Midterm I is due Wednesday!***Read in***[H]**:- §§3.1, 3.2, 3.4

- Content this week:
*M:*More on analytic geometry: identification of the Euclidean plane with the set**R**^{2}. Vector space operations, geometric interpretations.*W:*More vector operations: dot products, norms. Law of Sines and Law of Cosines*F:*Proof of Law of Sines, introduction to the complex plane, definition of the extended complex plane.

*Exercises due Friday:*- 2.5.4 and 2.5.7

*Read:*- in
**[H]**: §§3.5 (but**not**3.5.3), 8.1 - Topology handouts

- in
- Content this week:
*M:*Intro to topology: a*topology*on a set, the*usual topology*on**R**, the topology on the extended real line,*continuous functions**W:*more topology: the*usual topology on***R**^{2}, the*subspace topology*, the*discrete topology*,*metrics*, the*taxicab metric**F:*more metrics and topology, the circle as a topological space — it's homeomorphic to the extended real line. the homeomorphism of the extended complex plane with the 2-sphere.*fractional linear transformations*

- Content this week:
*M:*geometric interpretation of complex arithmetic;*isometries*(in general, and Euclidean isometries): basic properties*W:**fixed points*, particularly how many an isometry can have.*reflections*.*F:*the "three fixed points means it's the identity" theorem for isometries. the idea of the*composition*(or*product*) of two isometries; the*isometry group*(written*Iso(X,d)*) for any metric space*(M,d)*. a transformation is called a*translation*if it is the product of two reflections.

*Exercises due Monday:*- in
**[H]**: 3.5.6, 3.5.8, 3.5.9 - Also the problems here.

- in
*Read:*- in
**[H]**: §§5.1—5.4

- in

- Content this week:
*M:*finishing translations. defining, characterizing*rotations*.*W:*more rotations;*glide reflections*.*F:*starting*symmetry*.

*Exercises due Monday:*- in
**[H]**: 5.1.2, .4, .7, 5.2.4, 5.2.11, .12, .13

- in
*Read:*- in
**[H]**: §§5.6 & 6.1

- in
*NOTE:*Friday (by**5pm**) is the last day to withdraw with a grade of**W**recorded

- Content this week:
*M:*[finite] symmetry groups. isometries of finite order. symmetries of the equilateral triangle*W:*symmetries of the square. Frieze groups*F:*symmetries of the*n*-gon. possible symmetry groups

*Exercises due Monday:*- in
**[H]**: 5.3.5, 5.4.8, 5.4.10, 5.4.12, 5.6.8-10

- in
*Exercises due Friday:*- in
**[H]**: 6.1.7 and 6.1.8

- in
*Read:*- in
**[H]**: §§6.1—6.5

- in

**Spring Break!**No classes, of course.

- Content this week:
*M:*Starting non-Eucliean (actually, 2-dimensional hyperbolic) geometry. Points and lines in the Poincaré disk. Starting the hyperbolic distance.*W:*More on the hyperbolic distance. The upper half-space model of hyperbolic geometry.*F:*Basic results in hyperbolic geometry. Triangles, ideal triangles, angles, areas....

*Read:*- in
**[H]**: §§7.1—7.3

- in
*Exercises:*- None, but start working on Midterm II, which is due next Wednesday.
- Also, start thinking about your course project, information about which can be found here.

- Content this week:
*M:*More hyperbolic geometry, in the unit disk and upper half-space models. Overview of some possible final project topics.*W:*More on the hyperbolic distance. Omega points, triangles.*F:*Saccheri and Lambert quadrilaters.

*Read:*- in
**[H]**: §7.4 & §7.5

- in
**Midterm II is due Wednesday!**

- Content this week:
*M:*Some post-Midterm II discussion. Outline of the big picture towards the goal of "triangle angle sums are always < 180". More on Saccheri and Lambert quadrilaterals.*W:*Finishing the "angle sums in hyperbolic geometry" result. Other interesting results for hyperbolic triangles.*F:*Starting hyperbolic isometries.

*Read:*- in
**[H]**: §7.4 & §7.5, start chapter 8

- in
*due Friday:*- in
**[H]**: 7.4.2 - revisions of your solutions for Midterm II, if you so choose.

- in
- Make sure you have talked to me
**by Wednesday at the latest**about your final project. The goal this week is to have a specific project topic and a good reference (or a few references, or an idea of where to go to find these references). By the end of the week, it should be clear what will be the scope and content of the introductory section of your final project paper, and most of an idea of what the main capstone result will be.

- Content this week:
*M:*Hyperbolic isometries:*parabolic*,*elliptic*, and*hyperbolic*.*W:*Hyperbolic isometries: together they form the group*SL*. Using isometries in various geometries to build 2-dimensional surfaces, so: the 2-sphere has spherical geometry; the torus has (or can be given -- this is not the usual geometry of a torus floating in 3-space) flat, Euclidean geometry; tori with two or more holes can be given hyperbolic geometry._{2}(**R**)*F:*More on building hyperbolic, many-holed tori: constructions with**Geometry Explorer**to build a*4n*-sided polygon and*2n*hyperbolic isometries (all*hyperbolic*) which define*gluing instructions*to build an*n*-holed torus out of the interior of the polygon.

*Purely mental homework:*- Try to visualize how an octagon with the side identifications we
described in class represents the surface of a
*two-holed torus*. Conversely, try to visualize how cutting open a two-holed torus, by starting at one particular point and cutting four times around each of the non-trivial loops on the surface, and then laying the surface flat, yields an octagon with certain implicit side identifications.

- Try to visualize how an octagon with the side identifications we
described in class represents the surface of a
*Read:*- in
**[H]**: chapter 8

- in
- Please keep in regular contact with me this week (at least once,
preferably more often!) about the progress of your final project.
It is important that we have agreed upon the scope of that project
and have also discussed how to deal with any potential difficulties
which may have arisen (
*e.g.,*if you are having trouble finding good sources I can likely help,*etc.*).

**Exam week**, no classes.- As discussed in class, we are using our final exam time slot, which is
**Thursday, May 1st, 2008, from 10:30am-12:50pm in our usual classroom**, for students to give their presentations of final projects.