MAT 360 Geometric Structures, Fall 2011.

The main goal of the course will be to study the foundations of the Euclidean geometry. We will learn how to work with geometric images and how to construct careful mathematic arguments. The first part of the course will be about the basics of plane geometry: lines and points, angles, congruent triangles, quadrilaterals (parallelograms, rhombi, rectangles, etc), circles, and their properties, as well as constructions with compass and straightedge and other related topics. The second part of the course will be about "isometries" (i.e. symmetries, rotations, etc) and homotheties of the plane. We will also discuss similar figures here. Then, we will talk about plane transformations that play an important role in Lobachevskian geometry, and will make a connection to non-Euclidean geometry later in the course.

As this is an upper-level class, familiarity with proofs is expected. Indeed, one of the goals of this class is to use your MAT 200 skills to build the theory and logic of the plane geometry. Much of the material from the first part of the course will be somewhat familiar from high school, but we will take it to a very different level. You will be required to think like a mathematician, and to write careful proofs in your homework. The second part of the course will be new to most people, and somewhat more abstract.

Read Lecture notes on non-Euclidean geometry.

Week 12 (11/28 - 12/2)
Read Lecture notes on inversions Theorems 1.A and 1.B were not covered in class and are optional; the rest of material is mandatory.
Homework 12, due Tuesday, Dec 9: pdf

Week 12 (11/14 - 11/18)
Read sections 170-174, 181-183, 194-198 of the textbook; review the similarity lecture notes.
Homework 11, due Tuesday, Nov 22: pdf

Week 11 (11/7 - 11/11)
Read Lecture notes on similarity. Please refer to these as the textbook's approach is different.
Homework 10, due Nov 17: pdf

Week 10 (10/31 - 11/4)
Read sections 143-155 of the textbook; start reading the similarity lecture notes.
Homework 9, due Nov 10: pdf

Week 9 (10/24 - 10/28)
Read Lecture notes on isometries. (These notes were written by Prof. Oleg Viro in Spring 2010, with some changes and additions by Olga Plamenevskaya, 2011.)
Homework 8, due Nov 3: pdf

Week 8 (10/17 - 10/21)
Read sections 88-91 (we covered quadrilaterals before; this time, focus on their symmetries), 121, 99-101; start looking at the isometries notes (see link above).
Homework 7, due Oct 27: pdf

Week 7 (10/10 - 10/14)
Read sections 122-126, 127, 129, 132, 133. We haven't covered section 125 in class, but please read it, you'll need it for one of the homework problems (the section is quite similar to what we did discuss).
Homework 6, due Oct 20
From the textbook: 227, 231, 237, 275, 301, 306. Doing homework *before* the exam would give you some extra practice.

Week 6 (10/3 - 10/7) Read sections 84-97, 105, 111, 112-114.
Homework 5, due Oct 13
From the textbook: 178, 179, 180, 205, 239, 245. As always, remember to prove the if-and-only-if statement in the geometric locus problem, justify your constructions in the construction problems, etc.

Weeks 4-5 (09/19 – 09/28) Read sections 56-83.
Homework 4, due Oct 6
From the textbook: 109, 121, 124, 133, 157, 158, 166, 168. It's longer because of the break.

Week 3 (09/12 – 09/16) Read sections 31, 44-49, 51-55.
Homework 3, due Sept 22: pdf This is a harder one, so start early! Most questions are on applications of the triangle inequality.

Week 2 (09/6 – 09/9) Read sections 23-24, 39-43.
Homework 2, due Sep 15
From the textbook: 72, 76, 77, 79, 81, 82. Please give detailed proofs/explanations as appropriate. In particular, justify your example in 81 (i.e. explain why the corresponding elements in your triangles are congruent and why the triangles are non-congruent; do not just draw a picture).

Week 1 (08/31 – 09/02) Read sections 1-8, 13-16, 21-22, 31-35. We have discussed most of this in class. If don't have the book yet, use the sample pages. The reading assignments are important because they teach you how to follow the proof carefully, and how to build your own proofs and to use correct notation.
Homework 1, due Sep 8
In all the sections from the reading assignment, find all the instances where properties of isometries of the plane are implicitly used.
One of the properties is stated explicitly on page 2, lines 2-5: (i) One can superimpose a plane on itself or any other plane in a way that takes one given point to any other given point...
Other properties were given in the first lecture:
(ii) One can superimpose a plane on itself or any other plane in a way that takes one given ray to any other given ray.
(iii) A plane can be superimposed on itself keeping all the points of a given straight line fixed. This "flip" can be done in a unique way.
In other words,
(ii) There exists an isometry which maps a plane onto itself or any other plane in a way that takes one given ray to any other given ray.
(iii) There exists a unique non-identity isometry of a plane onto itself keeping all the points of a given straight line fixed.

Your task is to list all the places in the sections 1-8, 13-16, 21-22, 31-35 of the textbook where these properties are used implicitly, ie without explicit statement or reference. Present your solution as a table with rows:
page number, line number, the property, how the property is used.
(Example: p. 27, line 2, property (iii) is used to "fold" the diagram along the line BD.)

Please also do the following questions from the textbook: 2, 40, 61, 67. In all questions except 61, please give a careful explanation/proof. (To refresh MAT 200 material, read sections 28-30.) In 67, try to use a "folding argument" as in section 35.

Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.