MAT 360 Geometric Structures, Spring 2013.

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.

Week 12 (05/06 – 05/10) Read lecture notes on the Lobachevskian plane

Week 11 (04/29 – 05/03) Read lecture notes on inversions (written by Oleg Viro, with changes and additions by Olga Plamenevskaya).

Homework 11, due Thursday, May 9: pdf

Week 10 (04/15 – 04/19) In addition to the lecture notes, read sections 172-174, 194-197, 181-182.

Homework 10, due TUESDAY, Apr 23: pdf

Week 9 (04/08 – 04/12) Read sections 93-97, 143-153, 159, and the lecture notes on similarity. (Please refer to the lecture notes for the notion of similarity, the approach in the textbook is different.)

Homework 9, due Apr 18: pdf

Week 8 (04/01 – 04/05) Homework 8, due Apr 11: pdf

Week 7 (03/25 – 03/29) We started the second part of the course, transformaions of the plane.

Lecture notes (written by Oleg Viro in 2010, with changes and additions by Olga Plamenevskaya, 2011, 2013)

Lecture notes are made available because this part is not in the texbook. The notes briefly recap the material, they don't replace the actual lectures!

Read sections 88-91 in the texbook (symmetries of the quadrilaterals). Begin reading the lecture notes for the material covered in class.

Homework 7, due Apr 4: pdf

Week 6 (03/11 – 03/15) Read sections listed in the homework sheet. This is a substantial reading assignment, please take it seriously and read carefully.
Homework 6, due Mar 28: pdf

Week 5 (02/24 – 03/1) Read sections 84-92. (Some of these weren't covered in class, but all are useful and should be read carefully.)
Homework 5, due Mar 5 (TUESDAY): Prove that a quadrilateral whose opposite angles are congruent is a parallelogram. Also do 191, 194, 211ab from the textbook. There are many ways to solve 194, you can chase congruent angles and triangles or exploit symmetries (or combine both). This homework is shorter and is due TUESDAY (as opposed to the usual Thursday) because of the Thursday midterm.

Week 4 (02/17 – 02/22) Read sections 56-69, 9. (We haven't done 68 in class, but please read it, it's a good extra example.)
Homework 4, due Feb 28: pdf I think this one is also on the hard side. It's always a good idea to start early.

Week 3 (02/10 – 02/15) Read sections 23-24, 48-49, 51-55.
Homework 3, due Feb 21: pdf This is a harder one, so start early! Some of the trickier questions are on applications of the triangle inequality.

Week 2 (01/4 – 02/08) Read sections 39--47.

Homework 2, due Feb 14
From the textbook: 72, 76, 79, 81, 82, 84, 85. 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 (01/28 – 02/01) Read sections 1-8, 13-16, 21-22, 31-35. We will 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 Feb 7
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.