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Class meeting times in Morton 314:
Office hours for the current quarter are posted on my webpage.
The course information on this page mirrors the information posted on the course's Blackboard site. Students' grades are posted only on Blackboard.
This course is a continuation of the introduction to axiomatic geometry that began in Math 330A. We will begin with a unit on triangles. Then, for a few weeks after that, we will leave axiomatic geometry and study maps of the Cartesian plane, including some discussion of linear transformations and their representation in matrices. But the real motivation for this diversion is to develop a facility with the terminology of isometries in order to return to axiomatic geometry and study the distance and ruler postulates. The final portion of the course will be a unit on symmetry groups.
Course Information Sheet (link)
Syllabus (link)
Homework List (link)
Each week, the students receive a photocopied packet of notes containing reading material, reference material (such as definitions and statements of theorems that will be discussed in class), and homework exercises. The weekly packets are designed to be stapled together, so that as the quarter progresses, a book-like thing will emerge.
That book-like thing is available here, as it develops, by clicking on the links below. The material is being written by myself (Mark Barsamian), but the sequence of theorems in the first two sections of Unit 1 is taken from material originally produced by Barbara Grover and Jeff Connor. (In their material, the unit was entitled Similarity.) I'm afraid that because of the haste in which the material is produced, it may contain horrific typos and damning conceptual lapses. Please e-mail me any comments.
The online version of the notes is extensively hyper-linked. In the table of contents, the entries and the page numbers are active links. In the body of the document, almost every reference to a numbered definition, theorem, or proof step is an active link. Clicking on the reference will take you back to the thing being cited. Once you're there, if you want to get back to where you were previously, good luck. If you are reading the document using Internet Explorer on a PC, pressing the Alt and left arrow keys on your keyboard simultaneously will work. (Pressing the back button on the web browser usually doesn't work, because it simply brings you back to this page.) If you are reading the document using Adobe Acrobat Reader, then make sure that you have the Navigation Toolbar visible. It has a back button that does work as you would hope.
Unit 1: Triangles (link)
In the first unit of Math 330B, we will study triangles. Our study will be organized into three parts, each taking about a week of class time. The first part introduces the concept of triangle similarity and presents several important Euclidean geometry theorems about similarity. The second part applies ideas about area to re-prove some of the theorems from part 1. Then, some of the same styles of proof are used to prove a Generalized Pythagorean Theorem for Euclidean geometry. In the third part, we will study trigonometry. In particular, we will compare different versions of the trigonometric functions.
Unit 2: Transformations and the Ruler Postulate (link)
In the lists of Euclidean geometry postulates found in most high school geometry textbooks, there are two postulates having to do with measuring distance. One postulate has to do with measuring distance between any two points in the plane. The other postulate has to do with coordinate functions on lines, the so-called ruler postulate. Being aimed at high school students, the wording of the postulates does not use specialized mathematical terminology. But to really understand the meaning and significance of the postulates, it helps to have some facility with more specialized terminology and to be familiar with some related mathematical concepts. In this unit, we will study some of the basic concepts and terminology of transformations of the plane, and then go on to see how these tools will allow us to better understand the distance and ruler postulates.
Unit 3: Symmetry (link)
Most of us probably first encountered the word symmetric sometime in grade school or junior high. At the time, it may have been used as a name for objects or pictures that look as if one side is a reversed copy of the other - a sort of mirror image. This kind of symmetry is sometimes called bilateral symmetry. Later, we learned that there are other kinds of symmetry, as well. For instance, the patterns on wallpaper are periodic in both the horizontal and vertical direction. This is called translational symmetry. Some objects or pictures have radial symmetry, meaning that the pattern is the same along any ray from the origin. Some have rotational symmetry, meaning that the pattern may not be identical along different rays from the origin, but the pattern does repeat in the sense that there is some angle of rotation about the origin that will cause the rotated pattern to lie on top of the original.
It is easy to see symmetry in pictures and objects. It is harder to describe symmetry in purely abstract mathematical terms, without reference to pictures. But such a description is worth the effort. It turns out that many powerful abstract mathematical concepts can be applied to an analysis of symmetry. These concepts give us a deeper understanding of symmetry and also a richer, more precise vocabulary for describing it. And conversely, by developing an abstract mathematical notion of symmetry, without reference to pictures, we become poised to find symmetry in abstract settings where there is no picture.
This unit will consider symmetry of sets in the Cartesian plane from an abstract mathematical viewpoint. The discussion in Chapter 1 begins by defining symmetry in terms of isometries of the plane. (Students will be familiar with the concept of an isometry from Unit 2.) Chapter 2 introduces the notion of a group, a concept from abstract albegra. In Chapter 3, we will prove a theorem claiming that the set of symmetries of a set of points in the plane is a group. The theorem and its proof articulate the relationship between a geometric concept (symmetry) and an algebraic concept (group). In Chapter 4, we will explore some basic groups, learning along the way some new terminology that will be useful when we return to discussing symmetry groups. Chapter 5 is the return to symmetry groups. The Dihedral group is introduced and investigated.
Homework Solutions have been removed from this website.
Midterm #1 was given in class on Thursday, April 13. It covered the material from Unit 1: Triangles.
Midterm #2 was given in class on Wednesday, May 10. It covered the material from Unit 2: Transformations and the Ruler Postulate.
The Final exam was given on Monday, June 5. It covered the material from Unit 3: Symmetry.
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