MATH 302 Spring 2013
DuSable Hall 302
Monday, Wednesday 4 - 5:15
Richard Blecksmith

INTRODUCTION TO GEOMETRY (3 semester hours)

Basic concepts in plane and solid geometry, measurement, congruence and similarity, constructions, coordinate geometry, transformations, tessellations, topology, and selected topics.

Not used in major or minor GPA calculations for mathematical sciences majors or minors. It is a required course for the Minor in Mathematics Education.

PREREQUISITE: MATH 229 or consent of department.

COURSE OBJECTIVES:

• To give precise definitions:
  • What are points and lines?
  • What is distance?
  • Can two points on a line touch each other?
  • Given three points on a line, how do we know which one is in the middle?
  • What are angles?
  • What is area?
• To state and prove the really famous theorems of geometry whose truth we have come to accept as a matter of faith:
  • the three angles of any triangle add to 180 degree
  • the well loved Pythagorean Theorem.
  • Why does the area of a triangle equal one half base times height? Why don't you get three different answers, depending on which side you use for the base?
  • Why does the area of a circle equal pi times radius squared?
  • What is $\pi$ anyway?
• To increase the students' awareness and appreciation of the many diverse areas of mathematics related to geometry which are not usually encountered in high school or even undergraduate math courses.
  Besides the standard topics of elementary geometry-such as triangles, polygons, circles, congruence, similarity, area-we discuss topics in
  • analysis
  • logic
  • topology.
  Since lines are such an important part of geometry, we spend a lot of time trying to understand the prototype of all lines, namely the real number line.
• More generally, to gain an understanding and appreciation of
  • reasoning in general and mathematical reasoning in particular
  • the logical relationships among various geometric concepts
  • how to construct, analyze, and place in context specific examples and facts
  • how to infer general conjectures from the study of examples
  • how to articulate and communicate problem solutions, verbally and in writing
  • the use of abstraction in mathematics
  • connections between geometry and art.
  • to have some fun.

SYLLABUS: The course will cover most of Modules 1-8 of the text.

• Module 1. Infinity
  The course begins with the question "Does $.999\cdots$ equal 1?"
  • We then launch into a discussion of infinity.
  • The concept of infinity intrigues many people. It is mysterious.
  • Car companies are named after it. (You never hear of a car named "Square Root" or "Logarithm."
  • Can we compare infinite sets?
  • If you got a dollar for every positive integer and I got a dollar for every point on the real number line, who would be richer?
  • Or are infinity-aires all equally wealthy?
• Module 2. Logic
  • Give precise meanings to such innocent little words as
    • and
    • or
    • implies.
  • Does an "or" question allow both alternatives to be true?
    The answer is "no" if you are offered coffee or tea, but "yes" if you are offered cream or sugar.
    What about the statement "Give me liberty or give me death!"?
• Module 3. Topology.
  • Topology, sometimes called "rubber-sheet geometry" allows us to bend, stretch, shrink, and twist topological objects, but we are not allowed to tear, cut, or glue them.
  • We begin with a precise definition of distance, which we use to define other very basic concepts such as
    • inside
    • outside
    • boundary
    • and connectedness.
  • Many of these ideas enable you to use the logic you just learned in the previous section and to play mathematician in a simple and safe environment.
• Module 4. A Foundation for Geometry
  • We define what points and lines are by specifying how they behave.
  • In geometry rules are called axioms.
  • One axiom, for example, specifies that a unique line passes through any pair of distinct points.
  • At the root of our system of geometry lie some very basic concepts:
    • What is a triangle? Answer: the three segments joining three non-collinear points.
    • What is a segment? Answer: the points lying between two endpoints.
    • Aha, we have arrived at the basic underlying concept: betweenness.
    • What does it mean to say that one point is between two others?
    • Similarly, what is an angle?
    • How do we measure angles?
    • How do we measure distance?
  • All these concepts are fundamental to laying down a solid foundation for geometry.
  • Some standard topics built on our geometry foundation:
    • dimension
    • angles and how we measure them
    • triangles
    • congruence
    • isosceles triangles
    • perpendiculars
    • parallels
• Module 5. Topics in Geometry
  • Some advanced topics:
    • polygons
    • area
    • similarity.
  • We spend some time with the Pythagorean Theorem
  • We extend the number of right triangles we know with integer sides from two examples (3-4-5 and 5-12-13) to infinitely many.
• Module 6. Regular Polygons and Circles
  • All the "pi stuff" comes in here.
  • What are tangents and inscribed angles?
• Module 7. Non-Euclidean Geometry
  • Do parallel lines exist?
  • If extended indefinitely, do the two rails of a railroad track ever come together?
  • We can, like the Greeks, create an axiom which says they never meet, but mathematicians for centuries thought that such a statement should really follow as a theorem from the other, simpler axioms.
  • With the discovery of non-Euclidean geometries we know now that the obvious and intuitive first impression is not always the truth.
  • Geometries can exist which have no parallel lines or else lots more than expected.
  • The Greeks would have never dreamed how weird it can get.
• Module 8. Constructions with Straight-edge and Compass
  • Another problem that perplexed the Greeks and future geometers involves constructions with a compass (to draw circles) and a straightedge (to draw lines) - the so called "instruments of math construction."
  • It is child's play to bisect an angle using a straightedge and compass.
  • Try trisecting an angle using the same two tools.
  • Bet you can't.
  • It turns out that nobody can trisect an angle with just a compass and a straightedge.
  • If you stick with the course you'll find out why.

WITHDRAWAL: The last day for undergraduates to withdraw from a full-session course is Friday, March 7.

GRADING: Grades will be assigned on the basis of 500 points, as follows:

2 hour exams worth 100 points each

Homework, attendance, and writing assignments: 150 points total

Total: 350 points

TIME and PLACE: