Northern Illinois University

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.

- 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.

- 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!"?

- Give precise meanings to such innocent little words as
- 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.

- Some advanced topics:
- 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

- Office: Watson Hall 355
- Email: richard@math.niu.edu
- Phone: 753-6753
- Office Hours: Mon, Wed 3-4 and Mon 1-2

**HANDOUTS FROM THE TEXT:**
The text

- Math 302 Introduction to Geometry: Lecture Notes
- by Richard Blecksmith

NOTE: Each module is in PDF format, so you will need Adobe's Acrobat Reader which is a free and useful download. Click on the above link to get the latest version of the Acrobat Reader.

If you like, I can supply a paper punch to use these notes in a 3-ring binder.

The other sections and modules will become available as we get to them in class.

- Overview (Handed out in class)
- Module 1. Infinity
- Module 2. Logic
- Module 3. Topology
- Module 4. Foundation for Geometry
- Module 5. Topics in Geometry
- Module 6. Regular Polygons and Circles
- Module 7. Non-Euclidean Geometry
- Module 8. Constructions

**SAMPLE EXAMS:**
I will try to make some old exams available to you.

**SPRING BREAK:**
Spring Break is from March 10 through March 18, immediately after
the last day to withdraw from the course, Friday March 9.

**FINAL EXAM:**
The Final Exam is scheduled for 4:00 - 5:50 p.m., Wednesday, May 14, 2008.
(Note the new date since final exams have been moved back by a week.)
The room is probably the regular classroom, DU 348. If it changes,
you will be told at the end of the semester.

** About the text:**
In a way this text will be one of the easiest math books you've
ever read. Many of the sections are written in dialogue form,
with a minimum of formulas and a lot of space spent on simple
explanations. Let's face it, most math students don't
read the textbook, except for the exercises and possibly the
examples - to be read only when you get stuck on the exercises.
Hopefully this book is different. Try reading it and see.

In another way this text will be one of the most challenging
math books you might have ever been exposed to. It expects you
to **think** and thinking - especially in mathematics - is
hard work. But it can also be fun. That's why the cross-word puzzle
is often found near the comics in the newspaper.
Thinking isn't so bad if you believe that what you're
thinking about is worth the effort.
The difference between mathematicians and the rest of the world
is that mathematicians actually **like** thinking about math.
It may surprise you to discover that mathematics is not as
boring and humdrum as you might believe. Many people who dislike
the subject can still get interested in topics such as
infinity, chaos theory, space-filling curves, and logical
paradoxes. In this course you will discover that not
all infinities are the same; that you can have curves with
a finite area, but an infinite perimeter; that you can
reorder the points on the number line so that they do not
take up any length.

One warning: Not all the questions posed in the text are answered for you; not all the theorems and facts are proved for you. You will have to work many of them out yourself - possibly with the help of your instructor or classmates. This is not your typical "how-to" course. Your job, as a student, should not be rote utilization of formulas, but rather reasoning, thinking, and possibly discovering.

**Advice about the course:**
Perhaps the single most important factor in your success
in this course is your ** study habits **.
Think of learning math as "working out" in the gym.
Study at least 3 times per week; do not wait until the day before the exam.
Learn mathematics like you would learn a language.
Work on the concepts until they make sense.
Don't just memorize facts and then forget them a few weeks later.
Master as many homework problems as you can - beyond just getting a
correct answer.
** Always come to class! **
While you're there, listen, think, and ask questions.

Last update: Feb 4, 2008