Math 510, Introduction to the Foundations of Geometry
(7:25-8:40 MW in C-8)


Syllabus: Here is the syllabus.

Online resources:
 Homework:

Assigned
 Due date Read
 Exercises
8/30
Syllabus,
Introduction, Ch 1, pp. 1-23
 Study your syllabus. There will be a quiz on it on Wed, 9/8.
9/1
 9/8
 Ch 1, pp. 23-34
 pp. 43-46, 1.1, 2, 4, 7
Turn in on 9/8: 1.6
9/8
 9/15
 Ch 1, pp. 29-40
 pp. 43-46, 1.3, 8-10
Turn in on 9/13: 1.12
Turn in on 9/15: Given a line segment of length a, construct (with straight edge and compass) a line segment whose length is the square root of a.
9/15
 9/20
 Ch 1, pp. 38-40
Ch 2, pp. 53-60
 Turn in on 9/20: Prove that the method I showed you in class to construct the perpendicular bisector of a line segment does indeed construct a perpendicular bisector. I.e. show that the line CD is a perpendicular bisector of the line segment AB in this diagram, if AC and BC are congruent and AD and BD are congruent.
9/20
 9/27
 Ch 2, pp. 53-66
 pp. 43-46, 1.13-15
p. 48, Major Exercise 1
Turn in on 9/27: 1.16
9/27
 10/4
 Ch 2, pp. 67-71
 p. 48, Major Exercises 2, 3
pp. 91-92, 2.2, 4
Turn in on 9/29: p. 91, 2.3
Turn in on 10/4: Prove Prop 2.4 (p. 71).
10/4
 10/11
 Ch 2, pp. 72-79 Homework holiday. Prepare for your upcoming exam. New HW will be assigned next week.
10/11
 10/18
 Ch 2, pp. 79-83
 pp. 91-93, 2.6-9, 11
Turn in on 10/18: 2.8 on p. 92
10/18
 10/25
 Ch 2, pp. 84-91
 pp. 93-95. 2.10, 13, 14, 19
Turn in on 10/20: 2.12ab on p. 93
Turn in on 10/25: 2.12cd on p. 93. If you have already turned in c (because you have the 2nd printing of the text), do b and d.

Here is the complete exercise 12 for those of you with the 2nd printing of the text:

(a) Let l, m, n be lines in an affine plain. Show that if l || m and m || n and l is not the same as n, then l || n. (This is called the transitivity of parallelism.)
(b) Why must we assume that l and n are not the same in defining this property?
(c) Show that, conversely, any model of incidence geometry with this property must be an affine plain.
(d) Exhibit a model of incidence geometry in which parallel lines exist but parallelism is not transitive.
10/25
 11/3
 Ch 2, pp. 84-91
 pp. 94-95, 2.15, 16a, 18
No exercises to turn in this time, but be sure to work your way through Examples 8 and 9 in Chapter 2.
11/3

No new exercises. But clean up your lecture notes. These will form the foundations of your discovery notes. You don't want to build on weak foundations, do you. They should look something like this.
11/8
 11/17
Complete your discovery notes through what we called Theorem 3 in class. The proof of Theorem 3 is homework and is due on 11/17.
11/10
 11/17
No new homework. Prepare for your upcoming exam. And note the skeletal lecture notes posted at the top of this page. You can use them as the basis of your (more detailed and complete with all proofs) notes.
11/17
 12/1
Prove Theorems 5-7 in the notes.
11/22
 12/1
Prove Theorems 11, 12, 14-16 in the notes.
11/29
 12/6
Prove Theorems 18 and 19 in the notes. Construct a few examples of lines and points in Rn, and test whether the points lie on the lines.
12/1
 12/6
Prove that Axiom 5 holds in our interpretation of geometry in Rn.
12/6
 12/14
Finish the proofs of Axioms 7-9 and 12-14 in our interpretation of geometry in Rn.

The Putnam Competition: This is not strictly related to this course. The William Lowell Putnam Mathematical Competition will be held on Sat, Dec 4 this year. It is administered locally at the Calexico campus. The exercises primarily test your ability to construct rigorous mathematical arguments to solve (difficult) unfamiliar problems and not lexical knowledge. If you like proofs and enjoy the challenge of thinking about such problems, I encourage you to sign up for the competition by contacting me by Oct 7. Registration must reach the organizers by Oct 14. Here is an archive of past exercises and solutions.

Discovery notes: During part of this course, you compile your own notes from the work we do in class and the work you do for the homework. You will use your own notes to prepare for the exams, so it is in your best interest to keep them neat, organized, and above all complete and correct. To start you off and give you an idea of what is expected of you, here is the first page of my notes with some of our first definitions, axioms, theorems, and proofs. I recommend that you use your computer to keep your notes. This will make it much easier to edit and revise them as you grow wiser about axiomatic geometry. The question is what software to use. While it's possible to use MS Word, typing mathematical formulas in Word is slow, troublesome, and the possibilities are quite limited. Also, the output is ugly, although this is probably the least of your concerns now. You are welcome to stick with Word, but I do have a better suggestion for you.

Most mathematicians (and physicists and computer scientists) use LaTeX. The basic principle of Latex (it should be LaTeX, but that's annoying to type) is quite different from MS Word. In Word, you see your document exactly the way it will look when you print it. You format it yourself. In Latex, you edit the content, but leave a lot of the formatting to the computer. If you have edited an internet page directly in html, you know what I mean. If not, never mind, you'll see it shortly. The disadvantage of Latex is that it takes some time to learn how to use it, the advantage is that once you've gained some experience with it, it makes it very convenient to typeset mathematics. It produces professional quality output too. In fact, this is the same software publishers use to print math textbooks and journals. Since Latex is the gold standard in typesetting mathematics, if you are planning a career in math, you might as well get started on learning it. Even if your career plans revolve around teaching math in high school, I don't think you'll regret learning Latex. It's easy enough to find a free implementation of Latex for your computer. In fact, if you are running Linux, you most likely already have Latex on your computer. But you're probably running Windows. No problem, just keep reading.

You can get a start by looking at the source code of the beginning of my notes. In fact, you may want to start your own lecture notes by editing this file. You can use any text editor, such as Notepad on Windows, to view and edit it. If you aren't ready to take the plunge, and would like a more gentle transition to Latex, you may want to check out LyX (free) or Scientific Workplace (commercial) below, which let you edit your documents a little closer to WYSIWYG style and produce the Latex code for you. Here are some links:
Exam solutions:
 Educational links:
Math links: The links below lead to sites with encyclopedias of math terms.  You can use them to find definitions, examples, and some theorems.
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