Geometry

Math 342 - Spring 2006

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Final Exam -Tuesday, May 16 at 11:00 am

 

Topics for the final exam

I have finished making the final, and I think that this topic guide will be very helpful.

 

I plan a 2-part exam, a shorter closed-book part followed by a longer open-book, open notes part.  You would start with the closed-book part, and when you hand that in, you’ll get the open-book part.

 

 

On the closed-book part:

 

 

1.         Expect a proof or two based on sets of axioms.  (Don’t panic; I’ll give you the axioms.)  For example, I might give you the four Incidence Axioms (p. 28) and ask you to prove Incidence Theorem 1.

 

2.         Expect me to ask for a couple of models for sets of axioms. For example, I might give you the four Incidence Axioms and ask you for a finite geometry (alt. infinite geometry) that satisfies the axioms.  Good answers would be those given in Examples 1.4.1 to 1.4.4 on pages 28 and 29.

 

3.         History of axiom systems

I think that list at the top of page 71 of the four axioms systems for Euclidean Geometry is really important.  The actual axioms themselves aren’t worth memorizing, but the nature and character of each system is important.

 

 

On the open-book part:  You may use your books and notes. 

 

 

 Student presentations

A thread of students’ own summaries of their presentations is now available by following the “Discussions” link on the WebCT Course Page.  These are all in the “Main” thread.

            I plan to base about 15% of the final on these presentations, and I plan to take note of how well people do the questions on each of your presentations, so you have a real incentive to do a good job describing your presentation on WebCT.

 

 

2.            Translate some theorems proved in the book into two-column proofs, for example Theorem 3.3.5, the Isosceles Triangle Theorem on pages 91 and 92.

 

3.         Use a given axiom system to prove theorems.  For example, I might ask you to use the Hilbert axioms to prove that any line contains infinitely many points, and then also use the Birkhoff axioms to prove the same thing.

 

4.            Properties of finite geometries

 

5.            Rectangles and triangles in hyperbolic geometry

 

Updated Saturday May 13 at 10:30 pm.

 

 

Semester Calendar

 

Week

M

T

W

Th

F

 

 

January

  

1

 

1/24 - Introduction - Intro to axiomatic systems

 

1/26 - The Early History of geometry

Ways of knowing, deduction

 

 

February

  

2

 

1/31 - More modern history, Euclid's postulates, Euclid's 5th postulate

 

2/2 - Rules of inference, particular axiomatic systems, some theorems, models

3

 

2/7 - WebCT quiz 1-1 due

 

2/9 - WebCT quizzes 1-2 and 1-2b due

4

 

2/14 -

 

2/16 -

5

 

2/21 -

 

2/23 -

6

 

2/28 -

 

3/2 - Test, Chapter 1

 

 

March

  

7

 

3/7 - Chapter 2 - Neutral geometry

 

10/13 - hyperbolic geometry

8

 

3/14 -

 

10/20 -

   

Spring

  

Break

9

 

3/28 -

 

3/30 - Dianne Sullivan on models of hyperbolic geometry

 

 

April

  

10

 

4/4 - Theorems of hyperbolic geometry - Sign up for presentation times

 

4/6

11

 

4/11 -

 

4/13

12

 

4/18 - 

 

4/20 - Department Assessment exam

13

 

4/25 - Projects

4 - Kathleen Keller - Saccheri

5 - John Galvao - Alfabeto

6 - Tim Luchsinger - Beltrami

7 - Kevin Geogheagan - Fermat

 

4/27 - Projects

8 - Frances Johnson - Young

9 - Caitlin Baird - Lobachevsky

10 - Mat O'Grady - Pascal

11 - Kristin Petersen - Birkhoff

 

 

May

  

14

 

5/2 - Projects

12 - Kelly Williamson - van Schooten

13 - Danielle Kunz - Lagrange

14 - Brittany Davis - Hilbert

15 - Antoine Billy - Gino Fano

 

5/4 -

16 - Diana Moore - Gergonne

17 - Victoria Steele - Poncelet

18 - Andrew Brown - Hull, etc.

19

15

 

5/9 - Projects

20 - Mike Lazzaro - Coxeter

21 - Matt Buchta - Steiner

22 - Linda Scoralick - Euler

23

 

5/11 - Reading Day

Make-up day. 8:00 am  Make all arrangements by 5/4

Exams

week

  Final Exam -Tuesday, May 16 at 11:00 am

 

  

 

 

  Clinic Opens !!

 

Be there!  Take a friend!

 

calendar updated May 11, 2005

 

Hit Counter

 

The reward of a thing well done is to have done it.

 - Jonas Salk

 

 

January 30 - Syllabus has been posted.  Follow this link: Syllabus

 

January 30 - Three WebCT quizzes have been posted.  See the Semester Calendar below for due dates.

 

List of modern (since 1600) geometers (in alphabetical order)

 

Link to MacTutor

 

  1. Beltrami - Tim Luchsinger

  2. Birkhoff, G. D. - Kristin Petersen

  3. Bolyai

  4. Coxeter, HSM - Mike Lazzarp

  5. Desargues

  6. Descartes (1596-1650) - Kevin Geoghegan

  7. Euler (1707-1783) - Linda Scoralick

  8. Fano, Gino (1871-1952) - Antoine Billy

  9. Fermat (1605-1665)

  10. Gauss (1777-1855)

  11. Gergonne - Dianna Moore

  12. Heath , Sir Thomas Little, FRS (1861-1940)

  13. Hilbert - Brittany Davis

  14. Klein

  15. L’Huilier

  16. Lagrange

  17. Lambert - Danielle Kunz

  18. LeGendre

  19. Lobachevsky - Caitlin Baird

  20. Pascal - Mat O'Grady

  21. Pasch

  22. Poincaré - Gary Treadwell

  23. Poncelet (1788-1867)

  24. Riemann

  25. Saccheri- Kathleen Keller

  26. Steiner - Matt Buchta

  27. van Schooten - Kelly Williamson

  28. Young - Frances Johnson

  29. Brazilian Mathematics - John Galvao

 

Nobody’s heard of my young friend

Tom Hull - Andrew Brown

Folding Flat theorem

Department Assessment Test is tentatively scheduled for 8am on Thursday, April 20 and Friday, April 21.

 

Stroke Update:

 

My Mother has moved to a rehabilitation center, where they expect she will stay for ten to 14 days.  I am going to go to Dallas to be there when she gets out, and to stay for a few days after to help her move home and to help my Dad.

So, I am leaving on Friday, April 21 and returning to Danbury on Monday, May 1.

 

I will miss two days of class, April 25 and April 27.  Dr. Burns will take videos of the project presentations on those days, and, if there is time, talk about Saccheri quadrilaterals and Lambert quadrilaterals.  I will get to watch the presentations when I get back.  It won't be the same, but it will be the next best thing.

 

 

About that project

...

At the bottom of this page, and also on the Syllabus, there's a list of "modern" geometers.  You can pick anybody on this list.  If you find somebody not on the list, ask and it's probably OK.  

 

Your project should include:

  1. A short biography of your geometer.  Google "MacTutor" for a great resource.  Be careful not to plagiarize; I've read all this stuff.

  2. Find some theorems that your geometer proved or discovered.

  3. Prove one of them.

  4. Do a 10-minute presentation in class.

The whole thing should be about five pages long.  

Presentations begin April 18.