#
Math 119: Ideas in Geometry

Spring 2010

** **__Instructor:__ Tom Cooney

** **__Classroom:__ 141 Altgeld Hall

** **__Class Time:__ 12 noon, Monday, Wednesday, Friday

** **__Office:__ 150 Altgeld Hall

** **__Office Hours:__ 10 am Monday, Wednesday, Friday, or by appointment. This may be changed after I hear what times suit the students in the class.

__Email:__
__ tcooney@math.uiuc.edu__

__Office Phone:__ (217) 333-3547. (Email preferred)

## Exams

There will be four in-class exams and one take-home exam. There will also be a final exam. The final exam will be held in 141 Altgeld Hall from 7pm to 10 pm on Wednesday, the 12th of May, 2010.
## Hour Exam 4: Friday, the 30th of April

## Take Home Exam: Handed out on Friday, 30th of April. Due Monday, 3rd of May

## Homework Assignment 4: Due Monday, 26th of April

## Hour Exam 3: Friday, the 2nd of April

## Homework Assignment 3: Due Monday, 29th of March

## Hour Exam 1: Friday, the 12th of February

This exam will cover Chapter 1. Homework assignment 1 will be due on the Monday, graded and returned on the Wednesday, and the exam will be on the Friday.
## Homework Assignment 1: Due Monday, the 8th of February at the start of class

## Hour Exam 2: Friday, the 5th of March

## Homework Assignment 2: Due Monday, the 1st of March at the start of class

As with the first exam, the homework is due on the Monday and will be returned on the Wednesday of the week of the exam. Wednesday's class will be review for the exam. The exam will be based on the material covered since Exam 1, which is mainly Chapter 2 of the Course Notes.

## Links and Handouts from Class

### Course Webpage links

Math 119 Webpage with
online course notes
Check your grades and discuss the course on Compass.
Tom Cooney's homepage
Course Webpage for Math 119 in Fall 2007
Course webpage for Math 119 from Spring 2008
Unofficial Guide to Homework and Exam Due dates Official confirmation of all dates will be given in class, on the webpage on Compass well in advance.
### Links mainly to do with Section 1.1

Hilbert's Axioms for Euclidean Geometry As we said in class, Euclid's Elements are basically error-free. However, if you want to be really careful and list out every one of the assumptions you are using, then you should include extra axioms like Side-Angle-Side and what it means for one point to be between two other points.
Euclid's Fifth Postulate and the nature of geometrical truth People are still arguing about Euclid's axioms 2,300 years later. (This is a philosophy thesis from 1995. I suspect that if the author was talking to a mathematician about this, they would each accuse each other of being wrong and missing the point of what the other is saying.)
Geogebra An online tool that you can use to help visualize and experiment with the ideas of Euclidean Geometry. Also keep this in mind for later in the semester when we will do some compass and straightedge constructions. You can use this online without downloading any files by clicking on Webstart and then Applet Start.
The Euclidean Toolbox An online tool to help you explore the world of euclidean geometry. See for yourself what theorems hold. It will also be useful later when we consider
compass and straightedge constructions.
Online tool to help you visualize Spherical Geometry There's a drop down menu where you can select whether you want to draw a point, or a line, or a circle, etc.. You can ``drag'' the picture around to see it from different angles and see what is going on.
Let me know if you have any questions.
Same as above but for Hyperbolic Geometry! One could argue that there are three different possibilities for how geometry can work out. This is the third one. Instead of working on a plane or a sphere, one can work on a saddle-shaped surface or, as in this link, on a disk where distances near the edge are much longer than the distances near the center. We will not discuss this at any length in class but feel free to come talk to me in office hours if you are curious about this one.
A picture showing the three possible shapes of the universe and whether the sum of the angles in a triangle is more than, less than, or equal to 180 degrees. The middle picture is "hyperbolic" geometry, on saddle-shaped surface. This comes from the Wikipedia article on the Shape of the Universe.
### Links mainly to do with Section 1.2

Picture of the conic sections showing how to obtain these curves from a cone and a plane.
### Links mainly to do with Section 1.3

An online Taxicab Treasure hunt game
Solutions to the Taxicab/City Geometry introductory worksheet
### Links mainly to do with Section 2.3

Tessellations Worksheet from February 19th
Solutions to Tessellations Worksheet
All possible regular (semiregular, demiregular) tessellations - very pretty!
Penrose tiling - tiles the plane, never repeats!
What is a tessellation?
Totally Tessellated
### Links mainly to do with Section 2.4

Our introduction to proof by picture worksheet
Solutions to Proof by Picture worksheet
A New York Times article/blog about Mathematics, which talks about a problem that we will solve in here. This one talks about what is "1+2+3+4+...+n?", triangular numbers and Gauss's trick. It also includes some suggestions for interesting reading about the entertaining side of mathematics.

### Links mainly to do with Section 3.1

Geogebra An online tool that you can use to help visualize and experiment with the ideas of Euclidean Geometry. Also keep this in mind for later in the semester when we will do some compass and straightedge constructions. You can use this online without installing any files by clicking on Applet Start.
The Euclidean Toolbox An online tool to help you explore the world of euclidean geometry. See for yourself what theorems hold. It will also be useful later when we consider
compass and straightedge constructions. §
If you are in one of the University's
computer labs, it should have at least one of the following computer programs:
(The Geometer's) Sketchpad, Dr Geo. You could also use one of these to experiment.
### Links mainly to do with Section 3.2

Monty Hall Problem Story from The New York Times The Monty Hall problem is still capable of tripping people up as this story shows. It also includes an online game where you can compare for yourself how switching and staying affect your chances of winning.
The Monty Hall problem is talked through in the New York Times with the help of Monty Hall himself!
### Links mainly to do with Chapter 4

Compass and Straightedge Constructions Worksheet
Compass and Straightedge Constructions Worksheet Solutions
Median and Opposite Angle Construction
### General Links (including links for later in the semester)

There is an interesting column/blog about Mathematics in the world around us at the BBC website. Click here. There is a dropdown menu in the top right giving you access to the complete list of topics. It frequently discusses how numbers are used to confuse and deceive us. Learn mathematics and defeat their evil plot to baffle and con us!
The New York Times also has an interesting blog/column about Mathematics. This one is more directly focussed on mathematics (what is a number) and may be of more interest to the future educators in our class than the above.
A mathematician appears on The Colbert Report!The more math classes you take in college, the more money you will make in your life (on average). No, most mathematicians cannot do arithmetic in their head like this. Yes, many mathematicians are a little odd.
If you are in one of the University's
computer labs, it should have at least one of the following computer programs:
(The Geometer's) Sketchpad, Dr Geo. All very interesting and all with the
potential to be very helpful. These could be especially useful when experimenting with compass and straightedge constructions.
Monty Hall Problem Story from The New York Times The Monty Hall problem is still capable of tripping people up as this story shows. It also includes an online game where you can compare for yourself how switching and staying affect your chances of winning.
All possible regular (semiregular, demiregular) tessellations - very pretty!
Penrose tiling - tiles the plane, never repeats!
What is a tessellation?
Totally Tessellated
Wikipedia
A good source for interesting and often also true mathematical tidbits. Try
searching for biographies of the mathematicians listed below.
Euclid, Erastothenes, Pythagoras, Heron, Brahmagupta, Descartes, Newton,
Euler
Lots of mathematical info from
Wolfram Research ,
one of the world leaders in mathematical computer software,
based right here in Urbana-Champaign.
Euclid's Elements: The book(s) that helped start it all.
The MacTutor
History of Mathematics archive : Mathematicians' biographies and more!
The Most Pleasing
Rectangle Web Poll Poke around the rest of Jim Loy's webpage
for some more interesting stuff.
Demetri Martin (a stand-up comedian) corrects the
Rock, Paper, Scissors game (among other things).
The Geometric Meaning of the Geometric Mean Plenty of pictures
showing the geometric mean in action.
When would you use the Geometric Mean? Average rate of return on
investments, averages of things that fluctuate (the stock market for
example), center frequencies, averages of things that you are multiplying
by.
Some examples of the golden ratio in nature
The numbers that appear (the Fibonacci numbers) are VERY closely related to the golden ratio. For example, 55/34 and 89/55 (and so on for the other numbers that appear on this webpage) are very close to the golden ratio.