I first want to remark that if there is a topic in geometry/topology that is not on the list and you think is interesting and we all might enjoy it, please stop by my office and we can discuss its appropriateness. You should write a report on your findings/topic that should be understandable to general audience (non experts in the field; do not rely on the fact that I might know what you are writing about). There are few links on the course website with general instructions on writing and with a list of what type of paper will yield which grade. You are not allowed to write about the same topic you covered in your midterm. Your paper should be 5-7 pages long. If you have any questions, let me know.

1. Infinity You all know infinity is a little funny. But how funny is it really? Funny like this: there are infinitely many points on the line, right? There are infinitely many points on a semicircle? The same infinitely many? Or different inifintely many? Are all infinities the same? Prove some of your claims.

2. Mathematical induction This is a special method of proving statements. It's sort of like playing dominos: if you put them down right, you knock down one you can knock them down all. Describe the method and give examples of proofs using this method.

3. Jordan Curve Theorem One of the fundamental concepts in topology is so simple that it sounds trivial. It concerns the way a simple curve divides a plane.

4. Bridges of Konigsberg In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once....

5. 4-color theorem How many colors does it take to color a map so that no two countries that share a common border (not merely a point) have the same color? Show me your observations and thoughts. Do not attempt to reproduce proofs. In particular do not write about things you do not understand.

6. Isometries of the plane Isometries are maps that preserve distances. This is a wide topic and you can explore and write about all sorts of stuff here.

7. Tessallations of the plane Basically, a tessellation is a way to put shapes into the plane so that there is no overlapping and no gaps. One wants to put different restriction to make things more interestring, so you may want to talk about regular, semiregular, periodic, or nonperiodic tessalations, wallpaper tilings, or Penrose tilings.

8. Symmetries of tilings This might be a little too advanced and is connected to both 7. and 8. These are collections of isometries that preserve a certain tessalation. The example are the wallpaper tilings: these are so highly symmetric that they can be built from a single tile by using a set of rules (symmetries).

9. Counting and area You are given a polygon whose vertices lie on the square lattice (square grid-points whose coordinates are whole numbers) in the plane. Your task is to find a formula that relates the area of that polygon, A, and the number of points of the lattice inside the polygon, I, and the number of points of the lattice on the boundary of the polygon, B. You can do this by doing lots of examples until you find a pattern, and then proceed to prove that your guess is indeed correct one.

10. Fractals A fractal is an object or quantity that displays self-similarity in a certain sense. These are spaces that have FRACTIONAL dimension--how is that possible? Explore some of them, the most famous ones are Sierpinski carpet, Koch's snowflake, Mandelbrot's set, etc.

11. Regular polyhedra Using Euler number show that there are only 5 regular polyhedra. Define all the terms and justify all your claims.

12. Dissecting squares Given N>1 squares of arbitrary sizes is it always possible to dissect the squares into pieces that will combine (without overlapping or holes) into a bigger square? Think first about two squares and then generalize (maybe induction?).

13. 3-utilities puzzleThe diagram below shows three houses, each connected up to three utilities. Show that it isn't possible to rearrange the connections so that they don't intersect each other. Could you do it if the earth were a not a sphere but some other surface, a torus for example?

14. Knots Give basic definitions and examples. What does it mean that two knots are the same? How can we pass from one to the other (Reidemeister moves)? How do we even know if there are any knots taht are not the unknot? What is 3-colorability?

15. Graph theory Give basic definitions and examples. There are lots of things that one might discuss. One is the Euler number. Is there one for graphs? If so, do you see a connection to the Euler number of any of the surfaces we talked about? How do you explain it?

16. Taxicab geometry In taxicab geometry the shortest distance between two points is not a straight line, but rather the number of blocks a taxi has to travel along the streets. We idealize our city and pretend that it is a perfect square grid such as this . Pick any point and draw a circle of radius 4 around that point. Draw the circle of radius 4 around the same point but in Euclidean geoemtry. Are these two the same? What is the equation of the circle in the traxicab geometry? If the graph of x² + y² = r² is not a circle in this geometry, what would its graph look like? Draw two points in the city: your house and your friends house. You make a plan to meet half way between your houses on a Sunday afternoon. Where will you meet? What are the shortest paths in this geometry? Is there only one shortest path between any two points? Good luck and have fun.