Objectives:

The overall purpose of these games is for the students to familiarize themselves with the fundamental domains of various surfaces. The best way to accomplish this familiarity is through experience and discussion. In order to induce this kind of discussion, you may want to break the students up into teams of three to six each. The strategic planning that will carry on while two teams compete against one another will cause the students to observe and discuss the properties of the surface on which they are playing.

The fundamental domain of a surface contains all the information about a surface that an inhabitant of that surface would have access to. Thus, the fundamental domain really does contain the fundamental (topological) properties of the surface, and to understand the fundamental domain is to understand the essence of the surface.

1. The students will learn the terminology associated with fundamental domain notation. Most importantly, they will learn how to use arrow notation to represent how two sides of a fundamental domain are glued together.

2. The students will learn what sort of properties are topological properties that an inhabitant can observe (such properties are called intrinsic.). These properties include (among others) boundaries, orientability, and Euler number.

3. The students will learn to use the intrinsic topological properties of surfaces to differentiate between two topologically distinct surfaces and to recognize when two surfaces are topologically equivalent.

Questions and Answers

Dots and Boxes

Glue opposite edges of the board together. How does this affect the game? The removal of edges allows one to connect dots that were previously unconnectable. In the case of the Möbius strip, top dots could be connected to bottom dots and other similarly unusual connections could be made.

Play some games where dots are placed on the glued sides. Play some games where each pair of opposite sides are glued together. How do these variations alter game play? Putting dots on the glued sides really doesn't alter game play.

There is one catch, though. Though a game board shows two glued sides, there is actually only one side. When two sides are glued together, they become the same side (and they lose their boundaries). Thus, any dot or path on one side also exists on the other side, using the same rules given in the instructions for determining at which height to put things on each side. The easiest way to see this and to determine where to place something that lies on a gluing is to actually cut out the surface and tape/glue the sides together so that the arrows face the same direction. (Make nonorientable surfaces long so that they are easier to work with.) Then put the dot/path on the glued side and unfold to continue regular game play.

When each pair of opposite sides are glued together, even more opportunities arise for connecting dots. Plus, one ends up playing on such fascinating surfaces as the torus, the Klein Bottle (the Möbius strip with the other two sides glued together with arrows facing the same way, as seen in the video), and the Projective Plane.

Snakes

Glue one, two, or three pairs of sides together. What new moves does this allow? This, of course, simply allows one to exit through one side an reenter through the corresponding glued side.

Can you tell which board is topologically equivalent to board 4? Board 1 is. They are the only two boards which are orientable (all of their arrows line up the same way.) This is evident in game play since these are the only two boards on which you can exit through any side and come back through the other side at the same height (in relation to the side).

To actually deform board 1 into board 4 is a little tricky. Grab the left side of board 1 and pull it through the top glued side. As you pull, the 1-arrowed side will go through the top and come up from the bottom and the 2-arrowed side will move up to the to. Now board 1 will look like board 4.

Are any of the squares topologically equivalent to these game boards? Yes, the square like board 2 from Hex is equivalent to boards 1 and 4. The square like board 3 from Hex is equivalent to board 3 from this game. The square like the example surface used in diagram (1) of Torus Tic-Tac-Toe is equivalent to Board 2.

Hex

Recalling the definition of topological equivalence, can you see why board 1 is a disc? Board 1 can be stretched and bent into a disc.

EXTENSION: The game which best demonstrates topological equivalence is Dots and Boxes. After completing a game on a surface, find these values: number of dots (v), number of line segments between two dots(a), and the number of boxes(r). Now calculate v-a+r. This value, known as the Euler number and represented by c , is the same for all topologically equivalent surfaces, as you will learn later. In fact, generally speaking, two surfaces are topologically equivalent if and only if they have the same number of boundaries, they have the same value for c , and they are both orientable or both nonorientable. This test will work for determining which surfaces in the games are topologically equivalent, and so it may be worth doing if people are having trouble determining topological equivalence by other methods.