Original Game by: Conway and Paterson

Objectives:

1. To become accustomed to what sort of properties are intrinisc topological properties.

2. To learn how these properties can be used to identify, compare, and contrast topological surfaces.

3. To become accustomed to the various topological properties, especially Euler number and the existence or nonexistence of finitude, boundary, and orientability on certain surfaces.

4. To become accustomed to using the fundamental domains of surfaces when studying their intrinsic topological propoerties.

5. To relate the intrinsic topological properties of a surface with its shape.

Launch:

The Rules:

(Note: If the students are not familiar with fundamental domains or with certain nonoreintable surfaces, you may want to go through the surfaces sheet first.)

1. First, players alternate placing a dot on the game board until there are 4-6 dots total.
2. Now each player takes turns connecting a dot to a dot with a path. The rules for paths are (see diagram):
   • A dot can have no more than three connections with paths. (A dot connected to itself has two connections.)
   • No path can intersect with a path path (except at the starting and ending dot of the path being drawn).
3. After forming a path, the player places a dot on the path
4. The last player that can play a valid move wins.

Playing Tips:

1. It's a good idea to label the dots or make them fairly large.
2. In order to keep track of which dots are out of play, start the game using empty bubbles for dots and fill them in when they are no longer in play.
3. It's a good idea for each player to use a different colored pen for later analysis.
4. If a line is between two lines (or a side and a line) on one side of a surface, then this relationship should be maintained when that line passes through a glued side to the other side of the surface. Watch out for this especially on the Mobius strip, Klein bottle, and projective plane. Use the grid lines and/or number your paths to make it easier to keep track of this.

Explore:

1. After playing some games on each of the surfaces, try to guess what properties a surface must have to be able to enclose the surface in one closed path. Test your guess by trying to draw such a path on each of the surfaces.
   • Was your guess right?
   • On which surfaces can you draw such a path?
   • How would you change your answer to the first question now (if at all)?
2. Now for each game board that you could enclose with a path, do the following:

   • Pick a point in one of the corners. Put a dot there if there is not one already.
   • Start tracing a path along one of the sides, watching out for the following:

     If you run into another path or a dot along the side, then connect to the path or dot, and place a dot at this connection if there is not one already. Then go to the other end of the part of the path which lies along th side. Put a dot here if there is not one already, and continue tracing a path around the sides.

   • Continue along the sides until you return to the original dot in the corner.
   • Repeat along other sides and/or with the other corners until the entire space is enclosed in loops.
   • Connect the dots in such a way that you can follow some curve to get from one dot to another.
   • Now make a table of the following for each game board:

     a) the game number (Number your games as you wish.)

     b) the game board's surface

     c) the total number of dots (vertices)

     d) the total number of curves (edges) between 2 dots.

     e) the total number of regions (faces) enclosed by edges.

     f) the value: c-d+e

   • Do you notice a pattern? Does this value look familiar?
   • Are there similarities only shared by surfaces that share the same value?
   • Are there differences between surfaces that share the same value?
   • Looking at surfaces with the same value, try to find some relationship between them. For instance, is there a way that you can make one surface by modifying property of another surface? After finding a relationship, look at another group of surfaces that all share the same value and see if this relationship still holds.
3. • What is the minimum and maximum number of different regions that you can make by making paths on the various surfaces?
   • What are the maximum number of walls or closed loops one can place in each surface before dividing it up into two or more regions?
   • For each surface, for any given number of different regions that you want to divide the surface into, how many ways can this be done? Try to categorize the different ways (Ex: two parallel lines vs. two perpendicular lines). Try these paths on other surfaces and see if they divide into the same number of regions.
   • What surface properties seem to determine how a given path will divide it?
4. • If you try copying the drawings from a klein bottle directly to a square torus what problems do you encounter?
   • Does this change the outcome of the game or any of the relative positions?
   • Do all of the closed paths remain closed?
   • Do any valid moves become invalid?
   • What about trying to copy the drawings from a torus to a klein bottle?
   • Are there any two surfaces such that you can copy the paths from one surface onto the other?
   • What if the paths go all the way around the surface?
   • Can you think of a way of describing the restrictions on the paths that are needed before one can copy those paths from any one surface to any other?
5. • First, copy your game on a new sheet.
   • Take one of your surfaces and draw a line across the surface.

     (Note: Copy the paths onto the back of the paper in order to properly recreate the 2-dimensionality of the surface.)

   • If you can, glue the surface together along its sides.
   • Cut through the new line and if you have not already done so, glue the surface together along its sides.
   • Have any of the relative positions changed?
   • Have any paths changed?
   • Has the outcome of the game changed?
   • What does this tell you about how we choose to glue the sides together?
6. Choose some of the game boards and try to see how many ways you can close a loop around all sides under the following conditions:

   a) You must have at least one dot for the loop to begin at.

   b) Your path must remain along one of the sides at all times.

   • How many ways can you make valid paths?
   • How many ways can you make invalid ones?
7. Make a Mobius strip (see the surfaces sheet for instructions how).Trace a path that connects to itself, following near an edge of the mobius strip.
   • Did you ever reach another edge?
   • How many edges does a Mobius strip have then?
   • When did your path flip?
   • In light of this and question 5, where do you think the flip is on a Mobius strip?
   • What about a Klein Bottle or a projective plane?
8. After playing some games on the mystery game boards, try to use properties that you've discovered about the various surfaces and techniques from above to determine which surfaces these mystery surfaces are essentially the same as.
9. You can physically cut and reglue (see question 5) the mystery surfaces in such a way that they closely resemble the surfaces which they are essentially the same as. Give it a try. Hint: Note that we can bend and twist the surface (although this may not be entirely physically possible) and still maintain this similarity.
10. (Easy question:) Pick a surface with glued sides that is easy to glue and physically glue its edges. Do any of the paths change? What about if you move this surface around the room? What if you crease the surface?
11. Is the torus tied in a knot? How can you tell? Are any of the other surfaces tied in a knot? Again, how can you tell?

Summarize:

We've been looking at surfaces almost entriely from one point of view, the intrinsic topological view (except in question 11). In the first question, we calculated what is known as the Euler number of a surface. We also examined whether or not the surfaces had boundaries(edges), how paths close on surfaces, how paths separate surfaces into regions, and which surfaces had would flip a path from top to bottom or from left to right (This property is called nonorientability. Surfaces without this property are orientable). These properties are intrinsic because you need not know if or how the surface is positioned in a higher dimesnion to analyze them (see question 10). They are topological because they do not change regardless of how you bend, fold or cut and reglue the surface (as long as you maintain proper alignment when regluing)(This is in contrast to geometric properties, which can change if a surface is bent, folded, etc.).

These properties are worth looking at because they are the properties we would have to look at to determine the shape of our own space or reality. We are limited to looking at intrinsic properties for two reasons:

1) Our physical reality may not exist in a higher dimension.

2) Even on the off chance that we do exist in a reality imbedded in a higher dimension, as inhabitants of this reality, we cannot discern extrinsic ('non-intrinsic') aspects of this reality, nor are we affected by them. This is analagous to the way the paths are not affected by crumbling or creasing the surface.

We choose to analyze topological properties because under the right circumstances (like that the universe has the same geometric properties everywhere), the topology of the reality essentially determines both its topology and geometry.

One of the most useful tools for investigating the intrinsic topology of a surface is the fundamental domain. This is a representation of the surface that can be placed in in a space of the same dimension as the surface. As a result, only intrinsic properties can be observed, and thus we do not have to differentiate between the intrinsic and the extrinsic. Furthermore, from the fundamental domain we get a direct analogy to one of the possibilities of our own space: a surface that exists only within its own dimension.

Variations:

• On surfaces with glued sides, each path must cross a glued side.
• On a Klein bottle, on every other turn, each player's path must pass through the sides glued with a twist.
• Draw some walls on the surfaces by connecting points on opposite sides. Play some games with the added rule that you can't cross the walls.
• Play with multiple players.
• Play with any number of starting dots.
• In an effort to induce critical analysis of surfaces and their properties, the game can be played so that each player can move in some way that violates properties of the surface being played on. The other player must catch the error before moving, or else the move is allowed. In this way, players with a greater understanding of the surface being played on will be rewarded. On the other hand, using this rule before the players understand the surfaces may cause problems.
• The purpose of the game is to get students accustomed to the shape of the space and to the consequencees of gluing and nonorientability. Therefore in early games you may require that lines/dots lying on a glued side be represented on both glued sides. You may even want to go so far as to actually physically glue or tape together the sides of the simpler surfaces(see surfaces sheet). This is especially useful when determining what is or isn't a valid move and why. Later on, you may want to remove this requirement in order to induce the students to make that relationship intuitively when playing.