If you take a sheet and try to glue the left edge to the right edge (the single arrows tell you to do this) and the top edge to the bottom edge (the double arrows tell you to do this), the paper will crumple up and you'll get a big mess.

But, we can do this identification of a square with a stretchy piece of rubber. First we identify the left and right hand side in order to form a cylinder. We use these arrows and the points that correspond to glue the top and the bottom together and this forms a donut or torus.

After you have gotten the hang of the game, you and your partner will then play on one computer by changing the game to "human versus human" and clicking on "new game". Play best out of 7 games where you are allowed to scroll the board during the game and then play best out of 7 games where you are not allowed to scroll the board during the game. Keep track of who is winning and tell me who the winner is!
Scrolling Games Winner - best out of 7_________________________

No Scrolling Allowed Games Winner - best out of 7 ________________________

2-Holed Torus - A Hyperbolic Universe

Here we glue the side with one line through it with the opposite side that has the same label by using the arrowheads to show us that we should glue the sides straight across. Similarly, we also glue together the 2s (II), the 3s (III) and the IVs. It is an exercise in visualization skills to see that the resulting figure is a 2-holed torus.

Klein Bottle - A Euclidean Universe

Notice that this identification labeling of a square looks similar to the one that resulted in a torus, but the top and bottom edges are glued with a twist - a reflection in the line between the midpoints of the sides. Just as a 2D Flatlander could not imagine how to construct a cylinder out of a piece of paper, we Spacelanders have problems figuring out how to put together this square, because when we label corresponding points (such as the green squares, which are the same in this space because they are identified via the reflection in the dotted line), there seems to be no way to glue them together. However, an inhabitant of 4-space would have no trouble because he would have enough space to glue the edges together.

There is a way to for us to put this space together using a slinky. We can match up the corresponding points (in the above picture) by having the slinky pass through itself. Make sure that you try this yourself and can visualize it (as the slinky makes its way around the room). This space is called a Klein Bottle.

Here are some additional Still Pictures of a Klein Bottle. The space that we have represented here has a nasty intersection when the slinky passes through itself. Our gluing instructions give no hint of this and in fact, the intersections only arose when we tried to glue the space in 3 dimensions. In order to properly see an actual model of the glued square, we would really need a 4th physical dimension. In the 4th dimension there is enough room to glue the edges together without creating intersections, and it is here where the glued square really exists.

Experience what it is like to live on a Klein Bottle by playing Klein Bottle Tic-Tac-Toe. Change the Torus selection to Klein Bottle (do not click on the link - just change the selection on the tic-tac-toe page on the part that reads "torus" that is just above "human vs. computer"), click on "New Game" and play a couple of games with the computer. The top left square is really just below the bottom right square in this Klein Bottle universe.

After you have played a few games, you and your partner will play on one computer by changing the game to "human versus human" and clicking on "new game". Play best out of 7 games where you are allowed to scroll the board during the game and then play best out of 7 games where you are not allowed to scroll the board during the game. Keep track of who is winning and tell me who the winner is!
Scrolling Games Winner - best out of 7_________________________

No Scrolling Allowed Games Winner - best out of 7 ________________________