We have previously explored the fascinating and surprising properties of Möbius strips. We now continue our exploration and enter the wonderful world of Klein bottles, which in addition to being fascinating and surprising pose a greater challenge to visualize!
In our exploration, we have seen how cylindrical shells and Möbius strips can be constructed from edge identification diagrams.
Continuing this idea, what happens if we identify all four edges in the rectangle? A torus is obtained by identifying the two parallel sides of a rectangle in the same direction. One example demonstrating this identification is the game of Pac-Man: if Pac-Man exits to the right (or bottom) of the screen, then he will enter on the left (or top) of the screen from the same position where he exited. So the game of Pac-Mac is actually played on the topology of a torus!
It is also possible to extend other games that are typically played on a square or rectangular board to a torus. How would your strategies change for tic-tac-toe or checkers on a torus?
A Klein bottle is obtained by identifying the two parallel sides of a rectangle, one pair in the same direction and the other sides in the opposite direction. Klein bottles also have many surprising and wonderful properties.
By drawing a horizontal line down the middle of the above diagram, you can see that a Klein bottle is obtained by gluing two Möbius strips together along their edges! We have seen that it is not possible to construct a Möbius strip without self-intersections in 2 dimensions, but a Möbius strip in 3-dimensional Euclidean space is non-self-intersecting. Similarly, it is not possible to construct the Klein bottle in 3-dimensional Euclidean space without self-intersections and it is only possible to glue all directed edges with no self-intersections if we move to 4 dimensions. The torus and Klein bottle are examples of 2-manifolds without boundary, topological objects such that the neighborhood of each point looks locally like Euclidean space, i.e. a plane.
As an exercise, imagine playing Pac-Man on a Klein bottle instead of a torus!
A -manifold is a topological space such that the neighborhood of each point looks locally like . All of the objects we have considered through edge identification diagrams are -manifolds. Note that manifolds may be bounded or unbounded, and may be orientable or non-orientable.
The topology of a manifold is the set of properties of it which are unchanged by stretching.
We conclude with a well-known Klein bottle limerick:
A mathematician named Klein
Thought the Möbius loop was divine
Said he, "If you glue
The edges of two
You get a weird bottle like mine."