For an enclosed volume with a door, a lock securing the door is used to keep what is outside from going inside, vice versa; it keeps what is inside from going outside. However, a key can be used to turn the lock. This key can open or close the lock from the inside or from the outside. On the one hand, to open the lock from the inside one needs to turn the key counterclockwise. On the other hand, to open the lock from the outside one needs to turn the key clockwise. Conversely, to close the lock from the inside ones needs to turn the key clockwise while to closed the lock from the outside one needs to turn the key counterclockwise.


This trivial daily occurrence happens so frequently for anyone to take any important notice. Nonetheless, it hides a mathematical principle of a locking topology. This is a Möbius topology. This topology does not allow the partitioning of a completely closed volume. Consequently, a locking mechanism as described above cannot be constructed unless there is a distinction between clockwise and counterclockwise twists. Incidentally, a complete cycle around the Möbius topology is the same as turning the key clockwise at the outset but at the end of the cycle the turning is completed by a counterclockwise stop. The important question is where and when during the cycle did the turning changes from clockwise to counterclockwise, or vice versa?