I have been trying to find a good way to explain the fact that the fundamental group of the group of rotations in dimension 3 is isomorphic to the multiplicative group {+1,-1}.

The first step is to identify the group of rotations with the configurations of a person standing vertically on the surface of the earth. In other words, each configuration is a point on a sphere and a direction (along the sphere) at that point.

The above statement above can then be translated as follows.

Consider all trajectories which start and end with me standing in Ecuador (on the equator) facing north. One such trajectory has me turning clockwise on my heels, while another has me turning anti-clockwise instead. Then the above statement says that there is a "continous deformation" of the first trajectory into the second!

One problem is that explaining the term "continous deformation" is not easy! However, the following proof explains what one has in mind.

Consider all circular tracks that have me travelling towards my right while facing a point (exactly) north of Ecuador.

When this point becomes the north pole, I am travelling towards my right on the equator facing north.

Similarly, consider all circular tracks that have me travelling towards my right while facing away from a point that is exactly south of Ecuador.

When this point becomes the south pole, I am again travelling towards my right on the equator facing north.

On the other hard, consider what happens as the centre of the circular track comes close to Ecuador and ends up somewhere near my shoes.

In the first case, I am turning anti-clockwise on my heels, while in the second, I am turning clockwise on my heels!

I am sure that there much better and more striking ways of explaining this to a lay audience, but I'm still looking for them ...

Perhaps, I need to travel to Ecuador to get a better idea. To do that and come back to India, I need to find a hospital that will give me a Yellow Fever vaccination. Anyone knows one?