It doesn't make sense to me because a glass of water is a discrete set. If we only had two molecules you could just interchange them (with a 180 degree rotation), without any fixed points.
Show me a mathematical continuum in the real-world. And this is just the domain, nothing said about the mapping yet (the continuous function). Every analogy has its limits.
I'd guess the example of the glass of water was made as an afterthought, or as a vivid example of the potential complexity of the theorem. As your example shows, it simply cannot be true. If you think of a vase with marbles, it is also clear. Or if you think of an (ideal) gas in a closed system: would there be at least one particle that always(!) stays in the same place?
mbeex|2 years ago
tomrod|2 years ago
The space is continuous even if we can only measure down to the Planck length.
tgv|2 years ago