Imagine a sphere (in our 3 dimensions). If you drip a bit of water on it, the water will run down to a particular point.
It goes down to the same point no matter where you drip the water.
If you rotate it, the water will still all drip down to a particular point (just not the same one as before).
The above is true for some shapes (eggs, lima beans) and not true for others (Lego bricks)
So the question is: what about a 4 dimensional sphere? If you could somehow drip water on it... would the drips all converge to a particular point?
The answer is "yes"
The Poincare Conjecture postulates that if any loop on a "nice" surface can be shrunk to a point, it's topologically equivalent to a sphere. ("Nice" here means connected, finite, and without a boundary -- like a sphere or pyramid, but not a disk or infinite plane.) For instance, if the conjecture is true, a cube is topologically equivalent to a sphere, because if you draw a loop on it you can always shrink it down to a point; but a torus (donut) isn't, because a loop around a vertical cross-section can't be shrunk.
Perelman proved the conjecture for three-dimensional surfaces (which are the boundaries of four-dimensional objects).
No, the cube and sphere are homeomorphic regardless. (Pf: Points on S^3 are unit 4-vectors. Projecting onto the unit cube is continuous and invertible)
If the conjecture is true, and if you find yourself on a nice 3-surface, then you can conclude you're on the 3 sphere.
Unfortunately, HN will not allow me to delete or even edit my previous post.