Lifting a toilet seat without breaking urine stream Yes, I know the title is bizarre. I was urinating and forgot to lift the seat up. That made me wonder: assuming I maintain my current position, is it possible for the toilet seat (assume it is a closed, but otherwise freely deformable curve) to be moved/deformed such that stream does not pass through the hole anymore without it intersecting the curve (in other words, spraying urine everywhere!)?
 A: If it is deformable, pass it under your feet, up your back, and over your head.  It takes a bunch of deformation to make that work.
A: Here is a topologically equivalent reformulation.

Suppose you are ice fishing on a frozen lake. Is it possible for the hole in the ice to deform in such a way that the rod and line no longer pass through it, without breaking either?

The answer is yes, but (in both cases) you'll wish you had never asked.
A: Purely topologically, the situation here is the same as having a ring (a toilet seat) surrounding a ray (with you at the end of the ray, urine in the middle, and the toilet anchored "off at infinity"). So, one can simply move the ring off the ray, since the ray has an end. Bringing this into real world terms, this looks something like the following:


*

*Firstly, step into the toilet bowl. Now the topology is quite clear - you are a ray pointing downwards and the toilet seat must be lifted up to escape. Thus:

*Secondly, duck into the toilet bowl. This will bring the seat over your head, hence off the end of the ray. Lift the seat at this point.

*Finally, stand up. You have successfully unlinked the urine stream from the toilet bowl.


I write this solution mostly to point out that we can introduce some amount of rigidity to this problem without making it unsolvable: In particular, suppose you are a line segment, and there is a parabola emanating from your midpoint and terminating at a fixed location. There is a ring around this. Now, if the parabola and the line segment are perpendicular, and the line segment is longer than the ring (which it probably is), you're not going to be able to unlink the ring. However, you can unlink the system by making the parabola and line segment parallel, so they look like a point in cross section to the ring, making it easy to pass through.
Now, there's a few positions that would achieve this - for instance, you could levitate horizontally in midair, or do a hand stand, but the easiest to achieve would be to make everything vertical and just stand in the toilet. Now, since the parabola and line segment are parallel, we're back to having a ray, which is nice. This also has the advantage that you do not need to pry the toilet seat off. In fact, it's conceivable that, if you have a small enough cross section, this solution would actually work without breaking anything.
