Quote (ass666 @ May 13 2010 08:28pm)
you would think o.o but this has been said many times before and thead still continues.
Dark or Taxi
About Gabriels Trumpet
think you could explain to me why it has infinite surface area nd finite volume?
Way I see it is if the volume has a limit, then when taking the limit of the surface area it just gets smaller and smaller and simply hits well a limit. It just starts to become a line an lines have zero area?
it hurts my head
So basically the idea is that in the "bottom" of the trumpet, the area becomes arbitrarily small very fast. So fast, in fact, that any positive quantity of a sufficiently dividable "liquid" (i.e. a liquid that has no minimal units like molecules) you wanted to fill it with would clog up all but a finite amount of the trumpet. This issue isn't something you can think about physically, because there is an actual physical boundary to how small an object can go (in theory anyway, it's called the Planck length if you're interested). This means that nothing can actually achieve this level of tightness in reality. This is where the counter-intuitive part of the problem pops up, since you require that this object be contracting arbitrarily fast to keep the volume finite.