A pool rack is an equilateral triangle, filled with 15 equal-sized balls. Seen from above, we'd see a triangle, with 15 circles within.

Imagine we used smaller and smaller balls. The more the balls, more area of the triangle would be covered.

In the limit, with infinite balls, would all of the triangle be covered?

I noticed that the answer would be fairly straight forward, albeit mathmatically complex, if you have infinitely smaller and smaller balls to cover a set area, you will come closer and closer to completely covering it, and yet you never will, since the size and shape are forced to leave space inbetween, the balls will simply, as afore mentioned, leave an infinitely smaller and smaller uncovered space interspersed.  The crux is that since they are spheres, there will never be none, just infinitely small :)

Posted by josh on 2004-09-18 23:24:52