by Lord Hatsuma » Sun Jan 11, 2004 1:15 pm
I'll go ahead and try. Correct me if I'm wrong.
2-D: Imagine some blob of a shape. Basically, can you draw it with just the one line around the perimiter? If not, then there are one or more 'holes' in the shape. We can find explicit formulae for determining whether or not something has these. Yay!
3-D: Same principle in 3-D. A sphere doesn't have any. A donut does. If you wrap some line around the outside of the donut (around the whole thing, not through the middle hole), then you can't touch every part of it. (If you wrap it through the middle of the donut, then your not really on the outside of it, since you'll never be able to pull the line off of the donut)
And of course we can extrapolate this to any dimension through magic math power.