I had a hard time translating the last sentence, you phrased it just fine
Yatir
| 2 | Ö |
pS |
I think the translation is alright, but you might want to, if
I've interpreted the problem correctly, reword the last sentence,
for clarity, to:
| 2 | Ö |
pS |
Ö |
S/p |
| 2 | Ö |
pS |
Got it! Thanks. This is how I thought of doing it.
I had a hard time translating the last sentence, you phrased it
just fine
Yatir
Lol that's actually what I did but something still went wrong - but I might have an idea of what it is. Thank you v. much!
Wait a minute...
The forest could have a clearing in the middle and so have area
less than S whilst still encompassing the path
When I said that it is fully covered by trees, I meant that
there is no clearing....
That's interesting, though; I think if clearings are allowed,
there is, in fact, no path he could follow that would always lead
him out - no matter how long the path was. The forest could just
cover the path and could be made of arbitrarily small width
around the path.
Brad
That would be the freakiest wood I ever saw - i guess this is another occassion where a mathematician has taken a common everyday word and stretched it into something arbitrarily long, thin, and unrecognisable...i like it!
Of course, a wood can't be of arbitrarily thin width... sooner or later you're going to reach a limit (either when you decide that what you've got no longer constitutes a "wood" or when you get to widths less than that of a tree). Of course this opens the whole "vagueness paradox" (is there a better term for this?) issue. I'm not being pedantic... just pointing out a limitation of the mathematical model being used.
Brad, what you say is true for any fixed finite length, but can you (or anyone else) think of a path that would work if you were allowed an infinitely long path? (Also, you'd have to assume that the wood is never so thin that it is actually 1 dimensional.)
Not quite, because you could have a wood that got thinner and thinner as you go outwards but is never actually of zero width. For example, the positive real axis as the path and the set of points (x,y) such that x> =0 and |y|< =1/x as the wood.
Sorry to cut down the pace of this discussion but I don't
really understand what you're talking about now... Could someone
please elaborate... If we allow arbitrary "clearings" then where
is the outside of the forest that we're trying to
reach?
Marcos
You could either say that all clearings count as "outside" the forest or you could say that the outside of the forest is any clearing which has an infinite area. There might be more than one "outside" with this definition though, for example if your forest was all the points (x,y) with |x|< =1 then there would be two outsides, the points (x,y) with x> 1 and the points (x,y) with x< -1. I was thinking about the first one but the example I have in mind would work for any region that you are thinking of as the outside of the forest.
Dan - Is there any such path? I don't think so; you could make
a forest such that, upon walking a distance of n along a path,
the amount of area that you've seen of the forest (perpindicular
[or normal, to be techincal] to the path, lets say) is S - 1/n.
Given the general ambiguity of this paragraph, I'll probably post
an image if anyone asks for one.
Brad
Brad, I'm pretty sure I could describe a path that would work. Think about either approximations to space filling curves (do you know about these?) or spirals.
Ah, a space filling curve would of course work! I'm not sure I see how a spiral could work, though...
A space filling curve would work
abstractly, but it wouldn't be much help to you if you actually
found yourself in a very oddly shaped wood because you'd never
get anywhere. In most of the standard space filling curves the
length of all subpaths is infinite so to move from your starting
position you'd have to travel infinitely fast.
The solution I have in mind is this: start off by traversing a
finite length path that comes within a distance of 1 of every
point within a radius of 1 of your starting point and ends up at
your original starting point (actually, not moving at all would
do this). Next, traverse a path that comes within a distance of
1/2 of every point within a radius of 2 of your starting point.
Continue in this way and eventually you will get as close as you
like to any point in the entire space after a finite (but
possibly very long) period of time.
Does that explain how approximations to space filling curves and
spirals come in?