Imagine that you are lost in forest. The precise size and shape of the forest
are known to you (maybe you have a map in your pocket,) but you have no idea
where in the forest you are, nor do you have a compass or any other means of
determining true directions. What strategy should you follow to escape from
the forest along a path of minimum length?
This problem, which may be of some interest to orienteers, trail runners, and
others who occasionally find themselves lost in the woods, is surveyed in
a recent edition of the American Mathematical Monthly. (The exact reference is:
S.R. Finch and J.E. Wetzel, "Lost in a Forest", Amer. Math. Monthly 111(2004),
645-654.) It appears to have been posed originally by Richard Bellman in the
1940s.
If the forest is finite in extent, as most forests are, then everybody knows
a surefire way to get out: walk along a straight line and eventually you will
reach an edge of the forest. In the worst possible case, you will have walked
a distance equal to the "diameter". (Diameter can be defined for
noncircular regions as the length of the longest line segment entirely
contained within it.) In the case of circular forests this turns out to be
the best strategy, and it also optimal for forests that are not too far from
circular. (The technical term is "fat".) General results of this kind are
surveyed in the first part of the article.
The more interesting collection of results concern less symmetrical shapes.
For example, consider a forest that is in the shape of an infinite strip that
is everywhere exactly one mile wide. (If you don't like thinking about
infinite strips, consider a sufficiently long one instead: once the strip is
long enough it turns out that the best strategy is the same as for
an infinite one.) Here, the strategy of walking in straight line might be
very bad. If you happen to be walking parallel to the edges of the
forest, or nearly so, you will have to walk a long time before you reach an
edge. A better strategy would be to walk a bit more than a mile in a straight
line, say two miles to be definite, and if you aren't out yet you can figure
you are following a path that is nearly parallel to the edge. Thus, making
a right angle turn and continuing in a straight line will get you out before
long. (Exercise: figure out the farthest you would have to walk if you
followed this strategy.)
It turns out that a mile wide strip forest can always be escaped along a
certain path approximately 2.278292 miles long. The path is shaped like
a pair of calipers and is interesting for another reason: it the shortest
path in the plane whose minimum width is equal to one.
Less is known for other geometrical shapes, even very simple ones. For example,
it is not known what the minimum escape length is for an equilateral triangle
1 mile on a side. Consult the article cited for references to other known
results and conjectures.