Lost in a Forest

<< 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
a 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. >>

So if lost, what should one do?
Always have a digital web tool,
To search the web for protocol?
Not very impractical if one low on blood sugar,
No food, dead battery

What should one do,
If lost in the forest?

_______
Blog, or dog? Who knows. But if you see my lost pup, please ping me!
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Proving once again that there is book smarts and real smarts. I knew a
guy who could quote chapter and verse of just about any book we read in
college. Put him in the field and he couldn't find those things he knew
everything about, even if it were staring him in the face.
I'll take real smarts over book smarts any day.

Andy

Terry R. McConnell said:

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?

I would have gone with the "go straight and make a 90 degree turn" strategy,
but use a distance of sqrt(2) miles instead of 2 miles.

Worst case with this strategy is that you start on one edge of the forest,
the first straight line takes you to the opposite edge, and then after you
turn, you get back to the edge you started at. Doing it this way, you can
get out in sqrt(2) miles.

I wouldn't use the other strategies -- I think I'd get disoriented if I
attempted too many turns (-; Besides, 2.8 miles is good enough for me.

If you use 2 miles on the first segment, then you end up possibly going a
little over 3 miles.

Cheers,
--
Donovan Rebbechi
http://pegasus.rutgers.edu/~elflord/

Terry R. McConnell said:

Imagine that you are lost in forest.

Don't try this in Alaska, for an assortment of reasons:

1. The edge of the forest might be a major river, a bog, or an alpine
site. Beautiful scenery, but chances are you're really looking for a
road, your car or house, which might be at some edge of the forest, but
not the one you find.

2. It could be a *very* long walk - think days or weeks - although
likely you'll find one of the above challenges first.

One "experienced elsewhere" hiker got lost off a trail near the Steese
Hwy north of Fairbanks. She followed the "standard" procedure of walking
downhill. Problem was, she walked down the side of the ridge away from
the road. Next stop Arctic Ocean (or possibly Chukchi or Bering Seas,
depending on track), IF she could get across major rivers. She was
rescued alive.

Dot

--
"Dream Big, and dare to fail." --- Norman Vaughn
who was with Byrd in Antarctica and whose 99th birthday was Dec 19

Terry R. McConnell said:

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.

Give me a map and I'll navigate anywhere, compass
or no 🙂 Seriously, if you know the size and shape
of the forest you already know approximately where
you are, yes? As Daniel Boone once remarked, he was
never actually lost, just a "might confused' as to
his exact location...

Quoted message said:

What strategy should you follow to escape from
the forest along a path of minimum length?

Use your cell phone and call Search and Rescue? 🙂

Quoted message said:

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.

Don't know what type of forest is being used for
theoretical purposes here but where I am (and we
have about 25 million acres of forest) it's
impossible to geograpahically walk a straight line
anywhere. If you're a "might confused" in the Rockies
the usual advice is to follow a drainage downslope.
Course offical advice is just to stay put so Search
and Rescue can find you.

Quoted message said:

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.

Forests I know come in all sizes, shapes, and densities.
I know a boy scout who walked 20 feet into a local
forest to take a pee and got lost. The more he tried
to walk his way out the more lost he got. Don't think
it was a geometically well defined forest..

Quoted message said:

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
a 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.

An interesting mathmatical concept. But as a sometimes
wilderness guide who's spent a good 30 years tramping
through forests, it completely ignores the human factor
(i.e., human physiological and psychological reactions
when lost) plus geographic and geologic variables. Not
to mention the weather. For one thing, people usually
quickly develop a distorted view of their surroundings
and themselves (disorientation) when lost, becoming
incapable of making their own rescue decisions, and two,
forests are almost never places where one can walk a
straight line. Physiologically, minus a compass or some
landmark to site on no human being _can_ walk a straight
line even on a flat treeless surface no matter how straight
they think they're walking. The tendancy is to arc.

Quoted message said:

************************************************************************
Terry R. McConnell Mathematics/215 Carnegie/Syracuse, N.Y. 13244-1150
[email hidden] 229B Physics Bldg http://barnyard.syr.edu/~tmc
************************************************************************

Terry R. McConnell said:

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?

For those who take up the issue that forests are not two dimensional 🙂,
there is another version of a swimmer lost in a body of water
http://pauillac.inria.fr/algo/csolve/las.pdf
"Abstract. What is the path of minimum expected length for escaping
a planar convex region *? We rigorously obtain best 2-segment and 3-segment
solutions when * is an infinite strip, and numerically examine 2-segment solutions
when * is a disk."

Read more if you wish by getting the pdf from the link- I tried and failed.
-J, lacking in both kinds of smarts

In article <[email hidden]>,

Dot dot.h@#duh?att.net said:

Terry R. McConnell said:

Imagine that you are lost in forest.

Don't try this in Alaska, for an assortment of reasons:

1. The edge of the forest might be a major river, a bog, or an alpine
site. Beautiful scenery, but chances are you're really looking for a
road, your car or house, which might be at some edge of the forest, but
not the one you find.

The original article mentioned other problems such as searching for lines
and circles that idealize some of these variations.

Of course, as several other posters have mentioned, my posting addressed
some mathematical abstractions of the real life problem a person lost in
a forest would face. The practical aspects of what to do in such
a quandary form a quite different, though interesting, discussion.

Quoted message said:

2. It could be a *very* long walk - think days or weeks - although
likely you'll find one of the above challenges first.

One "experienced elsewhere" hiker got lost off a trail near the Steese
Hwy north of Fairbanks. She followed the "standard" procedure of walking
downhill. Problem was, she walked down the side of the ridge away from
the road. Next stop Arctic Ocean (or possibly Chukchi or Bering Seas,
depending on track), IF she could get across major rivers. She was
rescued alive.

Around here, the strategy of walking down the fall line would most likely
quickly lead you into an alder swamp -- a nearly impenetrable thicket, often
flooded by beaver operations.

Quoted message said:

Dot

--
"Dream Big, and dare to fail." --- Norman Vaughn
who was with Byrd in Antarctica and whose 99th birthday was Dec 19

--
************************************************************************
Terry R. McConnell Mathematics/215 Carnegie/Syracuse, N.Y. 13244-1150
[email hidden] 229B Physics Bldg http://barnyard.syr.edu/~tmc
************************************************************************

Tom Phillips said:

Terry R. McConnell said:

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.

Give me a map and I'll navigate anywhere, compass
or no 🙂 Seriously, if you know the size and shape
of the forest you already know approximately where
you are, yes? As Daniel Boone once remarked, he was
never actually lost, just a "might confused' as to
his exact location...

I misplaced my car this fall when using a trailhead I'd never used
before 😉 And people wonder why I take extra stuff with me.

Quoted message said:

Don't know what type of forest is being used for
theoretical purposes here but where I am (and we
have about 25 million acres of forest) it's
impossible to geograpahically walk a straight line
anywhere.

How'd they brush out section lines? Or do you not have surveyed section
lines in some areas? (Lack of surveyed section lines is an issue up here.)

If you're a "might confused" in the Rockies

Quoted message said:

the usual advice is to follow a drainage downslope.
Course offical advice is just to stay put so Search
and Rescue can find you.

Assuming you told somebody where you were going, an expected time of
return, and followed that plan. (had a recent case up here where an
experienced pilot didn't follow his plan, but was rescued in good
condition after about 5 days, I think. SAR happened to have a drill
going on where he had holed up, and was outside his original route)

Quoted message said:

For one thing, people usually
quickly develop a distorted view of their surroundings
and themselves (disorientation) when lost, becoming
incapable of making their own rescue decisions, and

And the situation is compounded by hypothermia and dehydration, where
the brain goes first.

Dot

--
"Dream Big, and dare to fail." --- Norman Vaughn
who was with Byrd in Antarctica and whose 99th birthday was Dec 19

wave is the far one obi wan. dark matter.
zig zag are done for reason.
X Y Z and T.
gen relativity. space is warp.
sun, moon travel.
wind.
moss grows.
there is many signs grass hoper if one care to listen.
I seen yesterday 3 deer eating at terminated time. a leftover corn
carrot hunters meal, under 10 inch of snow in 25 m.p.h. artic winds.day
to night.
zero temps this night.
my mind ,hear s baby deer say Lets go. should we go.
mommies deer say. I can't take this no more. Why dear Why are they
killing us, Why are they cutting our food source, why are they doing
this to us. How can we live. and pa pa deer. no stay, dig dig we need
food. how. so cold. eat eat.
my mine thinks.
before season an cold. 14 deer,
bow hunt time. then 8 deer.
gun time. now 3 deer.

and I watch as maybe barn fall down if? from sub division here. all
apple trees gone. woods gone. wind barrel int barn. and fill up sub
entrance with snow drifts and high wind area. let along latter floods.
and say before the earth movers. nature here works......espn bottle
cap gone now too. thief evan replace with a try again one, in my
trailer, in my jewel box. and I still not know why..............
so as terminator come I still shovel way. like a infantry man. with no
job, no unemployment, and no help at all.....for 111 Weeks ow. and I do
this so a zillion place can be od were I not get a nickel. per say......
and after 5 driveway fronts done and 2 man who rides and plow for free
too......i say. now.......I am still . well
it dwell all be my fault again.
a 5 % cut of gov heed office news said is $88,000.00 for the cost of
her in 1 year.
yes.......and if miss a thorned is her.
well hell that a cut...........
so as the 3 dears heed......I say I don't say why no more.
Merry christmas all in rec running. and have a good new year too.
still going strong......:>😉 o.)....

Dot said:

Tom Phillips said:

Terry R. McConnell said:

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.

Give me a map and I'll navigate anywhere, compass
or no 🙂 Seriously, if you know the size and shape
of the forest you already know approximately where
you are, yes? As Daniel Boone once remarked, he was
never actually lost, just a "might confused' as to
his exact location...

I misplaced my car this fall when using a trailhead I'd never used
before 😉 And people wonder why I take extra stuff with me.

Only people I know who do that go to Grateful
Dead concerts 🙂

Quoted message said:

Quoted message said:

Don't know what type of forest is being used for
theoretical purposes here but where I am (and we
have about 25 million acres of forest) it's
impossible to geograpahically walk a straight line
anywhere.

How'd they brush out section lines? Or do you not have surveyed section
lines in some areas? (Lack of surveyed section lines is an issue up here.)

Probably by sight? anyway I've noticed they're
not always such straight sections.

Quoted message said:

If you're a "might confused" in the Rockies

Quoted message said:

the usual advice is to follow a drainage downslope.
Course offical advice is just to stay put so Search
and Rescue can find you.

Assuming you told somebody where you were going, an expected time of
return, and followed that plan. (had a recent case up here where an
experienced pilot didn't follow his plan, but was rescued in good
condition after about 5 days, I think. SAR happened to have a drill
going on where he had holed up, and was outside his original route)

SARs people have radar..

Quoted message said:

Quoted message said:

For one thing, people usually
quickly develop a distorted view of their surroundings
and themselves (disorientation) when lost, becoming
incapable of making their own rescue decisions, and

And the situation is compounded by hypothermia and dehydration, where
the brain goes first.

Some check their brains before getting lost 🙂

Hypothermia is usually what happens here; the
people who get lost tend not to be properly
layered, can't read a map, or are doing what
they have no training for. There was one case
recently where a mother/daughter managed to get
lost while backpacking a rather well used trail
system. SAR looked for them for days; until they
finally walked out some miles from their trailhead.
Foodless and hungry but had this big "adventure."

I still can't figure out how they got "lost" on a
trail. Takes talent to do that.

Quoted message said:

Dot

--
"Dream Big, and dare to fail." --- Norman Vaughn
who was with Byrd in Antarctica and whose 99th birthday was Dec 19

Tom Phillips said:

Dot said:

I misplaced my car this fall when using a trailhead I'd never used
before 😉 And people wonder why I take extra stuff with me.

Only people I know who do that go to Grateful
Dead concerts 🙂

A clue to my mindset: if I take a water bottle for a "30-min" recovery /
exploration run, you know down deep I probably have almost no intention
of coming back in less than 1 hr. By the time I had decided I had dinked
around enough and really should head back to the car, I found my way to
a trail I knew, then followed that to where I was pretty sure the
parking lot was (not shown on maps - hence I wasn't real sure where I
started). To this day, I don't really know what trails I was actually
on, but I knew the lakes when I got there. Just couldn't find my car -
hence *it* was misplaced 😉

Quoted message said:

Quoted message said:

Quoted message said:

Don't know what type of forest is being used for
theoretical purposes here but where I am (and we
have about 25 million acres of forest) it's
impossible to geograpahically walk a straight line
anywhere.

How'd they brush out section lines? Or do you not have surveyed section
lines in some areas? (Lack of surveyed section lines is an issue up here.)

Probably by sight? anyway I've noticed they're
not always such straight sections.

ok. I've noticed ours are pretty straight, even through some rough terrain.

Quoted message said:

Quoted message said:

If you're a "might confused" in the Rockies

Quoted message said:

the usual advice is to follow a drainage downslope.
Course offical advice is just to stay put so Search
and Rescue can find you.

Assuming you told somebody where you were going, an expected time of
return, and followed that plan. (had a recent case up here where an
experienced pilot didn't follow his plan, but was rescued in good
condition after about 5 days, I think. SAR happened to have a drill
going on where he had holed up, and was outside his original route)

SARs people have radar..

An individual wouldn't show up on radar though, would they? Thermal
scanner, I could see.

He had flown beyond his destination, landed on beach, went beach
combing, returned to plane, plane wouldn't start, tide came in, tide
washed plane away. He had some of his survival gear, but not sure why he
couldn't get ELT out or trigger it. He was experienced enough to know he
was outside normal search parameters for where he said he was going to
be. Hiked about 15 miles, iirc, found a cabin with food and fire. I
can't remember the details, but I think he lit a signal fire and that's
how SAR found him while they were on another drill in that area. I can't
find the link to the article right now.

Quoted message said:

Some check their brains before getting lost 🙂

Hypothermia is usually what happens here; the
people who get lost tend not to be properly
layered, can't read a map, or are doing what
they have no training for. There was one case
recently where a mother/daughter managed to get
lost while backpacking a rather well used trail
system. SAR looked for them for days; until they
finally walked out some miles from their trailhead.
Foodless and hungry but had this big "adventure."

Ever notice how there seems to be a high correlation between "adventure"
and "low experience". I'll admit I still have my share of adventures,
but many more "another day at the office", and I try to keep it that way 🙂

Quoted message said:

I still can't figure out how they got "lost" on a
trail. Takes talent to do that.

Really. 🙂

Dot

--
"Dream Big, and dare to fail." --- Norman Vaughn
who was with Byrd in Antarctica and whose 99th birthday was Dec 19

Terry, this is why you never get invited anywhere. Mellow out,
man...........you're killing our buzz.

A lot of it is common sense and not panicking. I remember hiking near
Mt Tam, an area I spend a lot of time hiking at. But, the area is big
and there are lots of areas where there are few, if any, people. Well,
I was following what I thought was the right trail, it wasn't. It
disappeared and I was unable to find it again.

It was too forested to see the sky, so I could tell which direction the
sun was moving, so I thought of other things. Fortunately, there was a
time brook, and I knew that it was run towards Like Alpine, which would
lead me eventually out.

I followed the brook and though I walked an additional 5 miles, I
maneuvered my way to the trail I thought I was on.

Martha

Quoted message said:

Subject: Re: Lost in a Forest
From: [email hidden]
Date: 12/23/2004 11:32 AM Eastern Standard Time
Message-id: <[email hidden]>

Proving once again that there is book smarts and real smarts. I knew a
guy who could quote chapter and verse of just about any book we read in
college. Put him in the field and he couldn't find those things he knew
everything about, even if it were staring him in the face.
I'll take real smarts over book smarts any day.

Street smarts are just as important as book smarts.

Quoted message said:

Andy

Quoted message said:

while lost in the forest.
How do you make a Blonde eyes light up?
shine a flashlight in her ears.

Lost in woods.
2.2. miles ÷ 1 mile wide = x
x= A car?

Categories?
Blonde Jokes
Blonde Rolls Back The Odometer
A blonde wanted to sell her car, but couldn't find any buyers. She
called her friend for advice, and her friend asked her how many miles
she had on her car.
"235,000 miles." Her friend told her that was the problem.  But the
blonde's friend told her that her brother is a mechanic and could put
back the miles to whatever she wanted. So the blonde went to the
mechanic and told him to put the miles at 40,000. Two days later the
blond's friend asked her if she sold the car since her brother dropped
the miles.  The blonde told her, "Why would I sell the car? There are
only 40,000 miles on it!"

"why would she sell her car?
x= straight line.

x=straight line.
2.2 miles
1 mile wide and infinity long.
2.2x1=x
x= square the rectangle.
x= angle the triangle.
x=circle the square.

Categories
Blonde Jokes
Three Girls in the Desert
A blonde, brunette and redhead are in a desert. The brunette says, "I
brought some water so we don't get dehydrated." The redhead says, "I
brought some suntan lotion so we don't get sunburned." Then the blonde
says I brought a car door." The other girls said, "Why did you bring
that?" Then the blonde says, "So I can roll down the window if it gets
hot."

Joke of the WeekPress F1 for help
My friend was on duty in the main lab on a quiet afternoon. He noticed a
young woman sitting in front of one of the workstations with her arms
crossed across her chest, staring at the screen. After about 15 minutes
he noticed that she was still in the same position, only now she was
impatiently tapping her foot. He asked if she needed help and she
replied "It's about time! I pressed the F1 button over twenty minutes
ago!"

<< He asked if she needed help and she
replied "It's about time! I pressed the F1 button over twenty minutes
ago!" >>

Autism, AD[Hi!]D, OCD,
Or was she stoned?

_______
Blog, or dog? Who knows. But if you see my lost pup, please ping me!
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