Desert Island Math

Avoiding politics, I thought I would do something fun. These are the kinds of essays that get the most attention anyways.

Last month a new movie came out called Cast Away. It starred Tom Hanks as a ordinary man who finds himself stranded on a deserted island with nothing to talk to but a volleyball. The same day I saw this movie, the History Channel was showing a program on the history of sea navigation. The two got me thinking of a little mathematical puzzle.

Lets say Tom the Castaway found a way to send a message, but he needed to tell people where he was (or maybe he is just bored and wants to figure it out for himself).  How could he figure out his Latitude and Longitude using only things he has on his island.

I am going to try presenting a method that should be accurate to within a degree or two, and should only require paper, something to write with, string, a wrist or pocket watch, a rock, a stick, and a tree with a view of the horizon. Tom the Castaway had all of these at his disposal.

Longitude

Historically, longitude has been the most difficult thing to measure. It was not until late in the 18th century that the problem of measuring longitude at sea was solved by John Harrison by building an accurate pocket watch called H-4. 

In the 21st Century, longitude is the easiest to figure out thanks to wrist watches. Now in the movie Tom Hanks had a pocket watch which apparently broke in the plane crash, but lets say it did not break, and he kept it wound. With this around he could figure out Longitude as follows:

So we can add six hours to the 6:10 and we get 12:10. Another thing we need to know (which should not be a problem) is whether 12:10 is am or pm. Since 12:10pm would mean that local noon by the shadow is only 10 minutes after London noon, it would put us just east of the prime meridian and we would be in the Atlantic Ocean. Since we know we are stranded in the Pacific, it has to be 12:10pm in London.

The formula for Longitude is fairly simple. If local time is ahead of GMT, then every hour ahead is 15 degrees East Longitude. If local time is behind GMT, then for every hour behind is 15 degrees West Longitude. If noon to us is 12:10am GMT, then we are either 12 hours 10 minutes behind or 11 hours 50 minutes ahead. Which ever is less is what counts, so we use the 11 50/60 hours and figure we are in east longitude. Multiplying by 15 gives us 177.5 degrees east longitude. The accuracy of this figure is dependent on how accurate our measure of noon is. 10 minutes off would be 2 and a half degrees off. But, even watching shadows hitting a mark should be within five minutes (assuming our watch is correct) so we can be certain our longitude is somewhere between 176 and 179 east longitude.

UPDATE: There is a slight logistical problem to the above method. It is only truly accurate on June 22nd and December 21st. Because of the Earth's non-circular orbit around the Sun, the sun moves in a figure 8 shaped pattern through the course of a year. Thus, the above method could be off by as much as 10 or 11 degrees depending on what time of year it was. But, in the movie Cast Away, Tom Hanks actually manages to map out this figure 8 pattern to create a calendar, so it is possible to take this into consideration as well.

Latitude

Now comes the hard part. If you happen to be in the northern hemisphere, then your latitude is the measure of the angle between the horizon to you to the north star (also known as Polaris, actually the north star is not perfectly over the pole but it is less than one degree off, so for our purposes it is close enough). If you are at the north pole then the north star will be straight up perpendicular to the horizon, so you are at 90 degrees latitude. At the equator the north star is right on the horizon, so you are at 0 degrees latitude.

Sounds simple enough, so why is this the hard part? First of all, the navigators of old conveniently had devices like the astrolabe to measure this angle, and these are kind of hard to find on a desert island. A protractor would do, but even math geeks like me do not carry one in my wallet. We could use the degree markings on a magnetic compass if we had one. Tom Hanks had nothing like this on the island, which means we will have to use old fashioned geometry. Our second and more devastating problem is that if we really are stranded somewhere in the Fiji Islands, we would be in the Southern Hemisphere, and Polaris is nowhere to be seen in the southern sky. This is a problem for linear algebra (I hope you remember it from High School)

Lets start with the Northern Sky procedure. Tie a string to a high tree branch (high enough so you stand under it) that looks over the horizon to the north. Tie a rock or heavy object to the other end like a pendulum, but you do not want it to swing, we just want to use it as a plumb bob. Now take a piece of paper (Tom had boxes to work with, as long as the edges are straight and square it will do) and line one edge with the string. Now line up the bottom edge with the horizon of the sky (as long as there is a moon, the horizon should be visible) and line up the paper facing north so you can see the north star. With everything perfectly aligned, make a mark on the paper where the north star lines up. Now all we have to do is find the angle between the line from the mark to the corner and the bottom edge of the paper.

Since we do not have a protractor, we will have to measure this angle a different way. One method is by geometrical construction. If you remember how to bisect an angle, we can bisect a corner of the paper to make a 45 degree angle. Bisect again to make a 22 1/2 degree angle, then a 11 1/4 angle then a 5 5/8 angle then a 2 13/16 and at this point we can pretty much guess and be as accurate as our longitude measure. Even easier is to make our own protractor by folding paper. We can fold a 45 degree angle by folding diagonally so the top edge lines up with the side of the paper. Folding in half again makes a 22 1/2 degree angle etc. So if we live in the northern hemisphere, we now know our latitude and longitude accurate enough so that search planes could find us.

Now what about the southern hemisphere. To show you what we are up against, I created an animation of the southern sky as seen from Fiji. Click here to view it (in avi format only 48K). Fiji is about 12 degrees South Latitude.

To calculate the latitude from the southern sky requires a bit more work and we will have to repeat the north star procedure four times. We will also have to create an x-axis observation, maybe by putting some evenly spaced rocks on the beach and observing how the star we are observing lines up with those rocks (again use the plumb bob as a vertical straight edge). What we need are two bright stars near the south celestial pole (but not too close or our observations will be too similar). Measure the x-axis position and the angle from the horizon (our y-axis) of both stars. Wait a few hours and repeat the observation with both stars again. (By the way, the further south you are the higher in the sky the South Celestial Pole is, and therefore the harder to measure accurately an x-axis component. This procedure assumes you are just south of the equator like our Fiji example)

From the animation above, we can note that the stars constantly go in circles around the south celestial pole. Any line that is perpendicular to the tangent of this circle will pass right through the south pole. (If you did not understand this last sentence, now would be a good time to review geometry and linear equations.) All we have to do is calculate the equation of two of these lines and we will know the point in space of the south pole, the y-axis measure being our latitude.

With our four observations we can create two linear equations that meet at the South Celestial Pole. For each star we have to find the midpoint position of lines between the observed positions which is simply the average of the x-coordinates and the y-coordinates. xm = (x1+x2)/2, ym = (y1+y2)/2  The equations will have a slope perpendicular to the line between the two observations (which if you remember from Algebra is the negative reciprocal of the slope). And what is the slope? Remember "rise over run", the slope will be (y2-y1) / (x2-x1), so the perpendicular slope is m = -(x2-x1) / (y2-y1). Now we have a point (xm,ym) and a slope m, which means we can write an equation: y - ym = m(x - xm). Repeat the procedure for the second observed star and we now have two equations whose common solution is the South Celestial Pole. The y-axis solution being the south latitude.

All of this makes sense from a math point of view, but I tried a couple of these and they are harder to do in actuality than I make them sound here. But, it is a fun mental problem nevertheless.

Altitude

Just for fun, here is another problem. On Tom Hank's Island there is a high peak, how do you figure its altitude with just a rope.

I actually did something similar on a trip to a lake where we were jumping off cliffs into the water. We wanted to know how high we were jumping. With no actual measuring devises I used a little trigonometry. We started with one person swimming directly under the cliff and holding one end of the rope. Another swam to where the shadow of the sun cast over the cliff, and tied a knot on the rope where the shadow hit. Then we went to the beach and compared the length of my own shadow to the cliffs shadow represented by the rope. By multiplying my height with how many times the rope stretches over my shadow, we had the height of the cliff, which turned out to be 43 feet (14 meters). Every one was guessing 70 feet (22 meters), so everyone was a bit disappointed.

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