Triangulation

However, let's say you have a friend in the field and both you and your friend have lightbulbs and radios. Then you can report to each other the angle between the friend's lightbulb and the other lightbulb. Is this enough to find the other lightbulb? Not quite, because there is no angle-angle theorem in geometry. In other words, if you adapt coordinates so that you are at the origin, then if you double the distances of the other lightbulb and your friend from the origin, both the angles that you and your friend measure will be the same. The only way to fix this is to measure the distance between you and your friend.

1. If the angle you see between your first friend and the lightbulb is 45 degrees and the angle he sees between you and the lightbulb is 90 degrees, where is the lightbulb?
2. If both your friends report an angle of 90 degrees between you and the lightbulb, where is the lightbulb?
3. What if both your friends report an angle of 0 degrees between you and the lightbulb?

The GPS, or Global Positioning System, uses a similar method to determine the exact location of a small handheld unit. Instead of having friends in a field the system has satellites in the sky that have very precise clocks in them. The handheld unit receives signals from the satellites saying what time they think it is and their current location, which allows the handheld unit to calculate the distance from it to each of the satellites. Then the handheld unit is able to mathematically determine its location. Interestingly, the clocks are so accurate that they are able to detect effects from general relativity, and correcting for these effects leads to improvements in positional measurement of tens of meters.

Of course, in the real world you can't measure exactly what the angle is between your friend and the lightbulb. Any measurement has errors, so all that you will actually be able to say is that the lightbulb is very likely to lie inside a cone whose point is your location. The more accurate your measurement is, the skinnier the cone will be.

The Prisoner's Dilemma

In this episode Charlie refers to the Prisoner's Dilemma, which is a particular game that shows that if the players in a game are only looking out for themselves and not colluding with the other players, then the results of the game might be worse for everyone involved. Here's how it works. Suppose that there are two prisoners that were involved in the same crime, and for the police to successfully convict either one of them, they need the testimony of the other one. If neither prisoner testifies, then they will each serve 2 months on minor charges. If one testifies and the other doesn't, then the testifier will go free and the other will serve 12 years. However, if both testify, then each will get 8 years. Now it's obvious that if the prisoners can collude and trust each other, neither will testify. This result would be the best possible outcome in the sense that the total time served would be minimized. However, if each prisoner only acts in his own interest, both of them will testify. This is because if one prisoner testifies, the outcome for him is better than if he didn't testify no matter what the other person does.

This situation can be conveniently described in a table where the first number listed is the incarceration time (called a payoff in game theory) of the first prisoner and the second number is the incarceration time of the second prisoner.