Cooperation and Small Probabilities

Watch this clip from the TV show Friends (a popular late-90's sitcom). Joey, one of the six "friends," announces his intention to enter the Connecticut Powerball lottery. Most of the rest of the group decide that it would be a good idea to pool their money to buy many tickets and split the payoff if any of the tickets wins. However, Ross (a college professor in a scientific discipline, I might add) opts out, citing the incredibly small probability (or "odds") of winning. Chandler jokingly replies that with six times as many tickets, the group will have six times the probability of winning.

Is Chandler right about that?

Hide and Peek

There is a game for the Nintendo Wii called Wii Party (see also Wikipedia), which largely consists of a set of four-player "minigames." In one of the minigames, shown below, three players hide on a playground and a fourth player tries to find them.

Hide-and-Peek minigame (see the Wii Party website, under "All Minigames: 1 vs 3," for a video)

The non-hiding player has five chances to find all three of the others. There are seven hiding places, and any number of players may hide in each one. However, strangely enough, anyone who chooses the small hiding place in the center, behind the blue and pink spring horses, will actually be visible there (so unless the non-hiding player is completely inexperienced, he or she will definitely find that person).

The non-hiding player wins if he or she finds all three of the others. If not, all of the others win (regardless of which of them were found). Strategize!

Geometry of the sextant

As I mentioned before, I've been working with a sextant lately. Not surprisingly, Wikipedia has an informative article on sextants, with many details about their operation. The diagram below shows the most basic aspects of the geometry by which a sextant works.

(Click for the interactive version, in which the red dots can be dragged.)

The fundamental idea is that you look directly at one object while adjusting the angle between the sextant's two mirrors so that your view of a second object coincides precisely with your view of the first object. That is, the path of light from object 1 to your eye overlaps exactly the path of light from object 2 to your eye after reflecting off the two mirrors, as shown above. When this is accomplished, the scale at the bottom of the sextant shows the visual angle between the two objects.

The sextant derives its name from the fact that its scale forms a physical arc of about 60° — or one sixth of a full circle. But one of the first things that struck me about the scale is that its marks run from 0° to about 120°. So, since the sextant's mirrors are parallel when the sextant is set to 0°, it seems that the scale shows two times the angle between the sextant's mirrors.

So here are my questions: Is the visual angle between object 1 and object 2 really two times the angle between the mirrors? Can we prove it?

Backyard trigonometry

I recently got a hold of an old sextant (it looks about as old as the tuning forks, but the very same model I have is available for sale here). To see what sort of practical measurements I could make with it, I decided to try to figure out the height of a tall tree in my yard.

From some distance away, the visual angle of the tree was 40°18' (or 40.3°). From 12 feet closer, it was 46°1' (or about 46.02°). From another 12 feet closer, it was 53°33' (or 53.55°). The angles were each viewed from about 6 feet above the ground. How tall is the tree?

Expofinity

Consider the equation

x ^{x x x x x x x x x · · ·}= 3.

Can you solve it? Can you check your answer?

Sound waves (part 2)

There's a room near my office full of physics demonstration equipment, including a box of rusty old tuning forks. I picked out two that looked and sounded similar.

These are probably older than me.

In the following recording, you can hear me strike one of the forks, and then the other, and then both at the same time. As in my previous post, you may need to use headphones to hear the low-pitch (i.e. low-frequency) part of the sounds, which is where the interesting stuff happens.



Here are some close-up shots of the graph produced by a sound editing program:

Vibrations of first tuning fork.

Vibrations of second tuning fork.

Both tuning forks vibrating at the same time.

What is this devilry? Well, judging carefully by the first two graphs above, it appears to me that the tuning forks vibrate with frequencies of about 258.8 and 265.2 cycles per second (or Hz), respectively. If we graph the sum of two sine functions with those frequencies, each with an amplitude of, say, 0.2, we obtain the following graph.

Sum of two sine waves with slightly different frequencies.

As mentioned in my previous post, we can use Octave/MATLAB to listen to this function. Here are the sounds of a 258.8-Hz sine wave, a 265.2-Hz sine wave, and the sum of the two (graphed above):



Now, some questions occur to me at this point:

1. What is the simplest, most straightforward explanation for this weird audio phenomenon?
2. Is there a simple relationship between the very low frequency of the pattern seen/heard here and the higher frequencies of the individual tuning forks?
3. The mathematical graph above matches up very well with part of the graph of the actual sound of the two tuning forks (from about 25.8 seconds onward), but the earlier part of the sound graph is a bit different. How can we explain this?

Sound waves (part 1)

I recently opened the audio from this music video in a sound editing program. Such programs (e.g. Audacity) produce a visual representation of the audio — essentially a graph of sound pressure versus time.

At certain moments during the song, the graph looks something like this:

A 0.06-second interval around 70 seconds into the video.

There is a clear sine wave pattern here.

But on top of those sine waves, there are other, less dramatic but faster sine waves. How can we combine sine functions in such a way as to produce a graph like the original one above? And where did these two sets of waves come from?

Leap days

One year — that is, exactly one cycle of the Earth's seasons — is approximately 365¼ days long. So if our calendars only counted 365 days in every year, for every four years we'd fall one day behind. Our seasons would shift; after many years, North America would receive snow in June and summer heat in December.

Unfortunately, a year is not exactly 365¼ days long. In reality, a better estimate for the average length of a year is 365.2425 days. Because of this, the established rules for "leap days" in our calendars are a bit more complicated than simply adding one extra day (February 29) every four years.

I wonder: Do we have the best possible system for adjusting to a 365.2425-day year? To answer that question, assuming you're not already familiar with all the details of our system, I would recommend working out a system of adjustments on your own first, and then checking to see if you come up with the same rules.

Three Circles

Okay, now I am officially obsessed with spheres and circles.

A while back, I introduced my trigonometry students to the beautiful problem of finding the area between three congruent and mutually-tangent circles (though naturally I didn't pose the problem in such technical terms). Since then, I've been thinking about the more general situation in which three circles of different sizes are mutually tangent. Here's a diagram.

(Click the image for a nice interactive version.)

Now, what questions does this diagram raise for you? In creating various incarnations of the figure, some things that I found myself asking were:

1. If I have two mutually-tangent circles of different sizes, how can I construct a third circle tangent to both of them? Or, to put it another way, where must the center of the third circle be? What sort of curve must it lie along?
2. If I choose three points that do not lie on the same line together, can they always be used as the centers of three mutually-tangent circles? If so, how do I construct those circles? In other words, how do I find their radii?
3. And of course, the standard question — given three mutually-tangent circles, what is the area between them (shown in blue above)?

Spheres again

When two balls lie on a flat surface, in contact with each other, there's a pretty neat relationship between their sizes and the distance between the points where they touch the surface. There's maybe a slightly less nifty relationship between their sizes and the height of the point where they touch (the "point of tangency"). See if you can find one or both of these relationships.