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. Could we have accurately predicted the exact behavior of the sum of the sine functions, as shown in the mathematical graph above, without going through the trouble of graphing it?
4. 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)?

Any other thoughts?

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").

Focus!

If you hold a lens some distance away from a bright object, and place some kind of "screen," like a piece of paper or light-colored wall, on the other side of the lens at just the right distance, you will see a clear, upside-down image of the object.

Some variation of this basic principle is used in many practical situations (for example, in cameras, and when projecting a movie onto a large screen). Armed with a dollar store magnifying glass for a lens, and some spare time, I tried this at various distances from a lamp in my house and, as carefully as I could, recorded the distances needed to produce a clear image. You might want to try this out yourself, but my results are given below, measured in inches.

Distance to lamp (p)	25	40	55	70	155
Distance to image (q)	12	10	91/4	9	83/8

Here's a graph of the data.

If you're as fascinated by this as I am, there are many questions you might have at this point. For example, what would happen if I held the lens closer to, or farther away from, the lamp? This and many other potential questions can be boiled down to the following: What kind of mathematical function / equation could describe the relationship between object distance and image distance?

Human height and weight

Below is a graph showing the heights and weights of a large group of people, including infants (each dot represents a person). Based on this, what is a normal weight for someone with a given height? In other words, if h and w represent height and "average" weight, can you find a formula for w as a function of h?


(If you have Java installed, click the graph to load an interactive version, with which you can try plotting your function against the data. For example, one not-very-good possibility is w(h) = 3.86h - 110.42, which you can plot by typing it into the input box. Also, if you're a GeoGebra whiz, you might like to know that the data points are stored as list1.)

Modeling with the normal distribution

(Prerequisite: beginning understanding of calculations related to a normal distribution.)

According to a 2008 study, the height and weight of American men are distributed as follows.

Percentile:	5th	10th	15th	25th	50th	75th	85th	90th	95th
Height (inches):	64.4	65.6	66.3	67.4	69.4	71.5	72.6	73.2	74.3
Weight (pounds):	137.1	147.0	154.6	165.7	188.8	216.8	234.5	245.8	270.3

Can you tell from this information whether the height and/or weight of American men are normally distributed? If either is, can you estimate the standard deviation?

Deranged

How many ways can I rearrange the letters ABCDEFGH so that no letter is in its original position?

How far can you see? — revisited

Here is a photograph of the Earth:

From how far away was this photo taken?

(Alternatively, as Dan Meyer might say, "Here is a photograph of the Earth. Any questions?")

How tall is the atmosphere?

I am borrowing this fun little problem from Max Goldstein. In this NASA photograph of a small portion of the Earth, a somewhat orange layer of atmosphere is clearly visible. Based on the photograph, approximately what is the relative thickness of the orange layer? In other words, what is the ratio of the Earth's radius to the thickness of the layer?

,000 as a unit

I happened to see this the other day during an episode of Prehistoric Predators on the National Geographic Channel:

Click to enlarge. Sorry for the poor quality; I literally took a picture of the TV screen, in a state of mixed amusement and dismay.

Modeling temperature

The NOAA keeps close track of various kinds of weather data at a number of locations around the U.S. For example, here is a description (including the web location) of some data on hourly soil temperatures for the past couple of years. The NOAA station closest to me is in Chillicothe, MO (home of sliced bread!), and a graph of the hourly near-surface soil temperature in Chillicothe, starting on July 30, 2009, is shown below.

A good exercise in modeling is to find a function that "fits" a set of data like this one. But for me, the graph above also inspires a number of other questions that can lead into further topics. What sort of questions does it raise for you?

The derivative as a topic in algebra

Is there something wrong with giving college algebra students the following as a prompt for group discussion?

On Earth, gravity causes most objects to accelerate downward at about 32 feet per second per second. Because of this, according to physicists, if we were to throw such an object upward at a starting velocity of v₀ feet per second from a starting height of h₀ feet, the height h (in feet) of the object after t seconds can be modeled by h(t) = -16t² + v₀t + h₀.

Suppose I throw a ball upward at 45 feet per second from 20 feet above the ground. Based on the model function,

1. What will the height of the ball be after 1 second? And after 3 seconds? After 5 seconds?
2. When will the ball hit the ground?
3. How high will the ball reach?
4. How fast will the ball be moving 1 second after it is thrown?
5. Is the ball really accelerating downward at 32 feet per second per second?

Keep in mind that the function h given above is almost always introduced in college algebra courses as a prime example of quadratics. Also note that the idea of average rates of change (and in particularly bad textbooks, difference quotients) is not uncommon in college algebra.

Update: Around the same time I wrote the above, David Cox reported a somewhat related (and very exciting) experience introducing the derivative to an eager group of students.

How far can you see?

Here's a question I love to pose to my trigonometry students:

If I look out a window 100 feet above the ground, how far can I see? (The Earth is roughly a sphere with radius 3960 miles.)

Ideally, the bit of information about the dimensions of the Earth is only given after the students have had at least a moment to ponder the question. That question, or some variant of it, absolutely fascinated me as a child. Depending on how you interpret the phrase "how far," it can be answered using only the Pythagorean theorem or, at most, some basic right-triangle trigonometry. But as it turns out, the answer is practically identical either way.

An alternative, and in some ways more reasonable, setting for this question is something along these lines:

If a satellite is 10 miles above the surface of the Earth, how much of the Earth can it see?

A more difficult, more interesting, and possibly more applicable variant of this could be:

How far above the surface of the Earth must a satellite be in order to be seen by two receivers 500 miles apart?

The Gateway Arch as a quartic function

As a scientist or engineer might tell you, polynomials can be indispensable in creating or approximating any number of smooth shapes. For example, if you plot the polynomial equation y = -3.87×10^-8x⁴ - 0.00346x² + 625, its graph matches very closely the shape of the Gateway Arch in St. Louis (with x and y in feet).

Because of the shape of the Arch (moving downward to the left and to the right) it makes sense that an even-degree polynomial with a negative leading coefficient should be used to describe that shape. But as it turns out, the graph of a second-degree polynomial (a quadratic function whose graph is a parabola) would be noticeably "pointier" than the Arch.

In reality, the exact shape of the Arch comes from a more advanced function than those that we deal with in an algebra course, but the polynomial above is close enough for many practical purposes. For example, suppose we wish to hang a banner from the Arch, 500 feet above the ground. How wide should it be?

Facebook is (like) an infectious disease

The table below gives very rough data (from here) about the growth of the website Facebook.com from December 2004 to February 2009.

Date	Months since December 2004	Active Facebook users
December 2004	0	nearly 1 million
December 2005	12	more than 5.5 million
November 2006	23	more than 12 million
April 2007	28	20 million
October 2007	34	over 50 million
August 2008	44	over 100 million
January 2009	49	over 150 million
February 2009	50	over 175 million

If you plot this information, you will notice that as time went on, the number of active users grew faster and faster. In other words, the larger the number of users, the faster that number grew. This makes sense, since the more people joined Facebook, the more other people were likely to do so. In this way, Facebook membership resembled the early exponential growth of an epidemic.

In fact, if we choose two of the data points, we can find an exponential function that fits them. For example, if we choose u(0) = 1 and u(50) = 175, we can find that the simple function

fits these points. The graph below shows the data given above along with this exponential function. Even though the function was created based only on the number of active users at 0 months and at 50 months, it fits most of the rest of the rough data surprisingly well! According to this function, the number of active Facebook users grew by about 10.882% each month.

Now, the table below gives the remaining available data about the growth of Facebook. If we compare these numbers with what the function given above would predict, it is clear that the exponential growth of Facebook has started to slow. This may be because most of the people worldwide who are willing and able to join the website have already done so. As an analogy, Facebook membership is like an infectious disease that is running out of potential victims.

Date	Months since December 2004	Active Facebook users
April 2009	52	over 200 million
July 2009	55	over 250 million
September 2009	57	over 300 million
December 2009	60	over 350 million
February 2010	62	over 400 million
July 2010	67	over 500 million

A logistic function is a relatively simple model that scientists often use for the long-term growth of a population, or the spread of a disease. For example, the logistic function

seems to be a very good model for the recorded growth of Facebook so far. The graph of this function and the data is shown below. It will be interesting to see whether the function remains accurate in the coming months. If so, when will Facebook reach 750 million active users?