Friday, March 13, 2015

Pi Day Eve (Day 131)

Today is Friday the thirteenth once again. Those who suffer from triskaidekaphobia must surely hate this quirk in the calendar. Because February, with its 28 days, is the only month that can be divided evenly into seven-day weeks, the first 28 days of month always fall on the same days of the week as the corresponding days in February (unless there's a Leap Day, since 7 doesn't divide 29 evenly). So if February has a Friday the 13th in a common year, so does March. It's the only combination where there can be two consecutive Fridays the 13th, and it won't occur again until 2026. (Well, at least we don't have the Cotsworth 13-month calendar, where all thirteen months contain Friday the 13th -- that would really cause the triskaidekaphobes to panic!)

I made a big deal about Friday the 13th last month -- I even designed my Chapter 13 Test that day to have thirteen questions, with many of them involving the number 13. But it's not today's special Friday the 13th that's on my mind right now -- it's tomorrow's special Pi Day that I want to discuss.

Not only is tomorrow 3/14, with the digits resembling the first three digits of pi, but in fact tomorrow is 3/14/15, so the first five digits are represented, 3.1415.... Because of this numerical alignment, many people are declaring tomorrow to be the "Pi Day of the Century." So the double Friday the 13th talk can wait until 2026 -- we must discuss the special Pi Day now, unless we want to wait until 2115 for the next Pi Day of the Century!

Let me begin the festivities with discussing my own discovery of pi. I first learned about the number pi when I was in the second grade. I remember learning about the rounded value of pi, 3.1416. Now I know that the last value has been rounded up (otherwise next year would be Pi Day of the Century), but back then, I didn't know that. I knew, though, that 3.1416 wasn't the exact value of pi. I once tried to guess the next few digits of pi -- I thought that the pattern continued, so that the digits would be something like 3.141618202224262830.... Now I know, of course, that this guess is wrong -- in fact, there is a rational number that starts out 3.141618202224262830... (it's 30791/9801), but of course, pi is irrational.

Other than that, I was never really interested in the digits of pi. I was actually more interested in the digits of rational numbers, such as 1/17, which is .0588235294117647... repeating. This was mainly because I knew how to generate the digits of 1/17 with a scientific calculator -- I got my first scientific calculator when I was a fourth grader -- but I didn't know how to generate the digits of pi.

I was in college at UCLA when I first heard of Pi Day. I stumbled upon it when I was searching for who knows what on the internet. During my last year at UCLA, the math department held a Pi Day party, to be held on Friday, March 14th at 5:00, after all classes had ended. Another student pointed out that the party should have been at 1:59, since the next few digits after 3.14 were 159, but I suspect that the university didn't want the party to occur one minute before 2:00 classes began.

Since then, I've tried to celebrate Pi Day in the classroom whenever I'm able to be in a math class on Pi Day. Two years ago, I brought in a pizza to the Algebra I class where I was student teaching. For after all, pizzas are round, so their circumferences and areas can be calculated using pi. Of course, as this was an algebra class, I used the pizza to have the students calculate how many slices of pizza they can eat if they wanted to consume no more than 600 calories. Unfortunately, that was the only year that I could really incorporate Pi Day in the classroom. Today, I subbed in an English class so I couldn't do anything about Pi Day with them. (I did warn some of them about Friday the 13th!) But of course I always tell the students about Pi Day when I tutor them. (My geometry student already knew that it was Pi Day of the Century!)

So now we reach today's Pi Day lesson. As I mentioned last week, today's lesson will be based on the lessons of Drs. Franklin Mason and Hung-Hsi Wu.

Let's start with Dr. M, who covers pi in his Section 9.2. Originally, Dr. M stated that Lesson 9.2 would be one of the three best lessons of the year -- the other two being the Orthocenter Theorem (i.e., the theorem that the orthocenters of a triangle are concurrent) and the volume of a sphere. But now when I read Section 9.2, that claim is no longer mentioned, and his Section 9.2 has been stripped down somewhat. Most likely, Dr. M realized that there wouldn't be enough time to teach pi the way he wanted to, so he had to cut it down. When I was planning out my Chapter 8 lessons last week, the full Section 9.2 was still posted, so this means that Dr. M covered pi in his own class fairly recently. I guessed that since pi was covered in Chapter 9 of his 13-chapter text, Dr. M would reach the lesson close to Pi Day -- and I was correct.

The constant pi is introduced very similarly in Dr. M's Section 9.2 and U of Chicago's Section 8-8 -- the main difference is that arc length also appears in U of Chicago's 8-8 while this topic is saved for 9.3 in Dr. M. Both Dr. M and U of Chicago define pi as C/d, which is, of course, the ratio of the circumference to the diameter of a circle.

But here's what Dr. David Joyce says about how pi is defined in the Prentice-Hall text:

The tenth theorem in the chapter [5 of the Prentice-Hall text -- dw] claims the circumference of a circle is pi times the diameter. It is apparent (but not explicit) that pi is defined in this theorem as the ratio of circumference of a circle to its diameter. So the content of the theorem is that all circles have the same ratio of circumference to diameter. This theorem is not proven. It would require the basic geometry that won't come for a couple of chapters yet, and it would require a definition of length of a curve and limiting processes. It would be nice if a statement were included that the proof the the theorem is beyond the scope of the course.

The idea that all circles have the same ratio of circumference to diameter may seem obvious, but Joyce is correct in that it must be proved. We can't simply say that it's true just because we defined pi to be that ratio! After all, what to stop us from saying that phi is defined as the ratio of length of a rectangle to its width, and then claim this proves that the ratio of length to width must be the same for all rectangles? (On the other hand, it is possible to define phi as the ratio of length to width of a golden rectangle, but not for a general rectangle.)

Joyce points out that limiting processes -- that is, Calculus -- is needed to prove the theorem. Dr. M also states that Calculus is needed to prove the circumference formula. Because we can't prove the formula, Dr. M states that it is a postulate:

Dr. M's Circle Similarity Postulate:
The value of pi is a constant, i.e., it is the same for all circles.

We notice how Dr. M calls this the Circle Similarity Postulate. After all, it's because all circles are similar that the ratio of circumference to diameter is the same for all circles. This is what the Common Core Standards expect students to know about circles:

Prove that all circles are similar.

Then again, notice that Common Core seems to expect a proof here. How does Common Core expect students to prove the similarity of all circles without Calculus?

Unfortunately, none of our sources actually prove that all circles are similar. What I'm expecting is something like this -- to prove that two circles are similar, we prove that there exists a dilation mapping one to the other. For simplicity, let's assume the circles are concentric, and the radii of the two circles are r and s. So we let D be the dilation of scale factor s/r whose center is -- where else -- the common center O of the two circles. If R is a point on the circle of radius r, then OR = r, and so its image R' must be a point whose distance from O is r * s/r = s, and so it must lie on the other circle of radius s. Likewise, if R' is a point on the circle of radius s, its preimage must be a point whose distance from O is s / (s/r) = r, and so it must like on the circle of radius r. Therefore the image of the circle of radius r is exactly the circle of radius s.

Of course, this only works if the circles are concentric. If the circles aren't concentric, then it's probably easiest just to compose the dilation with an isometry -- here a translation is easiest -- mapping the center of one circle to that of the other. Therefore there exists a similarity transformation mapping any circle to any other circle. Therefore all circles are similar. QED

We can't really fault Dr. M for not including this proof -- after all, remember he did come up with the clever algebraic derivation of the area of a rectangle from that of a square. It's difficult to come up with some of the cleverer proofs of these basic geometric formulas.

Dr. M's old lesson was interesting, but he had to throw it out to save time. Therefore the activity that I will post today is based on Dr. Wu's lesson. We've already started Dr. Wu's lesson when we covered Section 8-4 last week. As I mentioned that day, the U of Chicago could have easiest segued from Section 8-4 to Dr. Wu's definition of pi as the area of the unit disk.

So today's activity begins with a circle whose radius is divided into tenths, not fifths. We ask the students to count how many squares are filled in -- hopefully, the answer is close to 314. Since each tiny square has area .01, the area of the circle would be around 3.14 or pi.

To get from the area of the unit disk (pi) to the area of any disk (pi * r^2), we are basically using the Fundamental Theorem of Similarity from Section 12-6 of the U of Chicago. This time, though, we are using part (b) of that theorem:

Fundamental Theorem of Similarity:
If G ~ G' and k is the ratio of similitude [the scale factor -- dw], then
(b) Area(G') = k^2 * Area(G) or Area(G') / Area(G) = k^2.

We skipped this formula back when we covered Section 12-6 because at the time, we hadn't learned about area yet. Although Wu attempts to prove a special case of the Fundamental Theorem of Similarity using triangles, it's much easier to do it using squares, as the U of Chicago does. If G can be divided into A unit squares, then G' can be divided into A squares each of length k. And the area of a square of length k is clearly k^2, so the area of G' must be Ak^2. For the circle problem k is the radius r, and A is the area of the unit circle or pi, so the area of a circle is pi * r^2. We can do this right on the same worksheet -- there's already a circle drawn of radius 10 times the length of a square, so instead of the length of each square being 1/10, let it be 1 instead. Then the area of the circle of radius 10 is equal to the number of shaded boxes, or 314, since the old unit square has been divided into 100 unit squares.

We must now get from the area of a circle to the circumference. Notice that Wu actually performs the "limiting processes" that Joyce tells us is necessary to get the circumference! For Wu considers the circumference to be the limit of the perimeter of a regular n-gon inscribed in the circle as n approaches the value of infinity.

Wu calculates the perimeter of the regular polygon by dividing it into triangles. What he is actually using is the formula A = 1/2 ap, where a is the apothem of the polygon and p is its perimeter. Then in the limit, the apothem approaches the radius of the circle and the perimeter of the polygon approaches the circumference, so A = 1/2 rC. Then substituting A = pi * r^2 and solving for C gives the formula C = 2pi * r, which converts to C = pi * d.

The word apothem and the formula for the area of a regular polygon appear in many texts, but the word apothem never appears in the U of Chicago at all. Using the apothem here would save time as it would get us directly from A = 1/2 ap to A = 1/2 rC.

And so this is how I set up my worksheet. If it's possible, I wouldn't mind bringing in some pies (either fruit or pizza) and having the students calculate arc length and possibly even the area of a sector -- to my surprise, the U of Chicago doesn't give the formula for area of a sector either -- by using a slice of pie. The arc length and sector area would have to be given before the student is allowed to eat the slice. Of course, it will be expensive to have enough pie for every student in every class -- but it may be worth it to have a Pi Day worth celebrating.

Now let's move on to some fun stuff. Here are some Pi Day links:

This is the Exploratorium, a science museum located right here in California. I live in Southern California and the Exploratorium is up in San Francisco, so I've never been there. But it is the museum where Pi Day was invented 27 years ago.

This is the National Pi Day site. National Pi Day was, believe it or not, declared an actual holiday by Congress back in 2009. This site is definitely celebrating this year's Pi Day of the Century.

This is a Pi Day site from a few years ago. This site also contains a link to some suggested activities for teachers. Some students had the opportunity to put a pie in their math teacher's faces -- indeed, a student I tutored last year pied his own teacher on Pi Day. It gives more links to Pi Day activities.

But my favorite Pi Day sites are all about music. There are, in general, two types of songs that one plays on Pi Day. The first type takes the digits of pi -- 3.1415926535897932384626433832795... -- and converts them into notes using the C major scale. So 3 becomes E (the 3rd note of the C major scale), 1 becomes C, 4 becomes F, 1 is another C, 5 becomes G, and so on.

Here are some YouTube links to songs formed by converting the digits of pi into notes. The first one was posted by Michael Blake:

This one does the same trick, except it uses the A minor scale rather than C major:

Here's another version that uses base 12 instead of 10:

With so many versions of the pi song out there, there was some controversy a few years back. Blake, whose video I posted above, had to take down his video for some time because someone else claimed that they had copyrighted the pi song. Another YouTube user, Vi Hart, criticized the lawsuit:

And she's absolutely correct -- the plaintiff was hardly the first to write a song just by converting the digits of pi. Indeed, one of the first websites I read on Pi Day has a pi song -- and it was created by a high school girl more seventeen years ago:

I credit Elizabeth, the author of this Sailor Moon parody page, with being one of the first people to inform me about the existence of Pi Day so many years ago. This page is so old that many of the links are to AOL and Geocities pages! As this was well before YouTube, many of her background songs are in MIDI format, which no longer plays on modern browsers. Yet the following link gives evidence that Liz had posted two songs (one major, one minor) that convert the digits of pi into notes:

And the teenager herself admitted that this idea wasn't original to her! She links to another page, created by Daniel Cummerow, who did the same thing with not only the digits of pi, but other constants such as e as well! He posted it on Geocities, which no longer exists, but it has been archived on the Reocities site:

Once again, none of the songs are playable without an old browser that can play MIDI files. But it goes to show that to be consistent according to Hart, Cummerow -- wherever he is -- should sue the plaintiff of the Blake case. Fortunately, the case was laughed out of court -- you can't copyright the number pi!

The other type of song that frequently comes up on Pi Day is the parody song. Basically, one takes a popular song and change the lyrics so that it becomes a song about pi. Two of the most frequently parodied songs are Don McLean's American Pie -- since "pie" sounds like "pi":

and Rebecca Black's Friday -- since it sounds like "Pi Day":

One of my favorite parody songs is based on Eminem's Lose Yourself:

One mathematician whom I've mentioned earlier on this blog, Danica McKellar, created her own video for pi. Here McKellar calls those who create songs about pi "nerds" -- then of course, she turns right around and creates her own song about pi:

The old Sailor Pi site also contains some parody songs. Here's a link to a parody of Rainbow Connection by Paul Williams:

The following, while not actually songs, are some poems that Liz wrote about pi. Notice that one of the poems, "A Scientific American's Dilemma," mentions a "Grand Theory" -- that is, the Grand Theory of Everything that Stephen Hawking sought:

Here's Liz's obligatory American Pi parody. You can see how old it is by the way the song refers to TI-83 graphing calculators rather than the modern TI-84:

Liz also used to have songs about other constants such as e and phi. I still remember her old e song, a parody of Sugar Sugar (of Archies fame), but unfortunately all links to that song are dead. It was fun to go to the Sailor Pi site on Pi Day and have all of these songs -- the pi major and minor songs, American Pi(e), and even the Sailor Moon song -- automatically play in MIDI format as soon as the page had finished loading.

Sometimes I wonder where Liz is now. I believe that she's around my age. I'd like to believe that somewhere out there, she's getting excited about the Pi Day of the Century.

There are also sites that have the lyrics of parody songs, but no recorded music. Many of these are parodies of Christmas songs. (So apparently for math teachers, our Christmas is in March.)

This site has "Oh, Number Pi" ("O Christmas Tree") and "Ring the Bells" ("Jingle Bells"). There is even a song about Ludolph van Ceulen, the 16th century mathematician who calculated the first 35 digits of pi. Of course, the authors of this site couldn't resist changing "Ludolph" to "Rudolph"! Finally, there is a link to a "love song." As it turns out, it's just the digits of pi converted to notes again -- just like Michael Blake, Liz, and all the others.

One of the songs at this link is a parody of "This Old Man." Here is its second verse:

Verse Two:
Number pi
You're so fun
And [so much] better than one
If you think that pi are round Beware!
We all know that [pi r squared].

 For some reason, the last line is incomplete with the phrase "pi r squared" missing. I decided to add the words "so much" to the third line since it appears to be two syllables shorter than the melody.

There also exist math parody songs that have nothing to do with pi. Many of them have been created in high school math classes. For example, this high school in Ohio created a parody of DJ Khaled's All I Do Is Win, changing "win" to "solve" (systems of linear equations):

Indeed, there are several math parody songs by this school posted on YouTube. It's an activity that can be done in math classes around the country.

And of course, I can go on and on about Pi Day and give more and more links and videos. This is why I make every effort to make sure that Pi Day fits on my pacing guide for the year. Sometimes I wish that every day in math class can be Pi Day.

And so I wish everyone a happy Pi Day. Let's make it the best Pi Day this century!

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