Lesson 10-8: The Volume of a Sphere (Day 108)

Lesson 10-8 of the U of Chicago text is called "The Volume of a Sphere." In the modern Third Edition of the text, the volume of a sphere appears in Lesson 10-6.

Today is Day 108 -- at least in my old subbing district. It can be considered as the end of the fifth quaver, which we determine by noting that one-quarter of the way from Day 85 (the end of the first semester) to Day 180 is 107.75. Of course, in my new district today is only Day 99. Thus it's not yet the end of the fifth quaver in my new district, seeing as it's only the second week of the semester.

Oh, and this is a good time to mention another difference between the two district calendars. In my old district, Lincoln's Birthday and President's Day are on two separate Mondays. But my new district does something different -- this one and several other districts observe Lincoln's Birthday on the Friday before President's Day, so that there can be a four-day break in February.

(Three years ago at the old charter, by the way, students observed a five-day weekend, from Thursday to Monday. But the first two days had nothing to do with Lincoln's Birthday. Instead they were PD days -- in other words, we teachers had only a three-day weekend. President's Day itself is the only school-closing February holiday in the LAUSD.)

What does this mean for the blog calendar? Well, I'm already committed to the calendar of the old district on the blog. This means that Lesson 10-9 won't be posted until Tuesday.

Meanwhile, today I subbed in a high school Chemistry class. Since it's a high school class that isn't math, I won't do "A Day in the Life" today.

Today is also 2/7 -- that's right, it's e Day. The digits of e, the base of the natural exponential function, is 2.718281828459045...., and so today is e Day. Last year on e Day, I subbed in a middle school classroom where I didn't wish to mention the number e. But since today is at a high school, I actually mentioned the number e today.

Almost as if to pay for having long breaks earlier this week, today's regular teacher is scheduled for all six periods -- five sections of regular Chem and one section of AP Chem.

I won't say much about the AP Chem class, which just has a worksheet today. I will point out that the number e is indirectly mentioned in the AP text. Exponential decay appears not just with radioactive materials, but ordinary chemical reactions as well. The text gives the "differential rate law":

Rate = -delta[A]/delta-t = k[A]

Thinking back to Starbird, we realize that in the limit, this becomes -d[A]/dt. To solve this differential equation, we must integrate it, obtaining the "integrated rate law":

ln[A]/ln[A_0] = -kt

The presence of ln -- the natural log to base e -- is because the integral of 1/A is ln A. There's also something called a "second-order" reaction, where we integrate 1/A^2 instead of 1/A. This time, the integral is just -1/A, with no need for natural logs or e.

I'll say a little about the general Chem classes -- once again, we can make some comparisons to science at the old charter, but again, high school science isn't appropriate for middle school.

Just like Wednesday's subbing, today's students are taking a "group test," although the regular teacher insists that it isn't really a test. The questions refer to a previous lab where a mysterious yellow liquid dissolves in water, but another blue liquid doesn't, and they must answer why.

The teacher wants students to distinguish (in color) between work that the group members come up with on their own vs. work that they must research on Chromebooks or in the text. And so I do what other teachers would do -- insist on closed (Chrome)books until the midpoint of the period, and then allow them to do research.

I suppose I could have had group tests at the old charter school for science (even as I maintain individual tests for math). I'm not sure how that would have worked out, though.

In all classes, I sing the "Number e Song." This song, a parody of "Sugar Sugar" by the Archies, was first written by Elizabeth Landau, aka Bizzie Lizzie. Apparently Landau has recently left JPL to work at NASA headquarters near Washington DC. I notice that Landau still writes song parodies, although these are more related to NASA and outer space, not numbers like e or pi:

http://www.lizlandau.com/music/

My lyrics for the "Number e Song" are incomplete, and I was hoping that Landau would post her original lyrics again, but there's no such luck.

On the Eleven Calendar, today is Sunday, the third day of the week:

Resolution #3: We remember math like riding a bicycle.

Even though this isn't a math class, this rule fits the first part of the group assignment -- students must remember as much Chemistry as they can before looking at outside resources.

While the students are a bit talkative the entire day, one problem does occur in the last period of the day -- and it's indirectly related to the end-of-the-day clean-up yet again.

As in the earlier classes, I tell the students to work without Chromebooks until the midpoint of the period when they can use them. The problem is that so many students are talking that no one hears me at that time, and so no one takes out a Chromebook. This means that the groups are now at a disadvantage compared to the earlier periods. Earlier, all groups (of 3-5 students) finish except for one group of two, whom in hindsight I should have sent to other groups. But in the last period, several groups fail to finish, because none of them look up what they can't remember.

And of course, I exacerbate the problem by ending a few minutes earlier so that they'd have time to put away the Chromebooks -- the ones that this period never uses! It goes without saying that this is a struggle -- why should they put away laptops that they never use that day? (This problem also occurred last week at a middle school -- one class must charge another period's Chromebooks.)

What I probably should have done is at the midpoint of the period, walk around to make sure that groups are actually using the Chromebooks -- especially those groups that appear to be struggling to complete the assignment.

This is also part of a bigger issue in my classes -- I often need to give important instructions, but so many students are talking that no one hears me, and they all do the wrong thing. What I usually ended up doing was argue -- and we all know what problems that leads to.

Today I avoid arguing, so it's at least a step in the right direction. A student or two put Chromebooks away, and then I sing the e song, even though not as much work is completed as the other classes. As for what to do when I must give critical instructions and no one's listening, that's something for me to figure out another day.

Lecture 24 of Michael Starbird's Change and Motion is called "Calculus Everywhere." Here is an outline of this lecture:

I. A method using derivatives to solve equations is called the Newton-Raphson method.
II. We have taken a striking journey, exploring a bit of one of the greatest intellectual achievements in human history.
III. The potency of calculus is a testament to the power of abstraction.
IV. The development of calculus is a long story of human struggle and triumph. But the story is not over.
V. Not all functions are smooth.
VI. At this moment in history, computers and biology and biotechnology are having the biggest influence on human life.
VII. Newton said, "If I have seen farther than others, it is because I have stood on the shoulders of giants." In exploring calculus, we have stood on Newton's shoulders, on Leibniz's shoulders, on Archimedes's shoulders, and those of many other great thinkers. We have communed with some truly spectacular high points of human thought. All of us can share an appreciation for that giant conceptual development that shaped and continues to shape history -- calculus.

Since most of this lecture is just a summary of the course, I'll just write about how Starbird uses the Newton-Raphson method to find the value of an irrational number -- sorry, it's sqrt(5), not e:

f(x) = x^2 - 5 (to find sqrt(5), a zero of this function)
f'(x) = 2x

Guess: x = 2
f(2) = -1
f''(2) = 4
tangent line through (2, -1) crosses x-axis at 2.25

Guess: x = 2.25
f(2.25) = 0.0625
f' (2.25) = 4.5
tangent line through (2.25, 0.0625) crosses at 2.2361111...

Guess: x = 2.2361111...
next iteration gives 2.236067978...

checking for accuracy:
(2.25)^2 = 5.0625
(2.2361111)^2 = 5.000192852
(2.236067978)^2 = 5.000000002

The professor mentions that most curves aren't smooth -- yes, most curves are continuous everywhere but differentiable nowhere -- apologies to teacher Sam Shah and his blog name:

https://samjshah.com/

One thing I notice about Starbird's lectures on this e Day is that he says very little about e. I know he wanted to skip to the easier parts of Calculus, but what can be easier than the fact that the derivative of e^x is e^x? The only mention of e was in Lecture 22 on business and econ -- to make one of the integrals easier, he assumed that interest was compounded continuously, A = Pe^(rt). At no point does he explain what e is, or that it equals 2.718281828... approximately.

This is what I wrote last year about today's lesson. Here, I compared the treatment of the volume in two different texts, an old McDougal Littell California seventh grade text and the U of Chicago text:

The McDougal Littell text, in Lesson 10.7, demonstrates the sphere volume formula the same way that it does the cone volume formula. We take a cone whose height and radius are both equal to the radius of the sphere, and we find out how many conefuls of sand fill the sphere. The text states that two conefuls make up a hemisphere, and so four conefuls make up the entire sphere.

But of course, we want to derive the formula more rigorously. Recall that Dr. David Joyce states that a limiting argument is the best that can be done at this level -- but I disagree. Dr. Franklin Mason, meanwhile, enthusiastically gives another derivation of the sphere volume formula, and Dr. M's proof also appears in the U of Chicago text. Recall that Dr. M considers this day on which the sphere volume formula -- Lesson 12.6 of his text -- is derived to be one of the three best days of the year.

The U of Chicago text mentions that this proof uses Cavalieri's Principle. But it was hardly the mathematician Cavalieri who first proved the sphere volume formula. Indeed, according to Dr. M [and Michael Starbird -- 2020], this proof goes all the way back to Archimedes -- the ancient Greek mathematician who lived a few years after Euclid. (It's possible that their lives overlapped slightly.)

Here is a Square One TV video about Archimedes:

We mentioned earlier that Archimedes used polygons to determine the value of pi (also known as Archimedes' constant) -- hence the line in the song, "He was busy calculating pi." He was also famous for using the principle of buoyancy to determine whether the king's gold crown was a fake, and this is also mentioned in the song. Legend has it that the Greek mathematician was so excited when he discovered his principle -- he had been in a public bath at the time -- that he ran down the streets naked and shouted out "Eureka!" to announce his discovery. The Greek word eureka, meaning "I have found," is the motto of my home state of California.

But Archimedes himself actually considered the discovery of the sphere volume formula to be his crowning achievement -- to the extent that he requested it to be engraved on his tombstone. So let's finally derive that formula the way that Archimedes did over 2000 years ago. And no, he didn't simply drop a ball into water to determine the formula. Archimedes' sphere formula has nothing to do with Archimedes' principle of buoyancy.

We begin by considering three figures -- a cone, a cylinder, and a sphere. We will use the known volumes of the cone and cylinder to determine the unknown volume of the sphere -- thereby reducing the problem to a previously solved one.

Our cylinder will have the same radius as the sphere, while the height of the cylinder will equal the diameter (i.e., twice the radius) of the sphere. This way, the sphere will fit exactly in the cylinder.

Our cone, just like the cone mentioned in McDougal Littell, will have the its height and radius both equal to the radius of the sphere. Such a cone could fit exactly in a hemisphere. But we want there to be two cones, so that their combined height is the same as that of the cylinder. We set up the cones so that they have a common vertex (i.e., they are barely touching each other) and each base of a cone is also a base of the cylinder. The two touching cones are often referred to as a "double cone" -- Dr. M uses the term "bicone." (A bicone is also used to justify to Algebra II students why a hyperbola is a conic section with two branches. A hyperbola is the intersection of a bicone and a plane, such that the plane touches both cones.)

The focus is on the volume between the cylinder and the bicone. The surprising fact is that this volume is exactly equal to the volume of the sphere! Here is the proof as given by the U of Chicago:

[2020 update: Recall that this is slightly different from the proof given by Starbird in his Lecture 11.]

"...the purple sections are the plane sections resulting from a plane slicing these figures in their middles. These purple sections are congruent circles with area pi * r^2. At h units above each purple section is a section shaded in pink. In the sphere, by the Pythagorean Theorem, the pink section is a small circle with radius sqrt(r^2 - h^2). The area of this section is found using the familiar formula for the area of a circle.

"Area(small circle) = pi * sqrt(r^2 - h^2) = pi(r^2 - h^2)

"For the region between the cylinder and the cones, the section is the pink ring between circles of radius r and h. (The radius of that circle is h because the acute angle measures 45 degrees, so an isosceles triangle is formed.)

"Area(ring) = pi * r^2 - pi * h^2 = pi(r^2 - h^2)

"Thus the pink circles have equal area. Since this works for any height h, Cavalieri's Principle can be applied. This means that the volume of the sphere is the difference in the volume of the cylinder (B * 2r) and the volume of the two cones (each with volume 1/3 *B * r).

"Volume of sphere = (B * 2r) - 2 * (1/3 * B * r)
                            = 2Br - 2/3 * Br
                            = 4/3 * Br

"But here the bases of the cones and cylinder are circles with radius r. So B = pi * r^2. Substituting,

"Volume of sphere = 4/3 * pi * r^2 * r
                            = 4/3 * pi * r^3." QED

The sphere volume is indeed the crowning achievement of Chapter 10. We began the chapter with the volume of a box and end up with the volume of the least box-like figure of all, the sphere. The start of Lesson 10-8 summarizes how we did this:

"It began with a postulate in Lesson 10-3 (volume of a box). Cavalieri's Principle was then applied and the following formula was deduced in Lesson 10-5 (volume of a prism or cylinder). A prism can be split into 3 pyramids with congruent heights and bases. Using Cavalieri's Principle again, a formula was derived in Lesson 10-7 (volume of a pyramid or cone). In this lesson, still another application of Cavalieri's Principle results in a formula for the volume of a sphere."

So take that, Dr. Katharine Beals! After all, she was the one who derided Cavalieri's Principle as progressive fluff that the Common Core tests on instead of actual math. But without Cavalieri's Principle, we'd be stuck finding the volumes of only boxes and their unions. Well, I suppose if we simply declared the volume formulas by fiat (i.e., as postulates) rather than actually deriving them, then Cavalieri's Principle is not needed. But if we want to prove them, then the Principle gives us an elegant proof of the sphere volume formula that was discovered over 2000 years before there ever was a Common Core -- a proof that, if mastered, should permit one to date a mathematician's daughter (as Beals mentioned on her website regarding the Quadratic Formula proof).

Sadly, we don't know whether Archimedes ever dated anyone's daughter, or whether he ever had daughters of his own. His life ended tragically, being captured by an enemy army. Legend has it that he was busy working on a geometry problem when the Roman army captured him. His last words before he was killed are said to be, "Noli turbare circulos meos" -- Latin for "Do not disturb my circles!"

Today is an activity day. It's based on this lesson's Exploration question:

25. Unlike a cone or cylinder, it is impossible to make an accurate 2-dimensional net for a sphere. For this reason, maps of the earth on a sheet of paper must be distorted. The Mercator projection is one way to show the earth. How is this projection made?

Notice that the correct answer to this question is quite complex. Here's a link that describes both a misconception and the correct answer:

https://www.math.ubc.ca/~israel/m103/mercator/mercator.html

Let's get back to e Day. Here is a post from this year mentioning e Day:

https://www.ipwatchdog.com/2019/02/07/in-support-inaugurating-february-7-world-e-day/id=106100/

No math holiday is complete without music. Apparently e is a magic number. (Well, the singer does use e in the magic formula e^(i pi) + 1 = 0, so I guess that's magic.)

Here's another e song -- and this one actually explains why the magic formula works:

Here is Michael Blake's e song, based on the digits of e:

Here's how we can write our own song based on the digits of e in Mocha:

http://www.haplessgenius.com/mocha/

70 N=7
80 FOR X=1 TO 44
90 READ A
100 SOUND 261-N*(18-A),4
110 NEXT X
120 DATA 2,7,1,8,2,8,1,8,2,8
130 DATA 4,5,9,0,4,5,2,3,5,3
140 DATA 6,0,2,8,7,4,7,1,3,5
150 DATA 2,6,6,2,4,9,7,7,5,7
160 DATA 2,4,7,0

Notice that whereas we can avoid the zero problem in pi, zero appears early in e. Blake uses a rest for the zero in his song, but I decided to use 18EDL, with zero as the tonic. I included 44 digits of e so that the song ends on the tonic. Oh, and using N=7 places that tonic on red E. A song about the number e needs to be in the key of E. (The fundamental tonic for 16EDL would be E, but 16EDL contains only nine notes from tonic to tonic when we need ten.)

Since I lament how Starbird doesn't discuss the role of e in Calculus, let's make up for it with this 3 Blue 1 Brown video:

Numberphile, of course, has an e video.

Here's another new e video, by Zach Star:

Since I subbed in a Chemistry class today, here's an e video by "The Organic Chemistry Tutor":

Let me rewrite the lyrics to the Bizzie Lizzie e song that I perform today in class:

1st Verse:
I just can't believe the loveliness of graphing you.
I can't believe you're more than two.
I just can't believe the loveliness of graphing you.
I can't believe you're more than two. (to Refrain)

2nd Verse:
I just can't believe your digits go forever now.
As long as a number can be.
I just can't believe your digits go forever now.
As long as you're the number e. (to Bridge)

Bridge:
Put a little cash in the bank, money.
Put a little cash in the bank, baby.
I'll make more next year, yeah, yeah, yeah!
Put a little cash in the bank.

100% interest on my money.
Compound it continuously, baby.
I'm gonna take the limit now, yeah, yeah, yeah!
My cash is multiplied by you, e. (to Refrain)

The bridge is mostly mine -- Landau didn't mention anything about money in her song. I chose to include money since it rhymes with the original Archies lyrics ("honey") as well as retell the story of Jacob Bernoulli's discovery of this constant.

By the way, let's think back once more to e Day itself. Some people would suggest that in deference to Europeans and other little-endians, e Day should be 2nd July, not February 7th.

Meanwhile, e Approximation Day works out to be July 19th as 19/7 = 2.7142857... is approximately equal to e. So e Approximation Day would be just three days before Pi Approximation Day.

One last constant I wish to celebrate on the calendar is closely related to e -- eta. The number eta is used in tetration (which I explained in a June 2016 post). We define eta as e^(1/e), because if we define the sequence:

a_0 = 1
a_(n + 1) = eta^(a_n)

then the limit of a_n as n approaches infinity is e. But if we replace eta with any greater base -- even eta + epsilon for small epsilon -- the sequence diverges. The tetrater Gottfried Helms was the first to use the symbol eta for e^(1/e).

eta = e^(1/e) = 1.44466786...

Thus Eta Day would be January 4th -- yes, just two days before Phi Day. Little-endians might observe Eta Day in April -- on either the 1st (1/4) or 14th (14/4). As for Eta Approximation Day, we can clearly approximate eta by the rational number 1.444..., which is 13/9 for September 13th. We are more than two decades away from Eta Day of the Century.
Remember that Monday is Lincoln's Birthday in my old district (and on the blog calendar), and so my next post will be Tuesday.

END