Lesson 6-4: Miniature Golf and Billiards (Day 64)

This is what Theoni Pappas writes on page 16 of her Magic of Mathematics:

"Ever wonder about the f-stop number of a camera? Where did it get its name? How is it determined?"

This is the only page of the subsection "Mathematics & the Camera." Yes, the U of Chicago text mentions cameras in Lesson 15-4, but nothing about f-stop numbers. Well, Pappas tells us what the f-stop number means in the very next sentence:

"'f' stands for the mathematical term factor."

Here are some more excerpts from this page:

"Photographers use what is known as the f-number system to relate focal length and aperture. For example,

f4 = 80 mm lens/20 mm aperture.
f16 = 80 mm lens/5 mm aperture.

Working with f-stop numbers and shutter speeds, you can manually decide how much of the photograph you want in focus."

The only picture on this page is that of a camera lens, with its f-stop numbers listed. As Pappas points out, the size of the lens remains constant, and so it's the aperture, or opening, that changes.

I remember seeing f-stop numbers on Square One TV. In the Mathnet episode "View From the Rear Terrace," Kate Monday adjusts the f-stop number on her camera as she spies on the villain from her upstairs window. Apparently, most f-stop numbers are powers of 2, like 4 and 16 above. I also recall noticing the f-stop numbers 1.4 and 2.8 between 1 and 4 -- I suspect the intention is for the f-stop numbers to form a geometric series, and so 1.4 approximates sqrt(2).

Chapter 4 of George Szpiro's Poincare's Prize is called "An Oscar for the Best Script." Here's how the chapter begins:

"Even though Poincare was an outstanding engineer, his first love remained mathematics. The curriculum at the Ecole Polytechnique, with its classes in mathematics and physics, catered to his tastes."

In this chapter, Szpiro writes about Poincare's return to the mathematical world. His first is to earn his doctorate, but there are a few problems with his dissertation:

"When he presented another paper on differential equations to Jean Darboux, who later became secretary of the Academie des Sciences and would pronounce the eulogy at Poincare's funeral, the Sorbonne professor extolled the essay's virtues but at the same time strongly criticized its rigor."

Of course, Poincare fixes the errors and is awarded his doctorate. For him, 1881 becomes a big year:

"Poincare was appointed to the chair of mathematical physics and probability at the Sorbonne. Simultaneously, he obtained a teaching position at his alma mater, the Ecole Polytechnique. In the same year he married Louise Poulain d'Andecy. The couple had three daughters and a son."

At this time, Poincare works to solve a huge problem in astronomy regarding planetary motion. The problem goes all the way back to 17th century German mathematician Johannes Kepler:

"The observations that he had received, or rather stolen, from his predecessor at Prague, the Imperial Mathematician Tycho Brahe, were the most exact data at the time. But even so, they contained minute errors."

The reasons for the errors is that the known equations solve only the two-body problem, where the motions of only two bodies (such as the sun and a planet) are considered. But there are many more bodies in the solar system, and each one is affected by the gravitational pull of all the others.

The Oscar mentioned in this title is actually a king:

"When he was fifty-five years old, King Oscar II of Sweden and Norway, a great supporter of science, was wondering how to celebrate his sixtieth birthday, which would fall on January 21, 1889. Gosta Mittag-Leffler, a Swedish mathematician [...] proposed the establishment of a prize for an essay about the n-body (here n means any number, including 3) problem as an appropriate way to celebrate the king's birthday."

Let's learn more about Oscar's prize:

"The monarch charged Mittag-Leffler with the minutiae of organizing the prize. He was to work out the rules, choose the jury, announce the prize, and finally select the winner."

And so it was determined that the contest would consist of three questions. One mathematician who doesn't like the questions is Leopold Kronecker -- a known spoilsport. (Indeed, Kronecker is one of the first to challenge Georg Cantor's set theory.) Szpiro writes:

"Kronecker immediately criticized first one, then another of the questions. But his criticisms were recognized as the misgivings of a grouch and not taken seriously."

A grouch -- well, this Kronecker reminds me of another Oscar, the one who lives on Sesame Street.

Anyway, over the next three years, a dozen mathematicians submit entries for the contest. As for the winner -- well, there's no real suspense who the winner is, considering whose biography we've been reading in the past two chapters. As the author puts it:

"With great trepidation, the envelope that contained the name was opened in front of a hushed crowd of invited guests: 'And the winner of the Oscar for the best script is ... La La Land [oops, wrong envelope -- dw] Henri Poincare, for "The problem of the three bodies."'"

Okay, that was a bad joke on my part. (I should have said Hidden Figures, not La La Land, since the movie, like Poincare's paper, is related to astronomy.) But as Szpiro explains, the winner's name really is in an envelope, since the papers are submitted anonymously to avoid possible biases.

The author describes Poincare's Oscar-winning paper:

"He proved rigorously that no analytical solutions (i.e., no elegant formulas) exist that would describe the position of the bodies at all times. The surprising implication of this result is that the positions of the planets in our solar system cannot be predicted with total precision."

(By the way, I'm sorry for mentioning Hidden Figures again, but Katherine Johnson also encounters the impossibility of finding an exact analytical solution when one of the bodies is a capsule. This is why she must resort to finding a numerical solution via Euler's method.)

Szpiro continues:

"That could have been the end of the story and all participants could have gone on with their lives. But the troubles were only just about to start."

As it turns out, Poincare's paper contains a fatal flaw. Szpiro explains that it has something to do with the convergence of a series of numbers:

"A series of numbers is an infinite number sequence, such as 1 + 2 + 3 +.... It is said to converge if its entries add up to a finite sum. For example, 1 + 1/2 + 1/4 + 1/8 +... = 2.0, hence the series converges. If the sum tends toward infinity as the number of terms increases, the series is said to diverge."

Szpiro fails to note a coincidence here -- the sum of a series is exactly what Poincare fails to prove in his high school bac exam, and here it bites him again. The problem is when we have a slow-growing yet divergent series. The usual example given, of course, is the harmonic series -- which we've already discussed in reading Ogilvy's book.

"Thus, even if the terms of the series decrease so rapidly that the computation of a planet's orbit can be computed to a precision of several digits behind the decimal point for very, very long time periods -- we are speaking many millions and billions of years here -- convergence of the series must be proven mathematically even as the number of the series's terms tends to infinity."

Part of Poincare's problem is that he fails to explain his work properly. Now maybe you can see why the Common Core requires students to explain their work:

"As he had done in his childhood, when his speech was often unintelligible because ideas came to him faster than he could pronounce them, in his mathematical work too he did not always spend sufficient time formulating the details."

The mathematician tries to halt publication of his erroneous proof, but his efforts are in vain:

"On December 4, three days after Poincare posted the letter, Mittag-Leffler informed him by return mail that prepublication copies of the faulty paper had already been sent out to a few select personalities."

At this point panic ensues -- if word gets out that the prize-winning proof contains an error, the reputations of all people involved, including the king, would shatter into dust:

"Hoping against hope that Poincare would somehow be able to rectify things, Mittag-Leffler took pains to cover up all tracks of the error."

But one of the prize judges, Karl Weierstrauss, eventually hears rumors about the error. And of course, our resident "grouch" wants to know more about a possible flaw:

"And Kronecker was quite willing to listen. So when Weierstrauss was asked by his colleagues about a possible error in Poincare's prize essay, he found himself in an extremely uncomfortable position."

And so Weierstrauss writes a series of complaint letters:

"More to the point, nobody could demand from a juror that he guarantee the truth of every statement in such an extensive manuscript, especially when many computations were not actually performed but only hinted at. The latter was a thinly disguised swipe at Poincare's irritating habit of omitting whole sections of a proof because they seemed clear to him."

In the end, publication of the faulty proof is finally cancelled:

"Now it was up to Poincare. Work had to be done that went far beyond a simple revision of the essay. The new manuscript was truly pathbreaking. It provided the first traces of chaos theory, which would become so popular only a century later. The possibility of chaotic movement of bodies was just what he had overlooked in the erroneous first version of the essay."

Of course, Pappas mentions chaos theory a couple of times in her book. She hints at it yesterday in the discussion of complex systems (and it doesn't get much more complex than the solar system), and there's a further discussion later in the book (which we read earlier this year -- see my May 8th and 9th posts for more info).

As it turns out, Poincare's fix works, and his contemporaries never see evidence of his original paper filled with errors. Unfortunately, he must spend his entire prize money on removing the mistakes (though the title of Oscar winner is never revoked):

"Only many years later, when historians of science searched through the archives of the Mittag-Leffler Institute and compared the original version with the published paper, was the true magnitude of the differences discovered."

Szpiro tells us that after this fiasco, the Oscar prize -- intended to be given out every four years -- is never awarded again. But a few years later, while the science-loving Oscar is still king of Sweden and Norway, another prestigious award is established -- the Nobel Prizes. (Hmmm, I wonder whether the Poincare problem is a reason that there is no Nobel Prize in mathematics.)

Half a century later, Russian mathematician Andrei Kolmogorov lectures on the n-body problem:

"The subject was the thorny question of what happens to periodic orbits of bodies when small perturbations disturb their course."

Even though Kolomogorov gives a partial solution, it isn't complete:

"The stability of our solar system, which consists of nine planets, not just three, is far from guaranteed by science."

But what happens to Poincare? Szpiro explains:

"Poincare was led away from the manipulation of numbers and formulas and toward the visualization of curves and flows. The subdiscipline he thus pioneered was to become algebraic topology. It will concern us throughout the remainder of this book."

Notice that so far we haven't actually mentioned the Poincare Conjecture yet. As we now see, it is a conjecture in algebraic topology.

The author tells us about the mathematician's final years, after he suffers prostrate problems:

"Back in Paris, Poincare soon resumed his old schedule and everything was normal for another four years. But on July 9th, 1912, he had to undergo surgery."

And, as we found out yesterday, he dies eight days later at the age of 58. Szpiro writes of his legacy:

"The university in his hometown of Nancy, for example, is named after him, and so is one of the town's best-known lycees [high schools -- thank you, my French teacher!]."

Szpiro ends the chapter with a note about Poincare's famous relatives:

"While we're at it, there is also the Avenue du Recteur Poincare [in Paris], in memory of Henri's cousin Lucien, an important university administrator, and in the thirteenth arrondissement an avenue carries the name of his brother-in-law, Emile Boutroux."

Lesson 6-4 of the U of Chicago text is called "Miniature Golf and Billiards." In the modern Third Edition, we must backtrack to Lesson 4-3 to play miniature golf.

This is what I wrote two years ago about today's lesson:

Today we proceed with the next lesson in the text. Lesson 6-4 of the U of Chicago text is all about applying reflections to games such as miniature golf and billiards. I don't need to make any changes to the lesson, so I can just keep what I wrote last year for this lesson almost intact:

One of my favorite TV programs is The Simpsons -- I've been watching it for decades. One of its earliest episodes, having aired almost exactly 27 years ago, was called "Dead Putting Society." In this episode, Bart Simpson is preparing for a miniature golf competition. His sister Lisa shows him how he can use geometry to help him make a difficult shot. After saying this, Bart proclaims, "You've actually found a practical use for geometry."

Lesson 6-4 of the U of Chicago text discusses miniature golf and billiards. Just as Bart learns in this above video, one can use geometry to determine where to aim.

The key is reflections -- one of the important transformations in Common Core Geometry. It is often said that when a ball bounces off a wall, the angle of incidence equals the angle of reflection. The text describes where to aim a golf ball G so that it bounces off of a wall and reaches the hole H:

"In this situation, a good strategy is to bounce (carom) the ball off a board, as shown [in the text]. To find where to aim the ball, reflect the hole H over line AB. If you shoot for image H', the ball will bounce off AB at P and go toward the hole."

We can write a two-column proof to show why the angle of incidence -- the angle at which the ball approaches the board, which is BPG -- equals the angle of reflection APH:

Given: H' is the reflection of H over line AB.
Prove: Angle APH = Angle BPG

Statements                            Reasons
1. H refl. over line AB is H'   1. Given
2. Angle APH = Angle APH' 2. Reflections preserve angle measure
3. Angle APH' = Angle BPG 3. Vertical Angle Theorem
4. Angle APH = Angle BPG  4. Transitive Property of Equality

Notice that for this proof, I've skipped a few steps. Technically, we should write that the reflection images of both A and P are the points themselves, since they lie on the mirror (Definition of Reflection), and so angle APH' is the image of APH (Figure Reflection Theorem). But I'm tired of writing that over and over again -- how much less, then, will the students want to write that.

Of course, this only works if the ball caroms only once. The U of Chicago text describes the game of billiards, where the player is required to bounce the ball off of three cushions. The text writes:

"Pictured [in text] is a table with cushions w, x, y, and z, the cue ball C, and another ball B. Suppose you want to shoot C off x, then y, then z, and finally hit B. Reflect the target B successfully over the sides in reverse order: first z, then y, then x. Shoot in the direction of B"' [...]

"Notice what happens with the shot. [...] On the way toward B"', it bounces off side x in the direction of B". On the way toward B", it bounces off y in the direction of B. Finally it hits z, and is reflected to B. [End of quote]"

In the video clip above, four caroms are required for the golf ball shot by Bart to find the hole. But unfortunately, the path of the ball as drawn on the show is impossible. To see why, let's label the direction from the starting triangle to the hole "North," and all the walls appear to meet at right angles, so they are all oriented in the north-south or east-west directions. Bart begins with the ball slightly to the right side of the starting triangle, so the initial direction of the ball is northwest. After hitting the first east-west wall, the ball is now traveling southwest. But then, after hitting the second east-west wall, the ball should be traveling northwest again. Indeed, we can use the Alternate Interior Angles Consequence and Test Theorems to prove that the path of the ball after hitting two walls should be parallel to the original direction of the ball. Yet the show depicts the ball as travelling due north after hitting two walls.

In fact, we can show that the only correct path to make the ball arrive in the hole involves hitting a north-south wall -- most likely the wall to the far west (in front of "Do not sit on statuary"). Only by hitting a north-south wall can the direction change from anything-west to anything-east, which is necessary for the ball to approach the hole. Lisa's advice to her brother is sound, but the way that it is animated is geometrically impossible.

For this worksheet, I decided to reproduce Bart's golf course, but tilt two of the walls in order for the path of the ball to be geometrically correct. I used equilateral triangle paper to create this page [2017 update: yes, the isometric graph paper I used for another assignment last year], so that the students will be able to figure out the paths without needing a ruler or protractor, for classrooms in which these are not available. Just as for Bart, four caroms are needed to get from G to H.

Working backwards on this worksheet, we can determine the path from A to H by reflecting H to the point H', then aiming from A to H'. But to determine a path from B to H using two caroms, we can't reflect H to first H' and then H", because H" would be well off the page. It may be better to aim from B to A', the reflection of A in the necessary wall.

Notice that if there are two walls and the paper is large enough, it may be actually possible to perform both of the necessary reflections. If the two walls meet at right angles, it is fairly easy to perform both reflections -- because the composite of those two reflections is a rotation. So to find the direction to aim at, we take the target point and rotate it twice 90, or 180, degrees around the point where the two walls meet. And if the ball is bouncing off of two parallel walls, then the composite of the two reflections is a translation, so we can just translate the target twice the distance between the two walls.

Officially, this lesson is the Guided Notes for Lesson 6-4 of the U of Chicago text. But this lesson naturally lends itself into a group activity -- the teacher can provide additional golf courses for the students to solve, or even allow the students to make up their own via the Bonus Question. It is a nice activity to give right before the week-long Thanksgiving break.