Monday, November 27, 2017

Lesson 6-6: Isometries (Day 66)

This is what Theoni Pappas writes on page 27 of her Magic of Mathematics:

-- "So 24:00 hours would be 30:00 hours, 8:00 would be 10:00, and so on," Selath explained.
-- "Whatever works best for you," I replied, a bit confused.

This is the fourth page of the section "A Mathematical Visit." We're in the middle of the narrator's visit to the mathematician Selath's house, where there are so many strange objects around. On Black Friday, we read about was how water was stored "in" a Klein bottle, but then two pages of this story were blocked by the weekend.

Now the eccentric mathematician is in the middle of explaining that his clock tells time in octal. After all, a shift of work is eight hours, so why not tell time in octal?

Technically, we're about a month away from our annual Calendar Reform post, but some Calendar Reformers do wish to fix the clock as well. Pappas tells us that the narrator's watch reads 5:30 while Selath's clock reads 21:30 -- so the minutes are still apparently in decimal.

[8] (default octal -- Do you remember these change of base signs I explained last summer?)

Or maybe there are only 60 octal minutes in a hour instead of the usual 74. It may seem more logical in an octal world to round 74 up to 100 minutes per hour, but let's think about it:

  • With 30 hours in a day, the length of an hour hasn't changed. This means that we can keep all the old time zones.
  • With 60 minutes in an hour, the minutes are now longer. But then :30 retains its meaning as the half-hour, while the quarter hours are now :14 and :44.
  • There now be 113 seconds in a minute. This number isn't round or convenient, but it allows the second, and all SI units dependent on the second, to retain its value. The hour and second keep their old lengths, with only the minute changing.
The following link gives the time in several different bases, including octal:


It is a pure octal clock, with 100000 "seconds" in a day, each nearly thrice as long as the SI second.

Let's continue on this page:

-- "Now let's go to the master bedroom," [said Selath.]
-- And off we went, passing all sorts of shapes and objects I'd never seen in a home before.
-- "The master bedroom has a small semi-spherical skylight in addition to...."

Well, we won't know what the skylight is in addition to until tomorrow, because this sentence ends on the next page. The only picture on this page is another picture of the interior of the house -- which includes the Klein bottle, but not the skylight in the bedroom.

Today is Cyber Monday. So of course, I had to order something on Amazon today -- and that something is the Pappas Mathematical Calendar for 2018. After she disappoints us in 2017, she does indeed return for 2018. This means that we won't read her Magic of Mathematics in 2018. (But I do notice, in browsing on Amazon, that she wrote Math-a-Day: A Book of Days for Your Mathematical Year, back in 1999. That's the book I probably should have used as a replacement for her calendar this year, instead of Magic. But of course I enjoyed her Magic this year anyway!)

Chapter 10 of George Szpiro's Poincare's Prize is called -- hold on, Chapter 10? Oops, I forgot:

[a] (default decimal)

Chapter 8 of George Szpiro's Poincare's Prize is called "Dead Ends and a Mysterious Disease." Here is how it begins:

"The first person to take a serious crack at the Poincare Conjecture was the Englishman John H. C. Whitehead, who usually went by his middle name Henry."

This chapter is about the earliest attempts to prove the famous conjecture. I've mentioned Henry's uncle Alfred North Whitehead in previous posts, most recently in my October 13th post, as Hoffman mentions him in connection with Erdos. (Oh, and even in decimal base, Lesson 6-6 of the U of Chicago text lines up with Chapter 8 of Szpiro rather than Chapter 6. We read two chapters over Thanksgiving in order to finish his fourteen-chapter book by early next week.)

But today we're reading about his nephew Henry. Szpiro writes that the young Henry -- like several other mathematicians we've seen so far -- is somewhat of a "dren":

"Somewhat careless in his work and not very good at mathematical manipulations, he nevertheless managed to pass the entrance examination to Eton, the most prestigious of England's boys' schools."

During World War II, Whitehead works for Alan Turing -- of Imitation Game fame. Let's pick up the story after the war:

"In 1947 he was named Waynflete Professor of Pure Mathematics at Oxford's Magdalen College. Upon the death of his mother in 1953, Whitehead inherited some cattle from her estate, and he and his wife [Barbara Sheila Carew Smyth] established Manor Farm in the village of Noke, eight kilometers north of Oxford."

Of course, the reason we read about Whitehead is that he works on the Poincare Conjecture:

"At that time it was just one of a host of open problems; nobody knew how fiendishly difficult the proof would be."

At first he believes that he has a solution, but of course he is mistaken:

"The sinking feeling one gets in one's stomach with the realization of a published error is not to be wished on anybody."

The reason his proof is invalid is that it doesn't apply to a certain manifold -- which is now known as the Whitehead manifold:

"So...no proof of Poincare's Conjecture. End of story. So maybe the Whitehead manifold is a counterexample to the Poincare Conjecture. Unfortunately, it also falls short in this regard."

The next mathematician Szpiro mentions in this chapter is Christos Papakyriakopoulos -- known as Papa, for short. The author begins by describing Papa's entanglement in local Greek politics:

"Papa voted openly again the king's return, but the majority went the other way and the king returned to Athens."

And Papa must deal with the invaders during World War II -- and then with the Cold War. He joins a group of Communists as he works at the Institute of Technology in Athens:

"But the atmosphere at the institute was not friendly towards Communist sympathizers. When the professor for whom he had worked as an unpaid assistant was fired, he had to start looking for other pastures."

And those greener pastures are across the sea -- in the United States, at Princeton University:

"During his first ten years in America, Papa produced proofs to three important open problems: the Loop Theorem, Dehn's Lemma, and the Sphere Theorem."

Szpiro tells us that sadly, Papa never returns to his homeland, as he dies of stomach cancer at 62 (and the author adds that many topologists, including Whitehead and Poincare himself, die in their late fifties or early sixties.) He writes:

"The National Technical University in Athens established a prize in Papa's memory, to be awarded every year to an outstanding freshman in mathematics. It is a reflection of his solitude that throughout his career Papa never had even a single coauthor."

So in other words, Papa has an Erdos number of infinity. Now Szpiro returns to Papa's attempts to prove the Poincare Conjecture. The mathematician shows his paper to a young grad student, Bernard Maskit, and his thesis advisor Lipman Bers:

"The bright graduate student was successful, and Bers then told his son-in-law, Leon Ehrenpreis, at New York University, about this."

By now, you probably know the story -- Maskit finds an error in Papa's proof, but not until after the mathematician tries to publish it:

"Why did he not rectify his announcement of March 1962 to the Bulletin [of the American Mathematical Society] if he was already aware of the error in December 1961?"

Szpiro now moves on to Elvira Strasser-Rapaport, who is married to David Rapaport, a Hungarian mathematician and psychoanalyst:

"Strasser-Rapaport had come late to mathematics, obtaining her Ph.D. only at age forty-three after she had raised the couple's two daughters."

As it happens, Papa has reduced the Poincare Conjecture to two sub-conjectures -- prove them both, and Poincare itself is proved. And Strasser-Rapaport is able to prove one of the sub-conjectures. But as it turns out, the second sub-conjecture can be proved false:

"This is exactly James McCool from the University of Toronto did. His paper 'A counterexample to conjectures by Papakyriakopoulos and Swarup' was published in the Proceedings of the American Mathematical Society in 1981."

Szpiro's next subject is born in Oakwood, Texas -- RH Bing. And as the author points out, this is his actual first name:

"Some older readers may remember JR, the eternal villain in the TV soap opera Dallas. Apparently double initials often moonlight as given names in Texas."

Bing becomes a high school math teacher, but then attends UT Austin to earn his MA. He meets Robert L. Moore, who does not like him because he is slightly older than most grad students. As it happens, Moore does not like Jews, women, or northerners either:

"A black student who wanted to join his class recounted that Moore told him, 'Okay, but you start with a grade C and can only go down from there.'"

Oops -- I mentioned race in my second straight post. But I won't give this post the "traditionalists" label, because I'm jumping out of this chapter. To make a long story short, Bing tries to prove -- and later on, disprove -- the Poincare Conjecture, and he fails on both accounts.

Well, we've already seen many "dead ends" as mentioned in this chapter. But is the mysterious disease, suffered by Bing and perhaps other topologists? Is it the disease that causes several other topologists to die young?

"[German Wolfgang Haken] became so obsessed with the problem that he was said to suffer from Poincaritis, an affliction that befell many a good mathematician in the twentieth century."

Haken does ultimately prove the Four-Color Theorem (and in fact, his name is mentioned in the U of Chicago text, Lesson 9-8), but not the Poincare Conjecture.

Szpiro wraps up the chapter thusly:

"But in the meantime let us turn from this rather depressing state of affairs to some more positive developments."

Lesson 6-6 of the U of Chicago text is called "Isometries." In the modern Third Edition, we must backtrack to Lesson 4-7 to learn about isometries.

I didn't write much about Lesson 6-6 two years ago, since I used that day to write about Presidential Consistency and the Common Core. I choose not to repeat that discussion, and so instead I'll reblog what little I wrote about the final isometry -- the glide reflection.

What, exactly, is a glide reflection? Well, here's how the U of Chicago defines it:

Let r be the reflection in line m and T be any translation with nonzero magnitude and direction parallel to m. Then G, the composite of T and r, is a glide reflection.

Just as reflections, rotations, and translations have nicknames -- "flips," "slides," and "turns," respectively -- glide reflections have the nickname "walks." The U of Chicago gives the example of the isometry mapping the right footprint to the left footprint while walking as a glide reflection. Another name for glide reflection is "transflection," since it is the composite of a reflection and a translation.

I once tutored a geometry student who had a worksheet on glide reflections. The student had to use a coordinate plane to perform the glide reflections, which were given as the composite of a reflection and a translation. But the problem was that on the worksheet, the direction of the translation wasn't always parallel to the reflecting line! In fact, in one of the problems the translation was perpendicular to the reflecting line. That would mean that the resulting composite wasn't truly a glide reflection at all, but just a mere reflection!



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