-- "...movable geodesic skylights. They are designed to optimize the use of solar energy."
-- "Marvelous, but where is the bed?" I asked.
This is the last page of the section "A Mathematical Visit." OK, so now we finally find out what's in Selath's bedroom -- solar skylights shaped like spheres and, most likely, paraboloids (as we saw on page 13 in the November 13th post). OK, let's discover the mathematician's bed:
-- "Just push the button on this wooden cube, and you will see a bed unfold with a head board and two end tables."
Let's skip down to the bathroom mirrors, since this is relevant to our current U of Chicago lesson on mirrors and isometries:
-- To my surprise I saw an infinite number of images of myself repeated. The mirrors were reflecting back and forth into one another ad infinitum.
-- "Now turn around and notice this mirror. What's different about it?" Selath asked.
-- "My part is on the wrong side," I replied.
-- "To the contrary this mirror lets you see yourself as you are really seen by others," Selath explained.
What's going on here? Pappas provides a footnote here:
"Made from two mirrors at right angles to each other. The right-angled mirrors are then positioned so that they will reflect your reflection."
We can use the isometries of Chapter 6 to explain what's happening. In the first case, the mirrors are parallel, and the composite of reflections in parallel mirrors is a translation. Therefore the first reflection image has reverse orientation, and the second has the correct orientation. But you can't see the front of your translation image -- only the back.
The second case provides you with a correct-oriented image of yourself that you can see. The mirrors are perpendicular, and the composite of reflections in intersecting mirrors is a rotation. Indeed, the composite is a rotation of magnitude 180 (twice the angle between the mirrors) and centered where the mirrors intersect. Turning you 180 degrees means that you see the front of your rotation image.
Here's how this section ends -- Selath's two dinner guests arrive, and so the unexpected arrival of the narrator means that there are four for dinner that night:
-- It was hard to conceal my enthusiasm. "But your table is set for three," I blurted.
-- "No problem. With the tangram table I can just rearrange a few parts and we'll have a rectangle.
Tangrams are often mentioned on math teacher blogs as an extra activity to give students. For example, frequent blogger Sarah Carter wrote about tangrams on her blog two weeks ago:
https://mathequalslove.blogspot.com/2017/11/puzzle-table-weeks-1-6.html
Carter attributes the tangram puzzle to another famous blogger Sara(h) -- van der Werf, to be exact.
https://saravanderwerf.com/2017/05/29/you-need-a-play-table-in-your-math-classroom/
Chapter 9 of George Szpiro's Poincare's Prize is called "Voyage to Higher Dimensions." Here is how it begins:
"The situation at the end of the 1960's was miserable: All efforts to prove Poincare's Conjecture had been fruitless."
As the title implies, we read about higher dimensions in this chapter -- and not just the fourth dimension, but 5D, 6D, 7D, and so on. As it turns out, Poincare is easier to prove in these higher dimensions than it is in the third dimension.
Szpiro explains what the analogs of the Poincare Conjecture look like in higher dimensions:
"We also defined the second homotopy group using parachutes. Poincare's Conjecture for four dimensions says that if all loops and all parachutes laid out on a four-dimensional body can be contracted to points, the body can be morphed into a four-dimensional sphere."
As it turns out, Poincare for 5D and above is proved by Steve Smale, though some controversy exists regarding who is actually the first to prove it. Of Smale, the author writes:
"Remember that a biographer of Henry Whitehead's wrote how unspectacular that mathematician's life was, 'with little for the biographer to chronicle'? Smale must be placed at the extreme other end in terms of life stories; there is nearly too much to chronicle."
And so of course I won't write about all of that in this post. (The author tells us that Smale is still alive today -- this summer he celebrated his 87th birthday.) I will say that Smale is educated in a one-room schoolhouse for grades K-8 and goes on to attend the University of Michigan. But he's more interested in politics than in his studies, and applies to grad school in math because he thinks that it's easier than physics:
"Eventually he was warned by the chairman of the math department that he would be kicked out if his grades did not improve."
Fortunately, his grades do improve, and he goes on to earn his master's degree. In fact, he spends time at several other schools, including the U of Chicago (yes, home of our text) and UC Berkeley. He describes what it means to "evert" a sphere:
"This term refers to the turning inside out of a sphere if the skin is permitted to pass through itself, but no holes must be made and no ripping or creasing is allowed."
(Of course, the sphere is evertible only in a higher dimension.) According to Szpiro, Smale also writes papers on chaos theory and the "horseshoe":
"It neatly expressed the idea behind the sensitive dependence on initial conditions, which is one of the catchphrases of chaos theory -- together with the butterfly effect, which, in principle, express the same thing."
(Again, Pappas writes about chaos theory several times in her book as well.) Now Szpiro tells us that Smale earns a Fields Medal, and on the way to Moscow to receive the award, he meets none other than Paul Erdos, the "Man Who Loved Only Numbers," according to Hoffman's book. It's Erdos who tells him that Smale has been subpoenaed by the House Un-American Activities Committee. (As Erdos would put it, "Sam" and "Joe" are at it again!)
"[Smale] had the 'honor' of being summoned by the committee without the hassle of having to be there. The Congress [of the International Mathematical Union] was a usual a grand affair, with thousands of mathematicians in attendance from all over the world."
Later on, Smale receives a grant from the National Science Foundation, but they threaten to take the money back because the mathematician often spends more time at the beach than doing math:
"After all, no government agency wants to be accused of paying for vacations and encouraging laziness."
Just before the turn of the century, Smale formulates a list of problems for mathematicians to work on, similar to those given by David Hilbert at a century earlier:
"Hilbert's problems had determined the direction of much of mathematical research during most of the first half of the twentieth century."
For all his work, Smale receives many prizes, including the Wolf Prize ten years ago. At the ceremony in Israel, he is lauded with the following statement:
"Against mainstream research on scientific computation, which centered on immediate solutions to concrete problems, Smale developed a theory of continuous computation and complexity."
The author now describes Smale's proof of Poincare for dimension five and above. His proof involves a concept called a "cobordism," or boundary, between two manifolds:
"For example, if one manifold is a circle and the other manifold is a pair of disjoint circles, the cobordism would be something resembling a pair of pants: The single circle would be the waist and the pair of circles would be the exits for the legs."
Smale sends his proof to Samuel Eilenberg at Columbia University to look over before submitting it to be published:
"Sammy, as everyone called him, was a Polish Jew who had escaped his fatherland just before the Nazis invaded it and was considered one of the world's leading topologists."
And so Sammy publishes the proof. Meanwhile, others become interested in try to prove Poincare for higher dimensions as well, including Princeton student John Stallings. Szpiro begins with an entertaining story about how Stallings survives lectures by getting the speakers drunk with whiskey:
"Reality television shows and pseudodocumentary films have recently been accused of using the same technique. Another example of mathematics on the cutting edge of culture, one might say.... Stallings had arrived at Princeton at a portentous time when 'topology was God and the Poincare Conjecture was its prophet."
Ultimately, Stallings attends graduate school at Oxford, and he proves Poincare for dimension seven and above. But this is at the same time that Smale is working on the same problem:
"More important, [Smale] had just proved the higher-dimensional Poincare's Conjecture. Now was the time to present his work to a wider audience in Europe."
And so Smale and Stallings meet in West Germany. Smale presents his proof -- and Stallings has a big smile on his face when he spots an error in Smale's proof. And so Smale spends the next four months trying to fix the error:
"In the fall of 1960, Smale's proof was ready to go, and on October 11 the manuscript with the title 'Generalized Poincare Conjecture in Dimensions Greater Than Four' was received by the Annals of Mathematics."
It's a race to who completes the proof first. Szpiro describes how Smale begins publishing his paper first, but Stallings has a finished proof, and to this day it's controversial which mathematician is truly the first to prove it. It's definitely true that Smale's proof works for 5D and up, while Stallings has a proof that begins at 7D. It takes the English mathematician Christopher Zeeman to add 5D and 6D to the Stallings proof.
Szpiro writes a little about Zeeman's early life:
"Just how difficult life must have been for the young boy is apparent from his description of his subsequent service in the Royal Air Force (1943-47) as a breath of freedom."
In other words, to Zeeman, his boarding school is harsher than the military.
Most of the rest of the chapter is about the argument between Smale and the other mathematicians regarding who is the first to prove Poincare for 5D and up. At any rate, it is Michael Freedman, a Southern Californian, who proves the 4D version of Poincare. Szpiro tells us that it's easier to prove Poincare in higher dimensions like 7D, 5D, and 4D than in 3D because there's more "elbow room" to twist the manifolds into spheres.
"Furthermore the only manifold whose parachutes also stretch and shrink to a point is the four-dimensional sphere. Poincare had been vindicated also in dimension four."
Lesson 6-7 of the U of Chicago text covers the Corresponding Parts in Congruent Figures Theorem, which the text abbreviates as CPCF. But a special case of this theorem is more widely known -- corresponding parts in congruent triangles are congruent, or CPCTC.
When I was young, a local PBS station aired a show called Homework Hotline. After school, middle and high school students would call in their homework questions in math and English, and some would be chosen to have their questions answered on the air by special teachers. Even when I was in elementary school, I often followed the geometry proofs that were called in, and more often than not, there were triangle congruence two-column proofs where the Reason for a step was often CPCTC. So this was where I saw the abbreviation CPCTC for the first time. (By the time I reached high school, a few calculus problems were called in to the show. Nowadays, with the advent of the Internet, the show has become obsolete.)
Here's a link to an old LA Times article about Homework Hotline:
http://articles.latimes.com/1992-02-09/news/tv-3184_1_homework-hotline
When I reached geometry, our text usually either wrote out "corresponding parts in congruent triangles are congruent," or abbreviated as "corr. parts of cong. tri. are cong.," probably with a symbol for congruent and possibly for triangle as well. But our teacher used the abbreviation CPCTC. Now most texts use the abbreviation CPCTC -- except the U of Chicago, that is. It's the only text where I see the abbreviation CPCF instead.
Dr. Franklin Mason, meanwhile, has changed his online text several times. In his latest version, Dr. M uses the abbreviation CPCTE, "corresponding parts of congruent triangles are equal."
Well, I'm going to use CPCTC in my worksheets, despite their being based on a text that uses the abbreviation CPCF instead, because CPCTC is so well known.
Once again, it all goes back to what is most easily understood by the students. Using CPCTC would confuse students if they often had to prove congruence of figures other than triangles. But as we all know, in practice the vast majority of figures to be proved congruent are triangles. In this case, using CPCF is far more confusing. Why should students had to learn the abbreviation CPCF -- especially if they have already seen CPCTC before (possibly by transferring from another class that uses a text with CPCTC, or possibly even in the eighth grade math course) -- for the sole purpose of proving the congruence of non-triangles, which they'd rarely do anyway?
So it's settled. On my worksheet, I only use CPCTC.
Notice that for many texts, CPCTC is a definition -- it's the meaning half of the old definition of congruent polygons (those having all segments and angles congruent). But for us, it's truly a theorem, as it follows from the fact that isometries preserve distance and angle measure.
Another issue that comes up is the definition of the word "corresponding." Notice that by using isometries, it's now plain what "corresponding" parts are. Corresponding parts are the preimage and image of some isometry. Unfortunately, we use the word "corresponding angles" to mean two different things in geometry. When two lines are cut by a transversal and, "corresponding angles" are congruent, the lines are parallel, but when two triangles are congruent, "corresponding angles" (and sides) are congruent as well. The phrase "corresponding angles" has two different meanings here! Of course, one could unify the two definitions by noting that the corresponding angles at a transversal are the preimage and image under some isometry. I tried this earlier, remember? It turns out that the necessary isometry is a translation. This is one of the reasons that I proved the Corresponding Angles Test using translations -- it now becomes obvious what "corresponding angles" really are. I mentioned yesterday, however, that in many ways using translations to prove Corresponding Angles is a bit awkward since it took so much work to avoid circularity. (This is why some authors, like Dr. Hung-Hsi Wu, uses rotations to prove Alternate Interior Angles instead.)
Returning to 2017, today I post the first page from last year's, but not the second page, which is about an unrelated math problem I had on my mind that year.
Instead, I replace it with an activity. Just before Thanksgiving, I wrote about how many teachers often given multi-day activities, but I've never done so on the blog. And so today I post such an activity:
Monday: Lesson 6-6 (Day 66)
Tuesday: Lesson 6-7 and Begin Multi-Day Activity (Day 67)
Wednesday: Finish Multi-Day Activity (Day 68)
Thursday: Review for Chapter 6 Test
Friday: Chapter 6 Test
This activity is based on the Exploration Question in today's text:
a. Find three characteristics that make Figure I not congruent to Figure II.
b. Make up a puzzle like the one in part a, or find such a puzzle in a newspaper or magazine.
Students can perform part a today and part b ("Make up a puzzle") tomorrow. Of course, if they wish, they can do the "find such a puzzle" part tonight. With this plan, we return to having only one day for review -- a formal test review to be given on Thursday.
The following worksheet comes from the website:
http://kool.radio.com/2011/02/19/can-you-find-the-3-differences-in-these-2-pictures/
Unfortunately, this picture (used as the initial example for part a) doesn't print very well. Then again, most versions I find online are interactive versions that print even worse.
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