*Magic of Mathematics*:

"How can it be that mathematics, a product of human thought independent of experience, is so admirably adopted to the objects of reality?"

-- Albert Einstein

This is the first page of Chapter 2 of her book, entitled "Magical Mathematical Worlds." As in the other chapters, the first page is basically a chapter intro.

Here are some excerpts from this page:

"Mathematics is linked and used by so many things in our world, yet delves in its own worlds -- worlds so strange, so perfect, so totally alien to things of our world. One finds such worlds composed of points, equations, curves, knots, fractals, and so on. For example, one might ask how an infinite world can exist only on a tiny line segment, or a world be created using only three points. This chapter seeks to explore the magic of some of these mathematical worlds and delve into their domains."

There is only one picture on this page -- the list of the first thirteen natural numbers. Here's what the caption reads:

"As discussed later, the counting numbers form a mathematical world in themselves."

And some of these worlds are explored on the annual Putnam exam. That's right -- two days ago was the first Saturday in December, also known as Putnam Saturday. And thousands of college students took the world's hardest math test over the weekend.

It's been a tradition on my blog to post one of this year's Putnam problems. Usually I post it on the Tuesday after the exam, and this year will be no exception, so expect a discussion about a Putnam problem in tomorrow's post.

Each year, I claim that I'll post Problem A1, since that's theoretically the easiest problem -- one which bright high school students might be able to understand. Yet each year, I find a different problem to post instead -- indeed, sometimes the A1 question is about Calculus. (That shows how difficult the Putnam is, if the "easiest" question on the test is Calculus!)

This year, I'll post problem B1, because its main subject is Geometry -- and this is, after all, a Geometry blog. I might mention some of the other problems anyway -- this year's A1 isn't Calculus, but just number theory (with a little Algebra), so students might understand some of this problem.

Last year, I mentioned the Putnam exam to my eighth grade class. This is the youngest grade to which I've ever talked about the test -- usually, I prefer to mention it to high school students for whom I'm subbing or tutoring. The problem I chose happened to be more puzzle-like, so even eighth graders could at least understand the gist of the problem. Indeed, the Putnam puzzle is eerily similar to a puzzle posted on Sarah Carter's famous blog last week:

https://mathequalslove.blogspot.com/2017/11/color-square-puzzle-from-puzzle-box.html

And since Carter's freshmen enjoyed the puzzle, it's not a stretch to think that eighth graders could appreciate it as well. But of course, as usual, the problem I had with

*any*discussion last year was that students talked too much -- and so some students never heard what I was saying. Still, I hope that at least a few of them were inspired by my Putnam speech.

Oh, and speaking of last year, today's Google doodle is all about kids who code -- and indeed, the commands to make the bunny hop remind me of Logo and turtle graphics. (At my old school, today should be Coding Monday -- I wonder whether the coding teacher made any reference to the Google doodle today. Last year, my loud class of eighth graders prevented the coding teacher from presenting a true coding lesson. Of course, if I'd had stronger classroom management, this could have helped the coding teacher with his lessons as well.)

By the way, while I was looking up info about this year's Putnam, I found this old page on the website of UCLA, my old school:

http://www.math.ucla.edu/~thiele/putnam/history.html

And yes -- my name is listed here. My Putnam score in 2001 was 22 out of 120. This sounds bad, but this placed me among the top 500 test takers in the country. The most common score on the test, after all, is usually zero.

By the way, last year, a UCLA student scored 87 points, good enough for 13th in the nation:

https://www.math.ucla.edu/news/ucla-putnam-team-places-7th

Again, don't try to map Putnam scores to letter grades. In middle and high school classes, 87 out of 120 would be 72.5%, a low C grade, but on the Putnam it's the 13th best score in the whole country!

The Putnam problems are all about proofs. This year on the blog, our Putnam discussion lines up well with our reading of the proof of the Poincare Conjecture. And so in sitting for and talking about the Putnam exam, students from the undergrad level down to, well, eighth grade, can appreciate what mathematicians do all day.

Chapter 13 of George Szpiro's

*Poincare's Prize*is called "The Gang of Four, plus Two." It begins:

"Having finished his whirlwind tour of the USA and disappeared again into the Russian wilderness -- actually to the city of St, Petersburg, Perelman left behind a plethora of new tools and methods that was to keep many mathematicians busy for years."

In this chapter, Szpiro writes about the mathematicians who must

*check*Perelman's proof to make sure that it's correct. The Putnam graders, of course, are mathematicians who are doing the same thing -- checking the validity of the students' proofs. Putnam graders are known to be tough -- recall what I said earlier about what scores on the Putnam look like. But the mathematicians "grading" Perelman's proof must be even tougher -- if they're wrong, the proof could be invalid, and the Poincare Conjecture could eventually be proved

*false*later on.

According to Szpiro, Perelman's papers are dense and terse:

"In general, an advanced paper is no textbook; authors take for granted that the basic facts are known to the intended readers."

And these first four intended readers are "The Gang of Four." These four mathematicians are:

- John Morgan, Columbia University
- Gang Tian, MIT (yes, one of the four "gangsters" is actually named Gang)
- Bruce Kleiner, U of Michigan
- John Lott, U of Michigan

These four present a program on Perelman's proof in the summer of 2005:

"It was organized 'around Ricci Flow and the Geometrization of 3-manifolds, particularly the recent work of Grisha Perelman.'"

But verifying the proof takes a long time:

"But even with that limited objective, the effort grew to proportions they had not imagined at the outset."

Indeed, it takes three years to verify the proof. As Morgan and Tian explain:

"'Because of the importance and visibility of the results discussed here,' they wrote in the introduction, 'and because of the number of incorrect claims of proofs of these results in the past, we felt that it behooved us to work out and present the arguments in great detail. Our goal was to make the arguments clear and convincing and also to make them more easily accessible to a wider audience.'"

Meanwhile, Kleiner and Lott worked to show how Perelman's work leads to a proof of a closely related result, Thurston's Geometrization Conjecture:

"Quite aware of the material's difficulty, the authors took pains to warn potential readers. Their 'Notes' are definitely rated 'PG: professional guidance recommended.'"

Now Szpiro proceeds to describe the "plus two" mentioned in the chapter title. These are two Chinese mathematicians, Huai-Dong Cao and Xi-Ping Zhu:

"Even a lawyer had a walk-on part. The affair became ugly, but ever so enticing for blogs and even mainstream media around the world."

And of course, that includes the blog you're reading right now. To make a long story short, Cao and Zhu attempt to steal credit for proving Poincare away from Perelman!

The author tells us that Cao and Zhu are old students of Shing-Tung Yau, a Princeton mathematician mentioned in a previous chapter:

"Finally they were satisfied that they had a full proof of the Poincare Conjecture. To test-run it and straighten out any remaining obscurities, Yau organized a seminar at Harvard."

After they check the proof, here's how they justify taking full credit for formulating the proof:

"'This can be compared to the construction of a building. The forerunners have laid the foundation, while the last phase work, topping out, is completed by the Chinese,' said Qiu Chentong [Yau, Shing-Tung]."

Well, what does the Gang of Four think about the Chinese claim?

"Gang Tian definitely knew about Cao's work because he had been coauthoring with Cao recently. Cao had also been presenting his work at Columbia [where John Morgan works]."

But at another point, Cao and Zhu concede that Perelman has a proof as well:

"We will provide a proof...by only using [a different] result. In particular, this gives another proof of the Poincare conjecture."

This claim, though, doesn't endear them to mathematicians outside China:

"Nevertheless, many mathematicians who glanced only at the introduction and had heard the reports by the Chinese press became too angry even to read the paper."

For example, Joan Birman from Columbia laments to the American Mathematical Society that mathematicians are indeed eccentric, and yet:

"But there is another, and a darker, side to the same phenomenon, i.e., a tolerance for bad behavior, especially when individuals whose actions might be questioned are highly talented."

Cao and Zhu try to claim the topologist Richard Hamilton for their side of the debate:

"He had been there before the conference, but had left by the time Yau gave his talk. So Hamilton address the audience through a video recording that had been filmed earlier. In the rather amateurish recording, the lighting was bad, there were sudden cuts, some parts were played twice."

And notice that this is back when YouTube is in its infancy. Nowadays, even my eighth graders from last year -- in that same coding class when they were talking too much -- were able to edit a better video than this!

Well, the next evening, Gang of Four members Kleiner and Lott summarize their finding with a video of their own:

"The persiflage took the form of a movie based on the previous evening's video, complete with bad lighting, strange camera angles, and effusive praise for some imagined scientist."

Szpiro tells us that Hamilton may have played along due to his relationship with Yau:

"Yau is an old friend who had helped, supported, stimulated, and encouraged him for decades. Hamilton simply responded in kind."

In the end, though, the Chinese mathematicians are wrong to claim the proof for themselves:

"Only if Cao and Zhu had shown that one of Perelman's statements was wrong, say by providing a counterexample, and then demonstrated that their own formulation does not suffer from the same defect would their paper have to be credited as an original contribution. Otherwise it simply belongs to the collection of simplifications and expositions, as does the work of the Gang of Four."

During this time, the

"On September 20 a Webcast followed in which the attorney read large parts of the rebuttal for all the world to hear."

Meanwhile, Cao and Zhu finally admit defeat in another arXiv-posted paper:

"It carried the title 'Hamilton-Perelman's Proof of the Poincare Conjecture and the Geometrization Conjecture.'"

But the Gang of Four never completely reconcile with the Chinese mathematicians:

"A lapidary announcement on the AMS Web site stated, 'We regret that the special event on the Poincare Conjecture and Geometrization Theorem has been canceled. It became apparent that the continuing controversy was undermining this special event.'"

The author reminds us that priority disputes in science and math go back centuries -- for example, who invented Calculus, Newton or Leibniz?

"If scientists did not worry about who will be first, they would go about their research at a leisurely pace. And this would certainly not be beneficial to the advancement of science. The ICM in Madrid in August 2006 provided the appropriate endpoint to the century-old saga."

The author tells us that Cao and Zhu are old students of Shing-Tung Yau, a Princeton mathematician mentioned in a previous chapter:

"Finally they were satisfied that they had a full proof of the Poincare Conjecture. To test-run it and straighten out any remaining obscurities, Yau organized a seminar at Harvard."

After they check the proof, here's how they justify taking full credit for formulating the proof:

"'This can be compared to the construction of a building. The forerunners have laid the foundation, while the last phase work, topping out, is completed by the Chinese,' said Qiu Chentong [Yau, Shing-Tung]."

Well, what does the Gang of Four think about the Chinese claim?

"Gang Tian definitely knew about Cao's work because he had been coauthoring with Cao recently. Cao had also been presenting his work at Columbia [where John Morgan works]."

But at another point, Cao and Zhu concede that Perelman has a proof as well:

"We will provide a proof...by only using [a different] result. In particular, this gives another proof of the Poincare conjecture."

This claim, though, doesn't endear them to mathematicians outside China:

"Nevertheless, many mathematicians who glanced only at the introduction and had heard the reports by the Chinese press became too angry even to read the paper."

For example, Joan Birman from Columbia laments to the American Mathematical Society that mathematicians are indeed eccentric, and yet:

"But there is another, and a darker, side to the same phenomenon, i.e., a tolerance for bad behavior, especially when individuals whose actions might be questioned are highly talented."

Cao and Zhu try to claim the topologist Richard Hamilton for their side of the debate:

"He had been there before the conference, but had left by the time Yau gave his talk. So Hamilton address the audience through a video recording that had been filmed earlier. In the rather amateurish recording, the lighting was bad, there were sudden cuts, some parts were played twice."

And notice that this is back when YouTube is in its infancy. Nowadays, even my eighth graders from last year -- in that same coding class when they were talking too much -- were able to edit a better video than this!

Well, the next evening, Gang of Four members Kleiner and Lott summarize their finding with a video of their own:

"The persiflage took the form of a movie based on the previous evening's video, complete with bad lighting, strange camera angles, and effusive praise for some imagined scientist."

Szpiro tells us that Hamilton may have played along due to his relationship with Yau:

"Yau is an old friend who had helped, supported, stimulated, and encouraged him for decades. Hamilton simply responded in kind."

In the end, though, the Chinese mathematicians are wrong to claim the proof for themselves:

"Only if Cao and Zhu had shown that one of Perelman's statements was wrong, say by providing a counterexample, and then demonstrated that their own formulation does not suffer from the same defect would their paper have to be credited as an original contribution. Otherwise it simply belongs to the collection of simplifications and expositions, as does the work of the Gang of Four."

During this time, the

*New Yorker*publishes an article criticizing Yau, depicting the Ivy League mathematician as trying to steal Perelman's Fields Medal from his neck. Yau threatens to sue:"On September 20 a Webcast followed in which the attorney read large parts of the rebuttal for all the world to hear."

Meanwhile, Cao and Zhu finally admit defeat in another arXiv-posted paper:

"It carried the title 'Hamilton-Perelman's Proof of the Poincare Conjecture and the Geometrization Conjecture.'"

But the Gang of Four never completely reconcile with the Chinese mathematicians:

"A lapidary announcement on the AMS Web site stated, 'We regret that the special event on the Poincare Conjecture and Geometrization Theorem has been canceled. It became apparent that the continuing controversy was undermining this special event.'"

The author reminds us that priority disputes in science and math go back centuries -- for example, who invented Calculus, Newton or Leibniz?

"If scientists did not worry about who will be first, they would go about their research at a leisurely pace. And this would certainly not be beneficial to the advancement of science. The ICM in Madrid in August 2006 provided the appropriate endpoint to the century-old saga."

And this takes us right back to the ceremony in Chapter 1. Szpiro concludes the current chapter:

"The great adventure that had seen so many ups and downs since its inception in 1904, had kept hundreds of mathematicians busy for a century, and had made and nearly ruined many a career has now, finally, come to an end."

Lesson 7-1 of the U of Chicago text is called "Drawing Triangles." Now that we've reached Chapter 7, the Second and Third Editions finally sync up again. Indeed, Lessons 7-1 through 7-5 of both texts are the same in both editions. It's also fitting that our discussion of the Putnam and Poincare proofs lines up with Chapter 7, since this is the chapter most strongly associated with proofs in Geometry.

This is what I wrote two years ago about today's lesson. Actually, I didn't say much about the lesson per se, but I did write extensively about

*proofs*, the major theme for today's post:

I introduced the concept of low-, medium-, and high-level proofs. These categories aren't rigid, but here's an approximate division:

Low-level: Prove SAS Congruence from first principles (i.e. transformations, if it's Common Core)

Mid-level: Prove the Isosceles Triangle Theorem from SAS Congruence

High-level: Prove the Equilateral Triangle Theorem (i.e. that an equilateral triangle is equiangular) from the Isosceles Triangle Theorem

So we can somewhat see the difference among these levels -- in particular, we may use the lower-level theorems in the proofs of the higher-level theorems.

But there's a more important distinction among these levels in the Geometry

*classroom*. Teachers are more likely to ask students to prove higher-level than lower-level theorems. Many Geometry texts, especially pre-Core, don't expect students to prove our low-level theorems, such as SAS Congruence from first principles. Indeed, they absolve students from the responsibility of proving SAS completely by making it a postulate!

And now we see where the opponents of Common Core come in. They point out that geometry based on transformations is too experimental to appear in the classroom. Instead, they favor the pre-Core status quo -- just declare SAS a postulate and throw out transformations altogether,

Now here's the problem with this thinking -- low-level is to mid-level as mid-level is to high-level. I can now imagine a hypothetical class where not only do we avoid the low-level derivation of SAS from transformations, but we can avoid the mid-level derivation of the Isosceles Triangle Theorem from SAS as well. Instead, just declare the Isosceles Triangle Property to be a

*postulate*and throw out SAS Congruence altogether! Students can still prove interesting theorems from this Isosceles Triangle "Postulate," including the Equilateral Triangle Theorem -- even the first problem from that Weeks and Adkins page from two weeks ago can be proved using only the Isosceles Triangle "Postulate" (and its converse, which could be declared yet another postulate).

One may argue that no Geometry text actually does this -- but

*au contraire*, there really is a text that does something like the above. In Lesson 5-1 (old version) of Michael Serra's

*Discovering Geometry*, Conjecture 27 is the Isosceles Triangle Conjecture (and Conjecture 28 is its converse), while Conjecture 29 is the Equilateral Triangle Conjecture (stated as a biconditional), with a paragraph proof provided to show how 29 follows from 28. So even though all three of these statements are labeled as "conjectures," the net effect is that 27 and 28 are

*postulates*(as no proof is given), while 29 is actually a

*theorem*proved using postulate 28.

Of course, this may seem silly -- Serra doesn't avoid SAS altogether, but instead gives it later on in the same chapter (Lesson 5-4, old version). And ultimately when we reach the end of the book when two-column proofs are taught, students are asked to prove the Isosceles Triangle Theorem using SAS, just as in most other texts.

But it does show that the pre-Core status quo is attacked on two fronts. If you argue that students should learn SAS so that the Isosceles Triangle Property can be a theorem rather than a postulate, then why not take it further in the direction of more rigor, and teach the students about reflections so that SAS can be a theorem rather than a postulate? Or going the other way, if you argue that students shouldn't have to learn how SAS follows from reflections, then why not take it further and say that students shouldn't have to learn how the Isosceles Triangle Property follows from SAS? It's not at all obvious why the exact status quo (SAS a postulate, Isosceles Triangle Property a theorem) is neither excessively nor insufficiently rigorous.

Now there is an argument that, if true, would vindicate the status quo defenders. It could be that the mid-level derivation of the Isosceles Triangle Theorem from SAS is easy for the students to understand, thereby preparing them well for the rest of Geometry and subsequent courses, but the low-level derivation of SAS from reflections is too hard for them and makes them cry after the test, thereby discouraging them from taking subsequent math courses. If this can be demonstrated, then the status quo is exactly right. Then again, until such a demonstration can be made, it's just as likely that the Common Core transformation approach, or even Serra's conjecture approach, could be correct.

There's one more thing that we must take into consideration -- the distinction between the Common Core

*Standards*and the Common Core

*tests*. Much of all my chapter juggling has occurred because I'd originally set up my lessons to match the

*standards*, only to see something else on the

*tests*. The standards state that students should learn how SAS and the other congruence and similarity theorems follow from the properties of transformations -- but such low-level proofs don't appear on the PARCC or any other Common Core test. It's actually easier to test for medium- and high-level proofs on a test, and the PARCC is no exception.

The PARCC question where students have to derive the Alternate Interior Angles Consequence from the Corresponding Angles Consequence is a mid-level proof. The PARCC question where students have to prove that the sum of the exterior angles of a triangle is 360 is definitely a high-level proof (after all, the triangle sum ultimately goes back to alternate interior angles). A low-level proof in this tree would be to show how the Corresponding Angles Consequence goes back to transformations.

Mid-level proofs on the PARCC are problematic -- and it's these questions that drive most of the changes in my curriculum. We saw last week how although the Common Core Standards ask students to derive SAS Similarity from the properties of dilations, a PARCC question asks them to derive a mid-level property of dilations from SAS~ instead.

But high-level proofs cause the fewest curriculum problems. I consider the classic two-column proofs of U of Chicago's Lesson 7-3 -- where students use SAS to prove two triangles congruent, but the "S" comes from the Reflexive Property or the definition of midpoint and the "A" comes from the Vertical Angles Theorem or some other result -- to be high-level proofs. This is because they appear at the top of the proof tree, rather than branch out to be used in other theorems. Such proofs don't require the students to derive SAS as a theorem.

On this blog, I will present low-level proofs in worksheets in order to meet the Common Core Standards, but I don't expect students to reproduce them in the exercises or on a quiz or test. On the other hand, students will have to know and understand the mid- and high-level proofs.

Let's get to today's worksheet. Now as it turns out, not only did I begin Chapter 7 last year near the Thanksgiving break, but it was also when I was purchasing a new computer and working hard to get it installed and connected to the Internet. These two facts combined mean that I don't necessary have a great Lesson 7-1 worksheet from last year.

Last year I gave some sort of an activity, where students were given parts of a triangle such as SS, AA, SSS, AAA, and so on, to determine whether they are sufficient to determine a triangle. Then I would follow this with a discussion of the results followed by some "review" problems. Once again, juggling the lessons around means that the students would be "reviewing" concepts that I haven't taught this year yet, such as Triangle Inequality and Triangle Sum. (Notice that this are related to the SSS and AAA conditions, respectively.)

So I decided to keep the activity-like part of the lesson and replace the Triangle Inequality and Triangle Sum questions with some more information about isometries and polygon congruence. This is what the resulting worksheet looks like:

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