I Say, The Thing is Done

In the early days of this blog, I had some things to say about physics teaching and physics textbooks (here and here, for example). Then I got pretty quiet, because I was working hard on a book of my own - and it's finally here!

The title is Force and Motion: An Illustrated Guide to Newton's Laws (Johns Hopkins University Press, 2009). To find the book on Amazon, you can click here. To order instructor's examination copies, you can click here.

I should say that although this is a textbook, I did my best to ignore the conventions of the genre and write for human beings. Much of the story is told with diagrams, and even a casual reader might be tempted to try a problem or two; many of the problems ask you only to sketch, and there are answers in the back. (Physicists, don't be alarmed: the equations are all in there. Indeed, in some ways this book exceeds the usual ambitions of an introductory course.)

My thought is that the book could be a primary text for a suitably focused course in a liberal arts college, or a course at the high school level. It's also concise enough (70,000 words) to be a supplemental text for a university-level survey course - to help students who are having trouble or who just want to dig deeper into the essential material. A friend of mine who is a returning student at Cal State Hayward had a good experience using a rough draft of the book in this way.

A third goal is to put the book in front of current and future teachers of physics at the high school and college levels.

***

The title of this post refers to my own book of course, but it's also an allusion to the original textbook on Newton's Laws: the Principia of Isaac Newton himself. One of the things I always enjoyed about reading the Principia was the style in which its mathematical proofs were written. Often there would come a moment when, having arranged all his pieces on the chessboard, Newton would imperiously announce checkmate: I say, the thing is done. The last two or three sentences of would then play out the inevitable conclusion as the components of the strategy clicked into place. You can see examples on this page of the Principia.

***

I could say a lot about how my book differs from traditional presentations of Newton's Laws, but I'll save that for a separate post. Right now I want to celebrate being #5 on today's Amazon ranking of books about dynamics! (Friends and family, I owe you one.)

Majorana Representations and Platonic Solids

A few years ago, I was asked to contribute a paper to a book celebrating the centenary of the birth of Ettore Majorana (1906-1938), a gifted Italian theoretical physicist who died during World War II in mysterious circumstances. I got this call because in my graduate work under Sir Roger Penrose (see here for example), I had used Majorana's representation of quantum spin states to investigate the conceptual foundations of quantum theory. (Roger had effectively rediscovered the Majorana representation just prior to my arrival at Oxford.)

I agreed to contribute a paper to Majorana centenary, but I didn't have any suitable projects in the pipeline. So, while driving down to New York one night along the Taconic - the setting for more than one productive daydream - I started thinking about spin states. Reflecting on a particular variety of spin states called "coherent" spin states, I wondered "how far from coherent" a state could get. It immediately occurred to me to look at (inverse) Majorana representations of Platonic solids. These states would, in a sense, be the opposite of the familiar coherent spin states of quantum theory, and so I gave them the name "anticoherent" spin states.

Following this chain of reasoning leads to some geometric curiosities, including a basis for complex five-dimensional space consisting of five states whose Majorana representations form a dodecahedron's worth of interlocking tetrahedra! The resulting paper is available online here (see especially the figures at the end). It was my honor to receive the 2006 Majorana Prize for it, as part of the centennial celebration of Majorana's legacy in contemporary theoretical physics.

These anticoherent states were such beautiful little objects that I felt sure they must have some physical importance. After all, coherent states are often said to be "as classical as possible," so perhaps the anticoherent states would be useful for exhibiting exotic quantum phenomena such as one encounters in quantum information theory. In the paper, I alluded to a couple of potential physical implications, but I'm not an expert on quantum information theory so I couldn't say much. But I was happy to see recently that some physicists who do know a thing or two have found some interesting physical properties of the anticoherent states! See Kolenderski and Demkowicz-Dobrzanski, "Optimal state for keeping reference frames aligned and the platonic solids," Phys. Rev. A 78, 052333 (2008), available here (subscription required).

Pierre Menard, Author of the Pythagorean Theorem

The bad news is I'm having trouble sleeping tonight; the good news is that I came up with another proof of the Pythagorean theorem, to add to yesterday's.

Given a right triangle with sides a and b and hypotenuse c, construct the figure shown below.



By similarity, we will have (i) x/(y+b) = a/c, (ii) y/x = a/c, and (iii) x/a = c/b. From (iii) we have x = ac/b, so from (ii) we have y = ax/c = a^2/b, so from (i) we have (ac/b)/(a^2/b+b) = a/c, which simplifies to a^2 = b^2 = c^2. QED.

This approach, of course, turned out to be an oldie. Wikipedia presents a similar proof - which, however, takes the "outer" triangle as the object of study, rather than "tacking on" a triangle as I have done. The similar triangles approach is sometimes ascribed to Legendre, but is almost certainly older (e.g. Bhaskara).

Truth be told, I came up with this proof by a circuitous route. I realized that the usual cosine addition identity cos(m-n) = cos(m)cos(n) + sin(m)sin(n) can be derived independently of the Pythagorean theorem, and that if we put m=n we get cos(0) = cos^2(m) + sin^2(m). If we grant that cos(0)=1, we have the Pythagorean identity cos^2(m) + sin^2(m) = 1, from which the Pythagorean theorem obviously follows.

With this in mind, I found a nice proof of the cosine addition identity at MathWorld; see Equations (49)-(52). To make my proof of the Pythagorean theorem trig-free, I modified the MathWorld diagram by letting h approach x. This explains why my picture ended up looking different than the classic similar triangles approach.

Many people claim that trigonometric proofs of the Pythagorean theorem are necessarily circular (see e.g. Wikipedia). Based on the path I took to a valid proof, I think the situation is more subtle.

A Physicist's Proof of the Pythagorean Theorem

One night last week, as I lay in bed thinking about the challenges of the economy, terrorism, and global climate change, I thought to myself, What this world really needs is another proof of the Pythagorean theorem.

This pleasant diversion helped me to fall asleep for a few nights in a row, at least until I came up with the following approach. I call it a physicist's proof not only because it's less than mathematically rigorous, but also because the idea is to begin with a triangle of zero size and derive its law of growth. This is a more "dynamical" way of understanding the theorem than the traditional dissection methods.

Without further ado, here is the argument:



We label the sides by x and y, so the hypotenuse is some function f(x,y). We assume f(0,0) = 0. Now let y increase by an infinitesimal amount dy; then the small triangle is similar to the large right triangle, so that df/dy = y/f. Separating, we find f^2 = y^2 + C(x). Repeating the argument with the horizontal leg, we find f^2 = x^2 + D(y). Thus C(x) - x^2 = D(y) - y^2 = E, a constant, whence f^2 = y^2 + x^2 + E. But f(0,0) = 0 implies E=0, so f^2 = y^2 + x^2. QED.

I believe this can be made rigorous using (sin q)' = cos q and (cos q)' = -sin q (relations which can be proved without recourse to the Pythagorean theorem!)

Anyway, having found a solution, I poked around the web to compare it to the extant proofs. As I expected, my approach was not new. The Wikipedia entry on the Pythagorean theorem features this same basic argument, as does this page's amazing list. The idea is attributed to Michael Hardy of the University of Toledo, from 1988. Let this post serve to celebrate the idea's 20th Anniversary.

As long as I'm being unoriginal, let me also link to a cool math page from the website of Contra Costa Community College. Yesterday I was on this site by chance and happened to see two of my recent triangle puzzles!

Math Moment at Au Bon Pain

Last week I was in Washington, D.C., for a series of meetings. During a break, I went across the street to a bakery/cafe called Au Bon Pain. There I sat at my table, watching an employee as she restocked paper cups and plates. At one point, I saw the woman, about 25 years old, waving her hands over a tray loaded with paper cups arranged in nested stacks. Thinking she must have been counting the cups, I said, "How many?" She said, "No, I didn't count them, I was just wondering would they fit on this shelf."

A minute later I said, "OK, I can't resist, I have to know how many there are." So I stood up and counted: "16..." (the number of cups in one stack) "...times...7" (the number of stacks) "...so...that's 112."

"Wow!" she said. "I never could have figured that out."

***

A very informative report has just been published by the National Governors' Association, the Council of Chief State School Officers, and Achieve, Inc. The full report, titled "Ensuring U.S. Students Receive a World-Class Education," is here. Of particular interest are pages 20-21, which address many of the myths about why other countries outperform the U.S. on international exams.

In an earlier post, I wrote about the academic underperformance of poor children in this country. Figure 15 from the Achieve report is worth looking at in this connection. Click to enlarge:



(n.b. From Sweden on up, the country's performance is measurably better than the U.S. as a whole; from Italy on down, the country's performance is measurably worse than the U.S. as a whole.)

A Trio of Triangle Puzzles

Last week I was daydreaming out the window and thought of a few puzzles. The first one in particular doesn't require any specialized math knowledge.

1. Given two distinct points A and B in the plane, find all points C so that triangle ABC is isosceles.

(Note, a triangle is said to be "isosceles" when at least two of its sides have equal lengths. Or to put it another way, a triangle is isosceles when not all of the side lengths are different.)

The next one is a little more mathy:

2. Given two distinct points A and B in the plane, find all points C so that triangle ABC is a right triangle.

It makes a nice image if you carefully draw the answers to (1) and (2) on the same sheet of paper.

3. What would be the answers to (1) and (2) if the two points A and B were given in three-dimensional space, instead of a plane?

I'm using a new laptop that doesn't have any drawing software on it, so I'll have to post the answers later. Or, feel free to upload your answer to the Web someplace and link to it in the comments!

Follow-Up on Differences in Test Score Variability

In an earlier post, I described a study by Hyde et al. in Science showing that the gender gap in math has disappeared, as measured by average scores on state accountability exams. However, in that article there was also evidence that boys are overrepresented at the high end of performance, due to greater variability in their scores.

When the Hyde et al. paper came out, the press did a good job of making the point that the scores of boys and girls are now equal on average. Most reporters ignored the overrepresentation of boys at the high end. Some may have wanted to tell a simple, positive story; some may have been influenced by Hyde et al.'s rhetorical strategies. (A reporter from Reuters was an exception, but I can't find her story to link to.)

Today I'm just following up because the variance ratio has now been observed in another study: "Global Sex Differences in Test Score Variability," by S. Machin and T. Pekkarinen, Science Vol. 322, 28 November 2008. Click here to go to the article (subscription required).

The new study, based on PISA results, is both weaker and stronger than the study by Hyde et al. It is stronger in that it includes more countries than just the U.S. But it is weaker in that it only looks at 15 year olds. (Hyde et al. considered students in 2nd through 11th grades.)

Also unlike Hyde et al., the study by Machin and Pekkarinen looks at both math and reading. Interestingly, the variance ratios turn out to be similar in the two subjects, but with different consequences or for different reasons. In reading, boys are overrepresented at the low end of performance, whereas in math boys are overrepresented at the high end.

As of this writing, I've only seen one news article on the study, a Washington Post article by Ruth Marcus. Her article is more about Larry Summers than about the research itself.