Friday, December 11, 2009


For anyone who wants to know, I have just deactivated my Facebook account. I didn't use Facebook very much, so this is just a move to simplify.

Sunday, November 1, 2009

y = x - (x - y): a Consequence Thereof

Last year I discovered a trick for helping myself to fall asleep at night. When I lie down in bed, I close my eyes and think about ways to prove the Pythagorean theorem. Usually I'm asleep in no time! It's very peaceful, watching those little triangles float past my mind's eye, like so many tricorn sheep leaping a stone wall.

The first two of these narcotic proofs turned out to exist already (as I noted here and here). Amusingly enough, however, the third seems to be new. Claims of novelty are never certain when it comes to the Pythagorean theorem, but in any case, you can see the proof at Forum Geometricorum, an open-access, peer-reviewed geometry journal with a recreational slant.

The idea for the proof is to use the subtraction formulas for sine and cosine to derive the trigonometric identity cos^2(x) + sin^2(x) = 1. From this identity the Pythagorean theorem follows immediately.

There are some details involved, but here is the crux of the argument. Given any x with 0 < x < 90, let y be any number with 0 < y < x < 90. Then x, y, and x-y are all strictly between 0 and 90 degrees, so we may straightforwardly apply the subtraction formulas cos(a-b) = cos(a)cos(b) + sin(a)sin(b) and sin(a-b) = sin(a)cos(b) - sin(b)cos(a). We note that:

cos(y) = cos(x-(x-y)) = cos(x)cos(x-y) + sin(x)sin(x-y) = cos(x)[cos(x)cos(y) + sin(x)sin(y)] + sin(x)[sin(x)cos(y) - sin(y)cos(x)] = cos(y)[cos^2(x) + sin^2(x)].

Hence cos^2(x) + sin^2(x) = 1. QED.

Strangely, it does not seem to have been recognized before that the identity cos^2(x) + sin^2(x) = 1 can be derived independently of the Pythagorean theorem using the subtraction formulas. (Of course, to be certain of this, one would have to comb through three centuries' worth of trigonometry textbooks!)

A note on the proof can be found at the recreational math website cut-the-knot.


The algebraic manipulation used above can also be represented diagrammatically using triangles; this yields a more traditional style proof, as shown here. The recursion that is apparent in the algebraic argument gets rendered diagrammatically by nesting a little proof of the subtraction formulas inside a bigger proof of the subtraction formulas.

I'm not sure whether I think of the diagrammatic version and the trigonometric version as being distinct. The style of reasoning is quite different in the two cases, although the two arguments are of course direct transcriptions of one another in some sense.

In any case, it is interesting to note that because the y parameter is arbitrary, the diagram is in a sense "flexible" - it can be drawn in infinitely many ways for any given initial right triangle. In that sense the basic argument spawns infinitely many proofs of the Pythagorean theorem. I made a movie in which each frame represents one of these proofs; you can see it on YouTube or here.

Dorothy Jean Zimba, 1931-2009

On October 28, 2009, Mrs. Dorothy Jean Zimba, of North Bennington, Vermont, died in her sleep of natural causes. She was 77 years old.

Dorothy was born in Chattanooga, Tennessee, in 1931. In her first marriage, to Mr. Ray Bailey of Chattanooga, she had two sons. Later she married Mr. Richard Zimba of Dearborn Heights, Michigan; the couple had three children together and remained happily married until the end of Dorothy's life. For decades, husband and wife worked alongside one another in the family business, Rip's Drive-In, a suburban Detroit landmark during its time.

A gifted storyteller with a lively laugh, Dorothy was admired for her outgoing personality and incisive intelligence. She was a crossword prodigy and a formidable opponent in card games of all kinds. Her hobbies included needlework, gardening, collecting antiques, and painting landscapes and still lifes.

Dorothy's health began to decline in 2007, and at that time she relocated from Michigan to Vermont with her husband. She resided at Watson House and Prospect House in North Bennington, facilities in which she received kind and attentive care until her passing. Dorothy will be lovingly remembered by her husband, Richard; by her five children: Mr. David Bailey, Issaquah, Washington; Mr. Wayne Bailey, Southgate, Michigan; Ms. Intissar Greene, Mesa, Arizona; Ms. Jilly Dybka, Kingston Springs, Tennessee; and Dr. Jason Zimba, Pownal, Vermont; by her five grandchildren: Sgt. Erik Bailey, Camp Liberty, Iraq; Mr. Robert Bailey, Monroe, Michigan; Mr. Jonathan Bailey, Issaquah, Washington; Miss Abigail Zimba, Pownal, Vermont; and Miss Claire Elizabeth-Jean Zimba, Pownal, Vermont; by her two great-grandchildren, Miss Madalyn Bailey, Monroe, Michigan, and Miss Joanna Bailey, Monroe, Michigan; by her sister, Mrs. Patsy Ramsey, Ringold, Georgia; by her two nephews, Mr. Chris Ramsey, Tampa, Florida, and Mr. Barry Ramsey, Ringold, Georgia; by her grandniece, Ms. Valerie Ramsey, Tampa, Florida; by her great-grandniece, Miss Caitlyn Ramsey, Tampa, Florida; and by her many cousins.

Tuesday, July 28, 2009

The 'Get a Life' Principle

Like a lot of people with more important things to worry about, I've been paying rapt attention to the spectacle of the so-called "Birther Movement." For my money, getting the full experience requires going to the source, such as On this site, attorney Phillip Berg writes:
The Obama candidacy is the biggest ‘HOAX’ perpetrated on the citizens of the United States in 230 years, since our nation was established. Obama must be legally removed from office.

I believe that 15 to 20 million people are aware of the Obama 'HOAX,' and we must make 75 to 100 million people aware.
(Screaming uppercase in the original; ditto for the scare quotes.)

Alas, Berg might actually be correct when he estimates that 15 to 20 million people agree with him. I base this observation on the fact that in a 1999 Gallup poll, fully six percent of Americans expressed a belief that the 1969 moon landing was faked by the government. I’d like to see this as a glass 94% full. But what it would appear to indicate is that if all of the Americans who dispute the truth of the moon landing were to get together and form their own U.S. state, it would likely be the fifth largest state in the union, outranking Illinois (take that, Obama). So I’m thinking that in terms of his numbers, Berg is probably in the right ballpark. (Though "being in the right ballpark" hardly seems an appropriate metaphor for this situation.)

Another interesting site is On this page we are treated to a quote by Thomas Jefferson:
To restore ... harmony, ... to render us again one people acting as one nation should be the object of every man really a patriot.
(I guess they are taking the long way around to harmony, by seeking the removal of the President.)

A small 1994 study by Rutgers sociologist Ted Goertzel found that among the subjects studied,
Belief in Conspiracies was significantly correlated (r = .43) with a three-item scale of "Anomia" (alpha = .49) made up of items taken from the General Social Survey of 1990. These items measured the belief that the situation of the average person is getting worse, that it is hardly fair to bring a child into today's world, and that most public officials are not interested in the average man.

The Belief in Conspiracies scale was also significantly correlated (r = .21) with the item "thinking about the next 12 months, how likely do you think it is that you will lose your job or be laid off."

Volkan (1985) suggests that during periods of insecurity and discontent people often feel a need for a tangible enemy on which to externalize their angry feelings. Conspiracy theories may help in this process by providing a tangible enemy to blame for problems which otherwise seem too abstract and impersonal.

As the name implies, is full of FACTS. Conspiracy theorists love facts – and they love FACTS even more. In FACT, in a 1999 scholarly article on conspiracy theories, philosopher B.L. Keeley stressed the importance to the conspiracy theorist of certain very special facts, which Keeley called "errant data":
The chief tool of the conspiracy theorist is what I shall call errant data.

Errant data come in two classes: (a) unaccounted-for data and (b) contradictory data. Unaccounted for data do not contradict the received account, but are data that fall through the net of the received explanation. ... For example, ... the fact that no BATF employees were in the building at the time of the [Oklahoma City] explosion [is] unaccounted-for data with respect to the received account of the bombing. Contradictory data are data that, if true, would contradict the received account. McVeigh's manifest idiocy in fleeing the scene of the bombing in a car without license plates is a contradictory datum with respect to the official account of him as a conspiratorial ringleader capable of planning and carrying out such a terrorist operation.

It may be that there are certain propositions which simply do not reward one's sustained attention. Take, for example, the proposition "Your mother and your father were both faithful to each other during all the years of their marriage." To adopt a position of serious doubt on this question, and then, in such a frame of mind, to scour the family archives for bits and pieces of errant data, would not bring you any closer to the truth – it would only drive you crazy. Along similar lines, philosopher Lee Basham would counsel us to avoid conspiracy theories in order to preserve ourselves from harm:
A more solid ground for the rejection of conspiracy theories is simply pragmatic. There is nothing you can do. While it would be speculative (but reasonable) to conclude that this is why many people dismiss conspiracy theory, it is a considerable reason why we should. The futile pursuit of malevolent conspiracy theory sours or at least distracts us from what is good and valuable in life.
Later in the article Basham writes:
Any number of conspiracies might be worming their way through our world order. Now what? The 'get a life' principle kicks in with a vengeance.
Keeley has additional advice:
...[We] should be careful not to over-rationalize the world or the people that live in it. Rejecting conspiratorial thinking entails accepting the meaningless nature of the human world. Just as with the physical world, where hurricanes, tornadoes, and other "acts of God" just happen, the same is true of the social world. Some people just do things. They assassinate world leaders, act on poorly thought out ideologies, and leave clues at the scene of the crime. Too strong a belief in the rationality of people in general, or of the world, will lead us to seek purposive explanations where none exists.
It would be easy to view conspiracy theorists as irrational. But as Keeley points out, their beliefs sometimes depend on an extreme faith in the rationality of others; and their dogged pursuit of errant data is in some ways a model of sober detection, similar to the work of journalists, detectives, and scientists. Keeley says something similar when he writes,
I suggest that there is nothing straightforwardly analytic that allows us to distinguish between good and bad conspiracy theories. We seem to be confronted with a spectrum of cases, ranging from the reliable to the highly implausible. The best we can do is track the evaluation of given theories over time and come to some consensus as to when belief in the theory entails more skepticism than we can stomach. Also, I suspect that much of the intuitive "problem" with conspiracy theories is a problem with the theorists themselves, and not a feature of the theories they produce. Perhaps the problem is a psychological one of not recognizing when to stop searching for hidden causes.
It’s hard to read a website like or without becoming slightly concerned about the psychological well-being of conspiracy theorists. There is an obsessive quality to the undertaking, of course, but also a hint of something unmoored. I think of those theories of paranoid schizophrenia, according to which many sufferers from this disease may lack a "theory of mind."
[M]any patients are unable to represent or imagine the actions or intentions of other people. They lack a "theory of mind." Persecutory delusions may form because the paranoid patient cannot imagine someone else's perspective or psychological experience. Instead, idiosyncratic and ultimately sinister speculations are manufactured about the motives and intentions that govern the social world.
And now at the end of this exercise, I’m sensing that the sanity or insanity of the birthers is itself a proposition that doesn’t reward sustained attention. And the artifacts produced by people like Philip Berg on finally speak to me of pain. At the top of this page, we read:
12/09/08 – My brother, Norman Barry Berg, just passed away. My brother meant so much to me. I gave the attached Eulogy in loving memory of him at his funeral. I go forth in my efforts to find the truth of Obama in memory of my brother.
I guess it is time to start practicing the ‘get a life’ principle myself.

Friday, May 8, 2009

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 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.)

Saturday, March 28, 2009

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).

Wednesday, February 25, 2009

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.

Tuesday, February 24, 2009

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!

Saturday, February 7, 2009

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) "'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!