Archive for the ‘Math’ Category

Magical Mathematics

December 11, 2011 Leave a comment

One of these days, the Wall Street Journal will stop delivering the paper, which I stopped paying for back in October. But it keeps coming, and as long as it does, I’ll keep reading its book reviews. Like the lead review in yesterday’s book section of Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks, by famed mathematicians Persi Diaconis and Ron Graham. And as always, when a book on mathematics receives a review from a major mainstream media outlet, I’ll be there to cover it.

From Princeton University Press’s description of the book:

Magical Mathematics reveals the secrets of amazing, fun-to-perform card tricks–and the profound mathematical ideas behind them–that will astound even the most accomplished magician. Persi Diaconis and Ron Graham provide easy, step-by-step instructions for each trick, explaining how to set up the effect and offering tips on what to say and do while performing it. Each card trick introduces a new mathematical idea, and varying the tricks in turn takes readers to the very threshold of today’s mathematical knowledge. For example, the Gilbreath Principle–a fantastic effect where the cards remain in control despite being shuffled–is found to share an intimate connection with the Mandelbrot set. Other card tricks link to the mathematical secrets of combinatorics, graph theory, number theory, topology, the Riemann hypothesis, and even Fermat’s last theorem.

Diaconis and Graham are mathematicians as well as skilled performers with decades of professional experience between them. In this book they share a wealth of conjuring lore, including some closely guarded secrets of legendary magicians. Magical Mathematics covers the mathematics of juggling and shows how the I Ching connects to the history of probability and magic tricks both old and new. It tells the stories–and reveals the best tricks–of the eccentric and brilliant inventors of mathematical magic. Magical Mathematics exposes old gambling secrets through the mathematics of shuffling cards, explains the classic street-gambling scam of three-card monte, traces the history of mathematical magic back to the thirteenth century and the oldest mathematical trick–and much more.

Diaconis is the rare mathematician who has received extensive coverage in the popular press, thanks to his unusual background. As Alex Stone explains in yesterday’s WSJ review, “Mr. Diaconis has an especially unusual résumé for a mathematician. In 1959, at age 14, he ran away from home to study with the great 20th-century sleight-of-hand master Dai Vernon—a man who once fooled Harry Houdini with a card trick. After spending 10 years under Vernon’s tutelage, Mr. Diaconis returned home to New York and enrolled in night school, eventually earning a full ride to a Ph.D. program in mathematics at Harvard.” He visited Seattle just two months ago to give a major public lecture at the university, and has visited frequently before from his home base at Stanford.

Stone adds that

throughout the book, Messrs. Diaconis and Graham shuttle back and forth between magic and math, probing each trick for hidden mathematical insights and developing new magic based on what they find. In the process, they encounter a number of unsolved problems, some of which have prize money attached to them. It’s a fun ride, even if you don’t follow the nuances of every theorem and proof, and a refreshing change from the bombastic sort of magic one typically encounters on television.

I am intrigued by the notion that the Riemann Hypothesis and card tricks are related. I’ll have to get a copy of the book to learn more.

Categories: Books, Math

Forgotten Land

November 30, 2011 Leave a comment

I wrote last night about how much I’ll miss the daily book review in the Wall Street Journal when the paper’s delivery finally ceases. A month ago it led me to Robert Crease’s World in the Balance: The Historic Quest for an Absolute System of Measurement, the subject of last night’s post. Two and a half weeks ago, I was introduced by Andrew Stuttaford to Max Egremont’s Forgotten Land: Journeys Among the Ghosts of East Prussia. I’m now some 70 pages into Egremont’s book and thoroughly enjoying it.

Here is the opening of Stuttaford’s review:

In 1945, Stalin seized East Prussia, Germany’s venerable redoubt on the Baltic Sea, as a spoil of war. A portion went to the “People’s Republic” that the Soviets had just created in Poland. He kept the rest. The last surviving Germans were killed or deported. The cozy old Königsberg of the Teutonic Knights—the home, during the Enlightenment, of no less than Immanuel Kant—was transformed into Kaliningrad—a bleak Soviet place named after Mikhail Kalinin, the token peasant who was titular head of Stalin’s USSR.

Nearly 70 years later, the countries behind these borders have changed, but the frontiers have not, and will not. The Polish part is finally and truly Polish; the sliver of East Prussia given to the Lithuanian Soviet Socialist Republic is now a part of independent Lithuania; the rest of the old Soviet slice is a seedy Russian exclave surrounded by the European Union. The only Germans there are tourists, in search of an elusive land that lingers on in family lore and in the dreams of the dispossessed for a vanished, fondly imagined, past.

Max Egremont’s idiosyncratic, disjointed and beautifully written volume makes an ideal guide to this shifting, shadowy realm. In part a piecemeal history of the final half-century of German East Prussia, in part a travelogue through what was left behind, “Forgotten Land” is gently elegiac. Shifting constantly between present and a variety of pasts, it is as wistful as a flick-through of an old photo album, as melancholy as a rain-spattered northern autumn afternoon.

Immanuel Kant may be the most famous genius associated to Königsberg, but the one whom Königsberg brings to my mind is Kant’s near contemporary Leonhard Euler, the greatest mathematician of the eighteenth century. Euler, a Swiss native, did not actually live in Königsberg. He spent much of his career in St. Petersburg and Berlin. But he knew Königsberg’s layout, and one of his early successes was his 1735 solution of the Königsberg bridge problem.*

I learned as a child about the problem and Euler’s solution, prompting me to wonder where Königsberg was. I was puzzled on finding that it lay in the Soviet Union and was called Kaliningrad. In due course, I read some of the relevant history, but new puzzles were introduced, such as why Prussia had an outpost so far east, embedded in modern-day Poland, Lithuania, and Russia. On reading Stuttaford’s review, I realized that Egremont’s book offered me the opportunity, at last, to put the pieces in their proper places. Plus, I could wallow in elegiac, wistful melancholy. I wasted no time downloading and starting Forgotten Land.

*Perhaps a few words on the Königsberg bridge problem would be in order. Below you see a drawing of the seven bridges that crossed the Pregel River in Königsberg in Euler’s time. (I have taken this drawing from Wikipedia, where it was the picture of the month on the Mathematics Portal for September 2011 and credited to Bogdan Giuşcă.)

The problem is to find a route through the city that crosses each bridge exactly once. Euler proved that there is no way to do this. In so doing, he laid the foundation for the modern-day mathematical field known as graph theory.

Categories: Books, History, Math

Pasta by Design

September 25, 2011 Leave a comment

What do I love most in the world? Well, yes, Gail. But forget about people. And sports. Let’s try again.

What do I love most in the world? Tough one, right? Is it pasta? Is it math? Let’s just say they’re tied. Guess what? There’s a book about them: Pasta by Design, by George L. Legendre.

I might have missed this book if not for yesterday’s WSJ, whose Saturday Review section devoted most of a page to illustrations from it. I didn’t have to look for long before deciding to order a copy.

The publisher’s website provides the following description of the book:

The pasta family tree reveals unexpected relationships between pasta shapes, their usage and common DNA. Architect George L. Legendre has profiled 92 different kinds of pasta, classifying them into types using ‘phylogeny’ (the study of relatedness among natural forms).

Each spread is devoted to a single pasta, and explains its geographical origin, its process of manufacture and its etymology – alongside suggestions for minute-perfect preparation.

Next the shape is rendered as an equation and as a diagram that shows every distinctive scrunch, ridge and crimp with loving precision. Superb photographs by Stefano Graziani show all the elegant contours.

Finally, a multi-page foldout features a ‘Pasta Family Reunion’ diagram, reassembling all the pasta types and grouping them by their mathematical and geometric properties!

I love the idea of a pasta taxonomy.

If you follow the WSJ link, you’ll see some of the photographs and diagrams. More can be found in this announcement of a book giveaway competition by Dezeen magazine, which explains that the book “includes photographs, 3D diagrams and parametric equations of 92 different pasta types, grouped and analysed according to their mathematical and geometric properties.”

Check out this example, included in the WSJ:

Or this, from Deneen:

I can’t wait to see them all.

Categories: Books, Food, Math

On Historical Perspective

August 24, 2011 1 comment

Famed Babylonian tablet Plimpton 322*

[Christine Proust and Columbia University]

I read a marvelous passage earlier this week that I would like to share. It’s an old one, from a book published in 1952, but it’s new to me. The book: The Exact Sciences in Antiquity by Otto Neugebauer. Neugebauer is a giant of twentieth-century history, the expert on ancient Babylonian mathematics and astronomy. An Austrian, he studied engineering, mathematics, and physics in Graz, then Munich, and then Göttingen, where he began to work on the history of ancient mathematics. He left Germany for Copenhagen in 1934, then moved on to the US, where he spent the remainder of his career at Brown and the Institute for Advanced Study in Princeton.

Neugebauer died in 1990. I was aware of his name and vaguely of his importance to the history of mathematics when we spent the year at the Institute in 1987-1988, but unfortunately I didn’t think to meet him or learn more about his work. Talk about missed opportunities. But now, 23 years later, I’ve been looking at some of his work.

In the 1930’s and 1940’s, in a series of books, he translated and interpreted the mathematics on the Babylonian cuneiform tablets that can be found in many of the world’s great museums and that date to the time period of 1900 BCE-1600 BCE. The book on exact sciences in antiquity from which I am about to quote is more of an overview that grew out of lectures he gave at Cornell in 1949. In the Preface, he explains that the nature of the presentation led him to omit the qualifications he might ordinarily have offered in a more scholarly work, adding that he has “enjoyed the possibility of being compelled for once to abandon all learned apparatus and to pretend to know when actually I am guessing.” The paragraph concludes with the passage below, which I leave for your enjoyment without further comment.

This does not imply that I have ignored facts. Indeed I have consistently tried to keep as close as possible to the source material. Only in its selection, in its arrangement, and in its coherent interpretation have I permitted myself much greater freedom than is usual in technical publications. And in order to counteract somewhat the impression of security which easily emerges from general discussions I have often inserted methodological remarks to remind the reader of the exceedingly slim basis on which, of necessity, is built any discussion of historical developments from which we are separated by many centuries. The common belief that we gain “historical perspective” with increasing distance seems to me utterly to misrepresent the actual situation. What we gain is merely confidence in generalizations which we would never dare make if we had access to the real wealth of contemporary evidence.

*The photo at the top is taken from a NYT article by Nicholas Wade last fall on the occasion of an exhibition of cuneiform tablets at NYU’s Institute for the Study of the Ancient World. Unfortunately, the exhibition closed last December, or else I would be making plans to see it during our upcoming trip to New York. Wade explained that the “considerable mathematical knowledge of the Babylonians was uncovered by the Austrian mathematician Otto E. Neugebauer, who died in 1990. Scholars since then have turned to the task of understanding how the knowledge was used. The items in the exhibition are drawn from the archaeological collections of Columbia, Yale and the University of Pennsylvania.” Be sure to look at the accompanying slide show, which includes more tablets, a photo of Otto Neugebauer, and his hand drawing of both sides of a tablet.

Categories: History, Math, Writing

Today’s Crossword Delight

June 15, 2011 Leave a comment

Swiss 10-franc banknote

I had a happy surprise last night when I did today’s NYT crossword. (If you haven’t done it yet but anticipate getting to it later, read no further until you’re done.)  The clue for 5-across is “Subject with limits and functions, informally,” with the not-so interesting answer ‘calc’.  But this set up up 45-across:  “Swiss 5-across pioneer,”  the answer (of course) being my favorite mathematician, Euler.  How can I not love a crossword with Euler in it?

It’s a continuing source of wonder and sadness to us mathematicians that geniuses such as Euler are largely unknown.  If he were a composer, he would be Bach. If he were a baseball player, he would be Cobb.  But he’s a mathematician, and a cipher, except at least in Switzerland, where he is a familiar figure.  (See above.)

This short biography gives some sense of his greatness. Have a look. And next time you see mention in a crossword of a Swiss mathematician, remember Euler.

Categories: Crosswords, Math

Daniel Quillen

May 10, 2011 Leave a comment

One of the greatest mathematicians I’ve had the privilege to know — Daniel Quillen — died ten days ago. He was on the faculty at MIT when I was a graduate student there, and did some of his most famous work at the time.

It’s a funny thing about mathematicians. If Quillen were a physicist, say, or a composer, or a writer, his death would have been the lead obituary of the day. Perhaps the NYT will eventually get around to taking note of his death. In the meantime, I’ll refer you to a blog post a week ago by Steve Landsburg, the economist, some-time mathematician, and popular economics writer, whom I happened to hang out with thirty years ago when we were both at the University of Chicago. Steve writes:

I met Quillen only once, and very briefly, but great mathematicians, like great poets, reveal so much of themselves in their work that one comes to feel a certain intimacy just by studying them. In that sense, Quillen was my close companion many a year.

Dan Quillen died this week at the age of 70, after a five year battle with Alzheimer’s. Scouring the web for obituaries and other recent mentions, I found very little besides a brief article from a Gainesville newspaper about an Alzheimer’s patient named Daniel Gray Quillen who had gone briefly missing in June, 2010. Followup stories identify the missing man as “a senior citizen with Alzheimer’s”.

“A senior citizen”?!?!?! Part of me wants to scream: “Dammit, this is no generic senior citizen! This is Daniel Fucking Quillen, Fields Medalist, Cole Prize Winner, architect of higher K-theory, conqueror of the Serre conjecture, and one of the intellectual giants of the 20th century!”

Categories: Math, Obituary

Abel Prize

March 25, 2011 Leave a comment

The ninth annual Abel Prize was awarded to John Milnor two days ago. As I explained in a post two years ago and again a year ago, the prize was established in 2001 by the Norwegian government to be the counterpart in mathematics to the Nobel Prizes in other disciplines. It has been awarded by the Norwegian Academy of Science and Letters each year since 2003 to one or two outstanding mathematicians and honors the great, early-nineteenth-century Norwegian mathematician Niels Abel.

This year’s recipient is John Milnor. As explained at the Abel Prize website, his

profound ideas and fundamental discoveries have largely shaped the mathematical landscape of the second half of the 20th century. All of Milnor’s work display features of great research: profound insights, vivid imagination, striking surprises and supreme beauty. He receives the 2011 Abel Prize “for pioneering discoveries in topology, geometry and algebra,” to quote the Abel committee.

In the course of 60 years, John Milnor has made a deep mark on modern mathematics. Numerous mathematical concepts, results and conjectures are named after him. In the literature we find Milnor exotic spheres, Milnor fibration, Milnor number and many more. Yet the significance of Milnor’s work goes far beyond his own spectacular results. He has also written tremendously influential books, which are widely considered to be models of fine mathematical writing.

Milnor is indeed a fine mathematical writer. I own his beautiful Symmetric Bilinear Forms, a classic, and his tiny monograph Topology from the Differentiable Viewpoint.

Milnor spent much of his career in Princeton, as a student and faculty member at the university and later as a professor at the Institute for Advanced Study, before moving late in his career to Stony Brook. Between the university and the Institute, he was briefly at MIT. I overlapped with him twice, at MIT and then during my sabbatical year as a member of the Institute, but foolishly, I never talked with him.

Categories: Math

Fields Medals

August 19, 2010 Leave a comment

The International Congress of Mathematicians (ICM 2010), a quadrennial affair, began today in Hyderabad, India. The opening ceremony lacks the grandeur of recent Olympics extravaganzas, but it does bring the awarding of Fields Medals to the latest winners. As explained at the website of the International Mathematical Union, “The Fields Medal is awarded every four years on the occasion of the International Congress of Mathematicians to recognize outstanding mathematical achievement for existing work and for the promise of future achievement.” The medal is generally considered the mathematical equivalent of the Nobel Prize.

To confuse matters, the Norwegian government established a mathematics prize in 2001 to serve explicitly as the mathematical version of the Nobel Prize. It is called the Abel Prize and has been awarded annually since 2003 by the Norwegian Academy of Science and Letters. (The prize’s eponym, Niels Abel, was a Norwegian mathematician who would make anyone’s list of the greatest mathematicians of all time and who died of tuberculosis in 1829 at the age of 26.) I wrote posts about the two most recent Abel Prize recipients, Mikhael Gromov and John Tate.

Whichever prize is more Nobelic — and anyway, who cares? — they do play different roles in the profession. The Fields Medal’s dual mission of recognizing outstanding achievement and promise of future achievement has come to be interpreted to mean that recipients not be too far along in their careers. Specifically, they should not be over 40.

Enough background. You can read about this year’s four Fields Medalists by going to the prize webpage at the ICM 2010 website. They are Elon Lindenstrauss of Princeton University, Ngô Bảo Châu of Université Paris-Sud, Stanislav Smirnov of Université de Genève, and Cédric Villani of Institut Henri Poincaré.

I don’t yet see an ICM press release providing general background on the four recipients, so I can’t link to it. What you can read now, at the prize webpage that I just linked to, are descriptions of their mathematical work written by other mathematicians. Don’t expect to understand too much. Maybe return later when there are biographical sketches.

In any case, this is a day mathematicians around the world eagerly await, ending months of speculation about possible winners. I didn’t want it to pass without mention.

Categories: Math

Martin Gardner, RIP

May 24, 2010 Leave a comment

Martin Gardner died Saturday. I wrote about him just last month, at which time I noted that although “not himself a mathematician, Gardner is one of history’s great popularizers of mathematics, through his long-running “Mathematical Games” column in Scientific American. He is as well one of the great debunkers of pseudo-science.” I own several compilations of his Scientific American columns, plus his one novel, The Flight of Peter Fromm, an odd book about religion and theology.

You can read more about Gardner in the NYT obituary, or by clicking on various links at the Scientific American website. For instance, several people pay tribute here and a 1995 Scientific American profile is republished here.

I have little I can add to what others have said. Here are some of the thoughts of Douglas Hofstadter, famed polymath in his own right:

. . . so few people today are really aware of what a giant he was in so many fields—to name some of them: the propagation of truly deep and beautiful mathematical ideas (not just “mathematical games,” far from it!); the intense battling of pseudoscience and related ideas; the invention of superb magic tricks; the love for beautiful poetry; the fascination with profound philosophical ideas (Newcomb’s paradox, free will, etcetera etcetera); the elusive border between nonsense and sense; the idea of intellectual hoaxes done in order to make serious points (for example, one time, at my instigation, he wrote a scathing review of his own book The Whys of a Philosophical Scrivener in The New York Review of Books, and the idea was to talk about the ideas seriously even though he was attacking the ideas that he himself believed in); and on and on and on and on. Martin Gardner was so profoundly influential on so many top-notch thinkers in so many disciplines—just a remarkable human being—and at the same time he was so unbelievably modest and unassuming.

Categories: Math, Obituary

O’Hare Eames Chair

May 18, 2010 2 comments

When I think of the famed designer couple Charles and Ray Eames, what first comes to mind are their iconic lounge chair and ottoman, pictured below.

Two such chairs and the ottoman graced the house we moved into when I was ten, and are there still, decades later. I was unaware of the Eames at the time. (I’ll confess, once I learned of them, I imagined they were brothers. It hadn’t occurred to me that Ray was a woman and that they were, in fact, a married couple.)

In yesterday’s Wall Street Journal page one feature article, Daniel Michaels reported from the Passenger Terminal Expo 2010 in Brussels on the latest in terminal seating. Pictured inside the paper were some newer options, but I was stunned when I saw on page one the drawing at right of that most familiar of all terminal seats, with the caption, “The Eames seat.”

I had no idea. Is there a chair that has been sat in by more people in history than this one? Probably not. And it’s yet another Eames chair.

It turns out, as everyone in the trade must know, that the Eames designed this chair for the opening of the expanded O’Hare Airport in 1962 (at which point O’Hare became the principal Chicago airport and Midway was relegated to minor status). The chair is still available. At the Herman Miller website, it is called Eames Tandem Sling Seating and given the following description:

Eames tandem sling seating serves millions of travelers every day and does it comfortably and reliably.

Designed for O’Hare International Airport in 1962, the sleek, contemporary design remains in style for all kinds of public transportation stations. Around the world, people find it a comfortable, inviting place to wait. And terminal operators appreciate its space-saving flexibility, durability, and easy maintenance.

I’ve never been a fan of the Eames tandem. It looks inviting enough, and I suppose it’s comfortable enough, but not when I’m about to sit on an airplane. I’ve had enough back problems over the years that the last thing I want to do before being locked into a plane seat for hours is lie way back in a chair with minimal lower back support. Often, when confronted with the tandem, I have chosen to stand, or walk around. I eventually succumb and take a seat, but after a few minutes I force myself to get up again, more out of fear of what the position I’m in might to do to me than any actual discomfort.

Eames. I still can’t get over it.

I can’t write about the Eames without mentioning my favorite of all their work, their mathematics museum exhibit. I first saw it in 1966, at Chicago’s Museum of Science and Industry. I had always assumed it originated there, but yesterday I learned (here) that it was initially commissioned in 1961 by IBM for the new Science Wing at the California Museum of Science and Industry (now the California Science Center) in Los Angeles. A duplicate opened later that year in Chicago. The Museum of Science and Industry considers its opening sufficiently important that it is listed as one of the highlights in the museum’s history, the only highlight between 1956 and 1971.

The California Science Center re-mounted the exhibit temporarily in the summer of 2002, as explained at the time in a press release, which notes that “the exhibition . . . won the hearts of several generations of teachers and students during its tenure at the California Science Center (formerly California Museum of Science and Industry) from 1961-1997.” The New York Hall of Science installed a permanent version in 2004, and a case study of the mounting of this exhibition is provided here, along with photos. Below is one of the photos, depicting the famous history chart.

If you have a chance to see the exhibit, please do.

I’ll conclude with one more Eames production, the powers of ten documentary from 1968, which you can watch below.

Next time you’re in New York, after visiting the Hall of Science, head to the Rose Center for Earth and Space at the American Museum of Natural History and see Scales of the Universe, a wonderful exhibit that shares the theme of depicting relative scales from the sub-atomic to the extra-galactic. And next time I find myself waiting for a plane in an Eames tandem, I’ll be thinking about those scales.

Categories: Design, Math