→ November 21, 2011
→ By

Here’s Looking at Euclid.

In case you’re interested, amazon has the Blu-Ray edition of The Lord of the Rings trilogy on sale for $49.99 (almost 60% off). Not sure how long that sale will last.

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Alex Bellos’ Here’s Looking at Euclid (known as Alex’s Adventures in Numberland in the U.K.) is a little lighter than the last math book I read, focusing instead of numerical oddities and paradoxes as well as the history of basic math. He keeps the tone light by revolving each chapter around one or more interesting personalities, such as the English dentist who used &#981 (the golden ratio) to design more attractive dentures or the various people involved in the invention and rise of sudoku.

Bellos’ gift with this book is to take mathematical subjects that might seem intimidating, such as the nature of irrational numbers like &#981 and &#960 or the concept of the normal distribution, and wraps them in interesting, easily accessible stories that might be enjoyed even by the math-phobic. There’s also an undercurrent here, only mentioned explicitly in one chapter, of sentiment that we don’t really do a good job of teaching math in American public schools. He talks about the need for someone to develop the number zero, without which no numerical system can properly function, and discusses a tribe in the Amazon that has no word for any number larger than five. The chapter on probability revolves around – what else? – gambling, from a conversation with a slot-machine developer to stories of people who figured out how to beat the house and forced changes like more frequent shuffling of more decks at the blackjack table. The final chapter was a real rarity, as it brought together one of my interests (math) with one of my wife’s (crafting) with a discussion of hyperbolic crochet, a way of building models of surfaces with constant negative curvature using yarn, which leads into a discussion of infinity and, of course, a stop at the Hilbert Hotel.

The book is not a straight narrative, but a series of chapters that can stand on their own, although Bellos tries to put them in a logical order from smaller concepts to larger ones. Readers generally interested in math will likely read it straight through – and quickly, as I did, because it’s well-written and I love the topic – but the design does allow anyone frustrated by the mathier sections to just jump ahead to the next part or the next chapter. There’s very little in here that a high school junior wouldn’t follow, however; calculus is mentioned but never used, and the hardest conceptual material appears in the final chapter.

Sudoku fans among you might be surprised to read about the puzzle’s history in the chapter “Playtime,” about math-based puzzles (including comments from Martin Gardner, not long before he died). A square of n smaller squares containing all the integers from 1 to n where all the rows, columns, and corner-to-corner diagonals add up to the same total is called a “magic square,” and has been known and studied since antiquity in Chinese, Indian, and Arab cultures, even finding favor with modern mathematicians like Leonhard Euler. The closest predecessor of modern Sudoku was first designed in 1979 by an American, Howard Garns, but redesigned by a Japanese puzzle maker named Maki Kaji and popularized by a New Zealand man named Wayne Gould, who saw one of Kaji’s puzzles in 1997 and wrote a computer program to generate them en masse. (For whatever it’s worth, I can’t stand sudoku.)

I’d love to see Bellos tackle more difficult mathematical material, given how well he translated the subjects he covered here into plain English and his ability to build a narrative around one or more people that kept the book from ever becoming dry. But I can imagine a sequel to Here’s Looking at Euclid (although I shudder to imagine the potential titles – Are Euclidding Me?) that keeps the material on the same level, as the world of math and numbers has far more stories to tell than Bellos fit into this one book.

Next up: Write More Good: An Absolutely Phony Guide, written by the very funny folks behind the @FakeAPStylebook Twitter account. I’ve read 75 pages so far, but that’s enough to know that every writer in the world will find at least something in here that s/he finds absolutely hilarious, since it touches on all areas of writing and has enough one-liners and short sections that there’s a good mix of dry humor and crude. I received review copies of both this and Euclid from the publishers.

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→ November 18, 2011
→ By

The Poincaré Conjecture.

As you probably noticed, I’ve got a new design here on the dish, one that was long overdue. I’d like to thank (and credit) Thomas Griffin for designing and setting up the theme, and reader Sara Showalter for designing that awesome custom header image.

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The Poincaré Conjecture was one of seven Millennium Prize Problems identified by the Clay Institute in 2000 as the most significant unsolved problems (or unproven theorems) in mathematics, and at this point it is the only one of the seven problems that has been solved. Such a solution should have earned its developer, in this case a somewhat reclusive Russian named Grigori Perelman, a million-dollar prize, but Perelman rejected the prize and the Fields Medal he was to be awarded for his solution. (The Riemann Hypothesis, which I discussed in my review last year of Prime Obsession, is another one of the seven.)

In his 2007 book, The Poincaré Conjecture: In Search of the Shape of the Universe (still on sale for $6.38 as a bargain book on amazon), Donal O’Shea, Dean of Faculty at Mt. Holyoke College and a professor of mathematics, gives a brisk history of the Conjecture with a quick mention of its solution. The first half of the book, from Euclid and Pythagoras up to Henri Poincaré and the early 20th century, was relatively fast-moving (for a math book) and easy to follow, but when O’Shea got deeper into topological discussions of the Conjecture, his explanations became shorter and I found myself getting lost.

The Poincaré Conjecture states that:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

In lay terms – and I apologize if I get this wrong – it means that any four-dimensional shape that is internally continuous and has no boundary can be mapped, point for point, to the four-dimensional shape called the “3-sphere.” The 3-sphere contains every point in 4-space equidistant from a single center; a point in 4-space is defined the set of coordinates (w, x, y, z). Think of a three-dimensional sphere, defined by all points (x, y, z) 1 unit distant from a single point, such as (0, 0, 0); this sphere will include (1, 0, 0), (0, -1, 0), (0, ?2, ?2), and all other points such that the square root of their sums equals one. (This is similar to the Pythagorean Theorem, but with another variable added to the sum.) We can picture this sphere in 3-space, so while we can’t picture the 3-sphere in 4-space, we can at least follow the math – the 3-sphere of unit 1 and center (0, 0, 0, 0) will include the points (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and so on.

Henri Poincaré, a prolific and brilliant French mathematician who built on work done by Bernhard Riemann, conjectured but could not prove that any four-dimensional shape that is “simply connected” – where any loop including two points can be reduced to a single point, meaning there is no disruption in the overall shape inside of such a loop – and “closed” – meaning if you walked on its surface, you would never reach an edge or boundary because the space closes around on itself – can me mapped, point for point, to the 3-sphere. As it turns out, this conjecture was extremely hard to prove, requiring mathematial concepts that did not exist at the time of the conjecture, and relevant to the question of the shape of the universe.

O’Shea did a solid job going into the history of first Euclidean and then non-Euclidean geometry, with interesting digressions on the lasting nature of the mathematical works of the ancient Greeks, how discoveries by Arab and Indian mathematicians (who were often religious leaders as well) spread to Europe, and how much knowledge was lost along the way, including much of Euclid’s work lost in the fire at the library of Alexandria. Poincaré himself is not an ideal central figure for a work of non-fiction, only jumping off the page in the chapter outlining his rivalry (and flame war, in letters) with the Prussian nationalist mathematician Felix Klein.

Where O’Shea lost me was with very brief introductions of critical terms used to describe the search for the Conjecture’s proof, then repeated use of those terms without sufficient explanation. I never encountered tensors in any of the math classes I took in school, and I don’t know what Ricci flows are (they were only created/discovered in 1981), or Betti numbers, or Laplace operators, but you need to understand those terms – and I mean really understand them – to follow the descriptions of the various steps leading up to and including Perelman’s solution. This is no small task; I’m asking O’Shea to describe upper-level college mathematics topics to readers who may not have gone beyond first-order calculus in a way that they will understand it. I don’t think he achieved that goal here.

I’m also not sure that O’Shea managed to deliver on the book’s subtitle. That the Poincaré Conjecture’s answer might help us understand the shape of the universe does not appear to be in any doubt. That it pushes us further toward understanding the shape of the universe is unclear, both from the book and from what I could find online that didn’t exceed my understanding. There does seem to be some thought that the universe might be a Poincaré dodecahedral space (also called a Poincaré homology sphere), a closed 3-manifold that is not simply connected, formed by taking opposing faces of a dodecahedron, rotating one to align with its opposite, and then smushing the dodecahedron and gluing each pair of faces together to form a 3-manifold in 4-space that is not homeomorphic to the 3-sphere. And I’ll stop there before I get further out of my league.

If you’re interested in these great problems of mathematics, I’d recommend John Derbyshire’s Prime Obsession, which I mentioned above and found more accessible than O’Shea’s book even though the problem under consideration, the Riemann Hypothesis, remains unsolved and likely has no practical application. O’Shea’s book reminded me of Amir Aczel’s slim volume called Fermat’s Last Theorem, also rather tricky to follow because of its heavy use of topology but with a bit more drama to help the reader plow through the less scrutable parts.

Next up: Sticking with math, I’m halfway through Alex Bellos’ Here’s Looking at Euclid, sent to me by the publisher earlier this year. It’s a fun tour of mathematical puzzles and oddities with a few dashes of number theory thrown in, but nothing you couldn’t follow if you have a high school degree.

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→ March 31, 2011
→ By

Five Equations that Changed the World.

My predictions for 2011 went up yesterday. Podcast and chat on Thursday.

Somehow I forgot to review Michael Guillen’s Five Equations that Changed the World
, a very strong look at five equations and the scientists who developed them that’s explained with very little math at all. Guillen’s target is the lay reader, a term which, since I haven’t taken a physics class since 1990, would include me.

The five equations aren’t hard to guess – they are, in the order in which Guillen presents them, Newton’s Universal Law of Gravity, Daniel Bernoulli’s Law of Hydrodynamic Pressure, Faraday’s Law of Electromagnetic Induction, the Second Law of Thermodynamics (discovered by Rudolf Clausius), and, of course, E = mc2, courtesy of Albert Einstein. But rather than just give the reader the equations and their derivations, Guillen crafts a short story around each, with background on each scientist’s life before the discovery*, the process that led to the development of the equation, and a brief epilogue on some major event or subsequent discovery that hinged on the equation itself. (For example, Newton’s law led to the manned mission to the moon, while Einstein’s led, of course, to the atom bomb.)

* So, does a scientist discover an equation, develop it, or something else? He doesn’t invent it, certainly; these are, as far as we know, immutable laws of our universe. I thought about using “unearth” to describe this process, but it seems to mundane, especially for Clausius’ and Einstein’s contributions. I’m open to suggestions here.

Newton’s and Einstein’s stories are rather well-known, I think, so I would say the most interesting sections of the book were the three that those two bookended. Clausius’ story was probably the least familiar to me, as I probably couldn’t have named him if asked. And what made his story interesting was how many other discoveries or developments had to happen along the way for him to be able to articulate his equation – including the invention of the thermometer, the creation of the calorie as both a unit and as a theory for the source of energy, and the life’s work of Julius Robert Mayer, a Bavarian doctor who first expostulated that all the energy in the universe had to add up to the energy that existed at the universe’s start (that is, the First Law of Thermodynamics), only to find himself rejected and ostracized by both the scientific and religious establishments of the time.

The final section of the book, on how Einstein’s theory of relativity led to the development of nuclear weapons, is a bit poignant as Einstein lived to see the destruction and regretted his role in encouraging President Roosevelt to order the development of the bomb. (I would imagine Einstein realized, however, that since the Germans would have eventually developed it themselves, the Manhattan Project was as much as a defensive move as an offensive one, even though it became an offensive weapon when we figured it out first.) Slightly less interesting, to me at least, was the extent of the family squabbles in the section on Bernoulli, where a pattern of fathers becoming jealous of talented sons tended to repeat itself in a way that would probably land them on Maury Povich today.

If you like the sound of this book but want something mathier, check out my review of Prime Obsession, a book about the development of the Riemann Hypothesis, perhaps the leading unsolved problem in mathematics today.

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→ August 16, 2010
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Prime Obsession.

I admit it: I am not afraid of math.

And if you’re not afraid of math either – in this case, some fairly heavy math – you might enjoy Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics as much as I did. It’s a book about an obscure question in the field of number theory, one that remains unsolved after 150 years and probably has little to no practical application, but John Derbyshire manages to give the subject some real personality while doing his best to make it accessible to readers who haven’t taken a lot of advanced math classes or who, like me, are a good 13 years removed from their last one.

The subject of Prime Obsession is the Riemann Hypothesis, which states that the non-trivial zeros of Riemann’s zeta function are half part real. “Non-trivial zeros,” in this case at least, are complex numbers (a + bi, where i is the imaginary number defined as the square root of negative 1 and b is nonzero) that give the result of 0 when plugged into the zeta function. “Half part real” means that a in that complex number is equal to ½.

The zeta function is the crux of the matter, the sum of the following infinite series:

That is:

Riemann posed his hypothesis when studying the Prime Number Theorem, which states that for any random number N, the probability of N being prime (and thus the frequency of primes around N) is roughly equal to the reciprocal of the natural logarithm of N, that is, 1/ln(N). In his one paper on the subject, he hypothesized that the frequency of primes and the differences between the actual frequency and the predicted frequency in the Prime Number Theorem was connected to the zeros of this zeta function. He couldn’t prove it at the time, and even though David Hilbert declared it one of the great mathematical problems of the 20th century in 1900, one of a list that has seen all but two of its number* solved, and in 2000 the Riemann Hypothesis was named one of the Millennium Prize Problems by the Clay Mathematics Institute, it remains unsolved. Prove or disprove it and you’ll get a cool million bucks for your trouble.

As you might imagine, solving the problem isn’t easy; indeed, it stands unsolved more than a decade after Sir Andrew Wiles’ solution of the equally perplexing problem of Fermat’s Last Theorem, one that required the development of an entire new field of mathematics (topology) unknown to Fermat at the time that he wrote that he had a “truly marvelous proof” to the problem. (Current thought is that whatever proof he had was incomplete.) The difficulty of proving or disproving the Riemann Hypothesis has led many of the major figures in mathematics, particularly in number theory, to attempt to tackle all or part of the problem or to work on further theorems and conjectures that build on the assumption that the “RH” is true. (And it has at least held true so far for very large numbers, which is not a proof but is weak evidence in its favor.)

Derbyshire’s main difficulty, beyond the lack of a clear resolution to the story, is making the solution of a potentially useless mathematical conundrum interesting; Wiles’ proof of Fermat’s Last Theorem was momentous and newsworthy, but the practical applications have been nil – it’s merely interesting to people who like numbers. Proving the Riemann Hypothesis would likely have a similar lack of real-world effects, and the hypothesis itself is a lot harder to grasp than Fermat’s Last Theorem was; the latter problem had an incredibly complex solution, but the question itself was easy for anyone who’d taken algebra to understand. Derbyshire does a masterful job of walking through the history of the Riemann Hypothesis, from earlier work on prime numbers, including the PNT, through Riemann’s brief life and career in mathematics to the major developments in the 151 years since his seminal paper appeared.

The book alternates between chapters walking through the math and chapters on the history and personalities involved in the hypothesis’ history. Carl Friedrich Gauss has a starring role early, while G.H. Hardy, Leonhard Euler, J.E. Littlewood, Jacques Hadamard, and Hilbert appear at some length later on. Derbyshire sprinkles stories of their peculiarities, senses of humor, and non-mathematical interests to keep the text lighter while also highlighting the chance occurrences that made some of the progress on the proof possible and regularly pointing out the remarkable longevity of most of the major mathematicians he mentions.

His math writing, while clearly geared to a lay audience, still got fuzzy for me when he got deeper into the zeta function as he tried to map it to the complex plane. Derbyshire relies on these “visual” interpretations that don’t correspond to any sort of plane or graphs that I’ve seen elsewhere, and I felt it was the one time he presupposed some familiarity with higher math on the part of the reader. But to his credit, he relies largely on algebra and gives a brief (re-)introduction to differentiation and integration for the short periods where calculus is necessary to move the math story forward. He also hits many major touchstones that will unlock memories for those of you who took and enjoyed lots of math classes, from the Sieve of Eratosthenes to the amazing Euler’s Identity, the latter of which states that

And if you look at that formula and are amused, fascinated, or just generally intrigued, Prime Obsession is a book for you.

I also recommend a book about one of the mathematicians who makes a cameo appearance in Derbyshire’s book, The Man Who Loved Only Numbers: The Story of Paul Erd?s and the Search for Mathematical Truth. Erd?s was a Hungarian-born savant who lived most of his life out of a suitcase, traveling the world, arriving at the doors of mathematicians he knew and announcing that “my brain is open,” after which he’d settle in for a few days or weeks and embark with his host on a streak of problem-solving and paper-writing. He had his own peculiar vocabulary, consumed large quantities of caffeine and later amphetamines, and combined brilliance and prolificacy (that’s peak and longevity for you Hall of Fame watchers) to the point where other mathematicians are referred to by their “Erd?s number,” where a person who co-authored a paper with Erd?s has an Erd?s number of one, while others are marked by how many papers you must go through to create the shortest possible chain back to Erd?s.

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