Books (Math): Chance

I've been reading David Whitehouse's The Sun: A Biography, but on Friday I forgot to bring it to work. Stuck without anything to read, I dropped by the book store and picked up Chance: A Guide to Gambling, Love, The Stock Market, & Just About Everything else.

Chance is a rather slim book -- which was what I was after, something to tide me over until I got back to my copy of The Sun -- that covers a lot of the same ground as Struck By Lightning, though more it covers that ground more lightly (the chapter on game theory is all of one page long!).

The book is so slim, in fact, that each chapter feels like it's ending about halfway through. It's hard for me to tell if it actually has enough material to be a good introduction for someone unfamiliar with statistics -- it seems too thin to me. But it is well-written and accessible, so it may well be a good starting point for the subject.

My biggest problem with the book is actually with the typesetting -- specifically, the typesetting of equations. There are some serious gaffes. When talking about the Gambler's Ruin, the book presents this formula:

1 - (q/p)m

P = 1 - (q/p)m+n

Of course, what they actually mean is:

P =	1 - (q/p)m
	1 - (q/p)m+n

It's not so bad in this example, but in some cases it gets very hard to figure out just what's supposed to be equal to what.

Another example is in the chapter on statistical sampling. The formula for the standard deviation of the normal distribution is:

√(sample proportion) × (-1 sample proportion)

sample size

Of course, the correct formula is:

√(sample proportion) × (1 - sample proportion)

sample size

There's a big difference between (-1 sample proportion) and (1 - sample proportion)!

This isn't Aczel's fault (on the next page, when values for sample proportion and sample size are substituted in, the formula is correct), but it should have been caught before the book went to print.

In spite of the book's lightness and the typographic problems, it's still an enjoyable book. There was even a nice coincidence in the chapter on coincidences. Aczel mentions his friend Scott Petrack who, when flying from Paris to Boston, discovered that the person he'd been sitting beside had sung in the same choir in Tanglewood (the summer home of the Boston Symphony Orchestra) about twenty years before. A week later, Scott got a business call from a stranger, and discovered that that person had sung in the same Tanglewood choir.

As it turns out, my father was also a member of the Tanglewood Festival Chorus about twenty years ago. There's a good chance he sang with Scott Petrack and the other two people, and that I heard them all. Not bad for an example of coincidence. Read more!

Books (Science): Trying to catch up

The problem with posting here is that I end up burning an hour or two on each post. So, when I don't have an hour or two to spare, I put off the posting -- and then it never gets posted. Of course, I also don't know if anyone's actually reading any of this, so it ends up not feeling that urgent. I never put off reading, though. I can't function properly if I don't have something to read during quiet moments.

As a result, I'm now several books behind in posting. So I'm going to short-change the books with an attempt at being brief, so I can catch up.

Jared Diamond: Collapse

Collapse is Diamond's follow-up to his first two books, The Third Chimpanzee and Guns, Germs, and Steel. I haven't read The Third Chimpanzee yet, but I found Guns, Germs, and Steel fascinating.

Collapse deals with the collapse of civilizations -- both past and future. Diamond examines how various past cultures have failed due to environmental problems (often self-inflicted), attacks by hostile neighbours, the loss of friendly trade partners that provided essential resources, and so on. He draws examples from all over the globe: Easter Island and various other South Pacific cultures, the Anasasazi and Maya in the Americas, the Viking settlements in Newfoundland and Greenland, and so on.

Collapse also examines current crises, comparing them to the past ones: the genocide in Rawanda, the economic collapse of Haiti (contrasted with the more successful, but still troubled, Dominican Republic next door), the environmental problems caused by irresponsible logging, mining, oil drilling, fishing, and so on. Diamond's also careful to point out examples of how responsible logging, drilling, and so on can be enormously helpful, not only to the environment, but to the economies affected.

Robert Kandel, Water from Heaven

Water from Heaven: The Story of Water From the Big Bang to the Rise of Civilization, and Beyond is just what its subtitle says: a very wide-ranging examination of water from a number of perspectives: cosmological, astronomical, physical, chemical, meteorological, environmental, sociological, and so on. The subject matter is fascinating from many perspectives, but the writing is rather dry, nowhere near as engaging as Diamond's.

Mario Livio, The Equation that Couldn't Be Solved

Mario Livio's previous book, The Golden Ratio, did a great job of prsenting the history of the famous "Divine Proportion" while cutting through a lot of the mythology around the number. The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry is another engaging book of math history, this time dealing with the development of group theory and the mathematics of symmetry.

The book follows the quest for a formulaic solution (or a proof of the absence of one) for general polynomial equations. High school algebra covers the solution to the quadratic: for ax² + bx + c = 0, the solutions for x are x = (-b + √(b² - 4ac))/2 and x = (-b - √(b² - 4ac))/2. Cubics and quartics have formulaic solutions as well. But what about quintic and higher-order polynomials? It turns out that, while particular subsets of higher-order polynomials can be solved with formulas, there is no general solution.

Livio centers the story of the quest for the solution (and the proof that there is none) around biographies of two contemporary tragic mathematical figures -- Niels Abel, who died at 27 from tuberculosis resulting from his poverty-stricken living situation, and Evariste Galois, who died at 21 in an idiotic duel. Abel provided the basis for Galois's great breakthrough, group theory, which forms the foundation of modern mathematics, and has provided valuable insight into the other sciences, especially physics but even anthropology.

Like Diamond, Livio's an excellent writer, providing deep social and historical context for the mathematics he describes.

Alexei Sossinsky, Knots

Knot theory is not something you hear about a lot in popular writing about mathematics. I'd been curious about how one would go about creating a mathematical model of knots long before I found Knots: Mathematics with a Twist.

A mathematical knot is slightly different from an everyday knot. Mathematicians close knotted strings into loops, so that there's no way to slide the knot off the end. Mathematical knots are more like Celtic knots than anything you'd find on a sneaker.

The big trick in the mathematics of knots is figuring out how to tell if two knots are essentially the same or different. If you take a trefoil knot (the simplest kind of knot) and, say, pull one of the loops through another, you haven't changed the essential nature of the knot -- you can restore it to its basic shape without cutting it anywhere. So the plain trefoil and the distorted one are essentially the same. But if you had a mirror image of the first trefoil, there'd be no way to change it into the original without cutting it.

The basic story of Knots is the story of the hunt for a perfect knot invariant -- a value you can assign to a knot which will be the same for all equivalent knots and different for all non-equivalent ones.

The problem is, the hunt hasn't really been successful yet. Each invariant presented has a fatal flaw -- there are some differing knots it can't distinguish, for example. Each one's better than the previous, but each one has a flaw.

Or, at least, I think that's where we end up, because there's one big problem with Sossinsky's book: it looses me for a while in the middle.

In the first half of the book, you get very used to seeing the phrase, "the proof is too complicated to present here". This isn't a big book -- 119 pages with lots of pictures on 5"x7" pages -- so it makes sense that you couldn't get too deep into elaborate proofs.

Unfortunately, when the book gets to the point of Vassiliev invariants, Sossinsky decides to go into detail about how Vassiliev invariants came about but never seems to quite explain what, in the end, they mean.

According to MathWorld, we still don't know if Vassiliev invariants are actually able to discriminate knots properly. The advantage they have over previous invariants is that they haven't yet been shown to fail.

Which basically boils down to, knot theory is so new that no-one's figured out it's "fundamental theorem" yet. Which helps explain why you don't see a lot of popular books on the subject.

Steven Squyres: Roving Mars

In 1999, Donna Shirley, who managed the Mars Pathfinder mission that put the Sojourner rover on Mars, published a great book on the mission called Managing Martians. That mission paved the way for the enormously successful Mars Exploration Rover (MER) mission that put the Spirit and Opportunity rovers on Mars in 2004 for a mission which was supposed to last 90 Martian days, or sols -- about three months -- but is still going even now.

Roving Mars covers the story of the MER mission from the point of view of Steven Squyres, the principle investigator who developed and ran the mission. The book starts with almost a decade's worth of failed attempts to get a mission accepted by NASA, and the string of failed missions which put NASA (and JPL) on the spot for a sure-fire success.

Squyres goes through the development of the rovers (on an extremely tight schedule), which adopted some of the innovations from the earlier Pathfinder mission. The new rovers were bigger and heavier, which meant that even though they were using an earlier design, it had to be adapted and expanded, leading to all sorts of testing nightmares.

Throughout the book it feels like the mission is on the verge of collapse. I can't imagine how stressful it must have been for everyone invovled -- no-one wants to be known for blowing the $250 million mission that shut down JPL!

The book ends at around Sol 250, when the mission had already run almost three times its original length. Anything accomplished after Sol 90 was bonus as far as NASA was concerned, but everyone expected the rovers to die by around Sol 150, most likely due to dust build-up on their solar arrays.

It is now almost Sol 850. Spirit has a lame wheel and Opportunity's arm-stowing motor is stalling, but both are still functioning and still collecting scientific data. After the catastrophic failures of three missions before the MER, Squyers's project has revitalized Mars exploration. Read more!