June 27, 2006

Coincidences, Chaos, and All That Math Jazz - Book Review

I am 70% of my way through Coincidences, Chaos, and All That Math Jazz by Edward B. Burger & Michael Starbird but have to return it to the public library soon and so I figured I might as well compile and blog about what I have read so far... since I did glean some more interesting tidbits about numbers, learned about a poll debacle during the 1936 Presidential election due to survey of an un-representative sample of the population, and so on..

Here is a "review"...subject to change and modification as and when I read the un-read chapters and finish the book...

Overall a very good read. Here is a quick review of the book, divided into 4 parts, with about 2-3 chapters in each part.

First a preface:

“When most of us think of math, we first think of numbers. And when we think of numbers, we think of counting. While at first blush we may not view the act of enumeration as profound, it is certainly something we may count on. (Sorry.)”…(Chapter 5, p79).

The book is full of such punning and corny jokes but while they are cause for a pause, they are a welcome distraction for someone like me reading the book in a very technical mood and are probably a necessary diversion to lighten the load for people who are of a mathematical bent of mind - we are after all talking about math and numbers, a much dreaded subject for many in their high school days – although I doubt that despite the mighty goals of such books to reach out to people without a mathematical propensity, do such people really read these books? That said, while the book is definitely void of any mighty equations that may scare some people away, it is not dumbed down and did make for an enjoyable read.)

Part I - Undersanding Uncertainty – Coincidences, Chaos, and Confusion

Some of the initial stuff regarding the role uncertainty, luck, and sheer statistics plays in explaining “coincidences” was elementary, in the sense that it was too obvious to someone with my background and interests. Although I personally would have liked the first two chapters to be shorter, it is likely that for a non-science non-mathematical person, these chapters on unbridled coincidences we encounter in our lives and the origin of chaos and its role in preventing us from knowing the future were perhaps necessary introductions that helped set the mood for what is to follow. The third chapter, titled ‘Digesting Life’s Data – Statistical Surprises’, was a interesting read even for someone like me who has a decent background in statistics… enlightening me about the bias in polling that led to a historic doomed poll conducted by the Literary digest for the 1936 US Presidency poll and also does a good job explaining basic statistical concepts for readers without a statistics background through interesting examples like SAT scores, HIV-AIDS testing, Air vs. road safety statistics - pointing out how statistics an help us understand the world by highlighting random or unknown features of the data but at the same time can also be used to manipulate data and lie with the right (or wrong) interpretation and presentation. (Related aside: Also see my blogpost about nonsensical surveys.)

Part II – Embracing Figures: Sensing Secrecy, Magnificient Magnitudes, and Nature’s Numbers

Skipped Chapter 4 on Cryptography for later reading and also skipped over Chapter 6 on the Synergy between Nature and Numbers (skipped the latter after giving it a quick scan as I plan to read The Constants of Nature : From Alpha to Omega--the Numbers That Encode the Deepest Secrets of the Universe by John Barrow & Just Six Numbers: The Deep Forces That Shape the Universe by Martin J. Ree, which should give more detailson these synergies.)

Chapter 5 titled, ‘Sizing up numbers’ was a great read and succeeds in its goal to put a ‘face to a number’, using common examples that arise in daily life that help us get a more intuitive feel for what thousands, millions, billions, and trillions actually mean….with some funny and interesting tidbits and examples like “The number of hours a student spends in class during a college education is one or two thousand; that number is also approximately the number of hours we sleep in a year. Coincidence? We think not…”; that it is possible for all 6.4 billion people in the world to fit into one cubic mile; and asking interesting questions like “How much will a million dollars weigh” (answer is estimated to be about 1600lbs); and if Bill Gates would be better off going off the clock to pick up a hundred dollar bill lying on the floor, if we assume his annual pay is the twenty billion dollar personal wealth increase a few years ago (the answer is NO.. at 20billion a year, he earns 2800$ a second and so earns 100$ every 1/28th of a second…so, “he should not only not clock out to pick up the 100$ bill, he shouldn’t even stop to look at it”.) But the best and most interesting example to me was the one used for explaining what a quadrillion is… where the authors use an example of folding a paper repeatedly into halves… showing that after 10 folds, we have a thousand layered stack which is 4 inches thick, after 20 folds, it is one million layers and 350 feet thick, …after 25, it is 2 miles thick, after 30, it is 1 billion layers thick and 64 miles thick…and by 50 folds it has reached 1 quadrillion layers and goes for 64 million miles…with the thickness going beyond 128 million miles (well past the sun) at the 51st fold. Ofcourse, in practice, we cannot go beyond 7 folds with any piece of paper.. (try it!)… but this example boggles the mind as even I didn’t anticipate or think of how quickly explosive repeated doubling gets. The next example in the chapter is also an interesting one on arranging a deck of card on top of each other without any glue but such that they do not topple over…and apparently (see page 90-91) it is possible to arrange a deck of 52 cards such that the top card is more than a mile beyond the end of the table so that we can actually sit on that top card without collapsing the leaning pile. Again, this may be impossible to do but the authors provide a sound mathematical and physical argument for why this and other interesting card arrangement tricks should be possible.

Part III - Exploring Aesthetics: Sexy Rectangles, Fiery Fractals, and Contortions of Space

Even though I have read quite a bit of technical literature in the area of chaos (by no means am an expert or even claim I understand chaos theory though), the chapters that lead into the discussions on chaos made for new and exciting reading. New & exciting because, based on my previous readings, I already understood how chaos arises mathematically from Bifurcation Theory but to see the patterns and ordered chaos of something like paper-folding was an amazing revelation. Chapter 7 gives us a good introduction to the aesthetics of the golden ratio and its role in art & nature and the “beauty” in golden rectangles whose base to height is the Golden Ratio, 1.618 and a delightful method to construct one with a straightedge which is unmarked and a compass. However, there are other detailed expositions of the subject (eg: The Golden Ratio : The Story of PHI, the World's Most Astonishing Number by Mario Livio) that are on my to-read list (if I ever get to them) and probably will make for even better interesting reads on the subject.

However, what I quite enjoyed was the transition made in copying over the golden triangles repeatedly, leading us from the precise beauty of the triangle to reproductive chaos. And even though I didn’t quite understand this section perfectly, it left me wanting more… the last sentence of the chapter summarizes it well.. “We now see that aesthetics and mathematics are deeply related. There’s beauty in mathematics and mathematics in beauty”. Well said.

And more delightful discussions of organized chaos did come in Chapter 8, which challenges us to go back to the ideas of paper folding and leads us through a series of very interesting pattern recognition exercises involving the valley and ridges in the folds of the paper. The authors show that results are quite mind-boggling – flitting between “sheer chaos” and “complete regularity” and in fact go on to show that the paper-folding sequence for arbitrarily many folds is actually the output of an extremely simple five-line Turing machine program!! (Some of this may be elementary for someone familiar with Turing machines and computer science basics but again… I am not sure I followed this section perfectly well ..not that I couldn’t but I gave it a quick read and did not bother to wait and study the details. The same can be said of the next section on folded swans and leading up to the Dragon curve and the common fractal structure of the self-similar Sierpinski gasket (which you can create in Excel, btw and then measure the resistance in!)

Still to read Chapter 9.. which takes us on an exploration of an “elasticized world” followed by Part IV – Transcending Reality: The Fourth Dimension & Infinity.

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