Artful Geometry…

I'll confess to not being a big fan of coffee-table books; pictures of animals and exotic places are okay and appreciated in certain contexts, but outside of those two subjects, I'm usually not much enamored of large picture books collecting dust on a tabletop as a snooty bit of decor. And that goes double for math and science books pretending to be table-toppers… I want to read and ponder math and science, more than I want to stare at pretty pictures, so I'm always a bit skeptical of coffee-table-like books that approach these subjects... and appropriately such books are a bit rare.

Having said all that, I have another confession to make: I'm a sucker for geometry!! ...My initial thought when I first glanced at the new volume "Beautiful Geometry" was, "oh good a new Alfred Posamentier book!"… but it isn't from Dr. Posamentier; it is from Eli Maor (and artwork by Eugen Jost), another fabulous math-popularist. And if I owned a coffee table (I don't) this is a volume I'd be very pleased to plop upon it! Princeton University Press has done its usual splendid job of presentation with this large striking offering of indeed beautiful geometry (beginning with the gorgeously-rendered cover of a Sierpinski triangle -- which, though you might not recognize it from that cover, is characterized by the delightful fractal paradox of having an infinite perimeter bounding an area of zero!).
 
I could be mistaken, but it seems as if the wide popularity of Clifford Pickover's "The Math Book" has spawned a mini-proliferation of these sorts of artsy-math compilations, that combine math text with eye-catching art and graphics. The artwork in this volume is consistently exquisite -- sometimes its meaning being clear and obvious, and other times requiring closer examination or pondering. Maor's text is excellent (and very accessible) as well, and does raise the volume above the level of a mere look-see coffee table book -- you can definitely learn some interesting and real math along the way, which is beautiful in its own way, apart from the artwork.

Although the topics covered (over 50 gems) run mostly in chronological order, you can easily hopscotch around the volume jumping to topics which interest you most, or which are new to you. Of course there are many classic bits of plane geometry included here, but also some subjects that might be less familiar to many readers, like "Ceva's theorem," "Steiner's prism," "Lissajous figures," and the "Reuleaux triangle."
It's not often that math books can be described as "delicious," "scrumptious," or "delectable," but such words fit in this case, or, as Ian Stewart calls it, "a feast for the eyes."
Mind you, you're average neighbor who stops by for a beer on Sunday afternoon may not be much moved by these pages, but for math enthusiasts and especially young people delving into geometry this is A BUY! Geometry has, perhaps, never been rendered more beautifully!

From indications I've seen, it looks like another banner year ahead for popular math books (this particular volume is scheduled to hit bookstores around end-of-January)… I can't imagine it can equal 2013, but there's some great stuff coming down the pike (the one I'm currently most anticipating is Max Tegmark's "Our Mathematical Universe" due out in a couple weeks). Yeah, I'm already likin' the looks of 2014!
So a Happy New Year to all!!, as MathTango, which started as an experimental offshoot of Math-Frolic, essentially completes it's first year of postings.

Euclid... and His Explicators

A blurb today on "The King of Infinite Space: Euclid and His Elements," the latest offering from David Berlinski....

Sometimes for math book reviews I'm tempted to caution, "your mileage may vary"… meaning that you may get more (or, less) out of a particular volume than I do... we all have different inclinations. David Berlinski's books are especially subject to the varying proclivities of individual readers.

Berlinski is a controversial figure (some even call him a "provocateur")… but there is no debate, so far as I'm concerned, that he is an interesting, clever, even unique, writer. With the exception of David Foster Wallace's "Everything and More" I don't know anyone who writes math-related material with more panache than Berlinski does -- and I mostly enjoyed 3 of his prior books, "The Advent of the Algorithm," "Infinite Ascent," "One, Two, Three," and some parts of "A Tour of the Calculus," but this latest work left me flat, despite my being a geometry fan.

In tackling Euclid and his axiomatic "Elements" Berlinski may have met his match of subject material that just doesn't lend itself to enthralling explication or word flourishes. At some turns I had a feeling that the author may have churned out parts of the volume just to meet a deadline. This offering seems to lack the richness or fullness of his prior writing, and in the end falls between some cracks… it certainly isn't a textbook, but neither is it really tuned to a mass market. It is not even necessarily suited for a general audience of mathematicians, but at best perhaps for some subset of those who enjoy popular math with a narrow focus on geometry and logic.  The elements of Euclid's "Elements" are just too dry and tedious to be turned into scintillating reading even by Berlinski's deft wordplay talents. At ~150 pages, it is at least a brief and fairly easy read, if not an enticing one.

One thing that may be telling (or not) is that there are no individual endorsements for the book on the back cover (as is commonplace in today's publishing world), nor even in the promotional letter I received with the book. The volume gets a favorable nod from both "Publishers Weekly" and "Kirkus Reviews" (which are perfectly respectable sources), but oddly lacks a positive thumbs-up from any mathematician/writer. "Publishers Weekly" says, "...even the most math-averse [will be] enthralled by Berlinski’s rich, vibrant language" -- did they read the same book I did!? I certainly don't believe this volume will appeal to the "math-averse," and "rich, vibrant" is a stretch as well, even though occasional passages here-and-there might satisfy that criteria. And "Kirkus Reviews" states Berlinski "brings Euclid’s little-known life alive" -- huhhh? there's actually not much here about Euclid's life at all (in fact early on, the author mentions that little is known of Euclid's life), but rather the volume is a highly-focused analysis of the content of Euclid's "Elements" bit-by-bit.
Axioms/postulates, theorems, proofs are fairly dry topic matter (even when there is beauty in them), and analysis or close scrutiny of axioms/postulates, theorems, proofs, doesn't much change that, even with a few interesting stories sprinkled in along the way. Perhaps tellingly, in recent years there have been several volumes on Pythagorus and Fibonacci, and certainly more modern mathematicians, but not so much on Euclid -- possibly a reflection of the difficulty involved in making the subject engaging. At one point, Berlinski quotes the poet Edna St. Vincent Millay, "Euclid alone has looked on beauty bare" -- Berlinski's book often captures the 'bareness' of Euclid's work, but only weakly illuminates whatever beauty might be gleaned.

Yet, having said all this, the volume does contain a passage that is among the favorites I've ever read in a popular math book, and I'll quote in its quirky entirety (coming at the very end of Chapter 3):

"Like any other mathematician, Euclid took a good deal for granted that he never noticed. In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means.
"Euclid expressed himself in Greek; I am writing in English. Neither Euclid's Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn.
"These examples suggest what is at work in any attempt to describe once and for all the beliefs 'on which all men base their proofs.' It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact."

Hmmm.... "proof," as little more than artifact... (in fact one might even go further and argue that "evidence" in science is ultimately little more than artifact, so riddled is it with assumptions, but, I won't go there).

Even this passage brings us back, I think, to 'your mileage may vary'… because I suppose what I view as a keen, insightful passage, will strike others as 'ho-hum, what's the big deal about that,' but that may often be the case with Berlinski's commentaries (which can be quite philosophical).

Anyway, my advice would be to look for this book in your public library and give it a free read before plunking out dollars for it. Its appeal may be narrow (specifically, for hard-core geometers, or outright Berlinski fans), and there are likely some better current choices out there for your math bookshelf (an older book, by the way, which covers some of the same material but is more expansive, and which I did enjoy, was Leonard Mlodinow's  "Euclid's Window").