Kudos again to Sol, this time for introducing me to Dr. James Tanton, a creative mathematician with his own YouTube channel of interesting videos here (okay, I'm a sucker for an Aussie accent):

In other news, another recent "tweet" (from Twitter) that caught my eye:

"It is known that e is irrational and that pi is irrational, but it is not known if their sum is irrational."

I'm wondering (maybe someone out there knows the answer) is it EVER the case that 2 irrational, transcendental numbers are known to sum to a rational???

Want to win a few coins... The "Penney Paradox" is a very intriguing though less-discussed paradox than some of its more famous counterparts (it's named after its discoverer Walter Penney, though it is also often discussed using a penny as the working example).

If one flips a fair coin 3 separate times, there are 8 equally probable (heads/tails) triplet-results: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. In this game a first player selects one of these triplets, and then a second player chooses a different one. The coin is then flipped repeatedly until one of the selected triplets appears as a run and the player having chosen it wins the game (and coin). For example, if the chosen triplets are HTH and THT and the flips go THHHTH, the last three flips mean that HTH has won... the first triplet appearing matching a player's choice, wins.

One might first think that any one triplet is just as likely to occur as any other. However, upon reflection it will probably be clear that given a series of 4-or-more flips there are more ways for a triplet like say HTH to occur than the triplets TTT or HHH to appear. But what is far more intriguing is that NO MATTER what triplet the first player chooses, there are triplets that player #2 can select giving him/her a probabalistic edge of winning.
"Futility Closet" site mentioned this a few weeks back (and how player 2 can make his/her choice), but without elaborating much on how the mathematics of it works.

I should also mention that IF you do have Martin Gardner's "Colossal Book of Mathematics" on-hand he covers the subject well in his chapter 23 on "nontransitive paradoxes" (it is the "nontransitivity" of the relationships involved that result in the differential probabilities for the triplets).

It's annoying (and expensive) when, while awaiting the arrival of certain books in a bookstore, I suddenly happen upon a volume I've never heard of that looks enticing for the money I've been saving for the other specific books!...
This weekend I stumbled upon "Mathematics 1001" by Brit Richard Elwes, a new book that does a great job of covering in a nutshell a huge number of ideas/concepts/terms cutting across a broad swath of mathematics. Several such books are already available, but this looks to be the best one I've seen yet, written at a layman's level... a great little reference source and quickie introduction to various math nuggets, for your shelf. It is a sort of glossary of mathematical ideas, organized not alphabetically, but generally from simple and basic ideas that hang together to more complex and abstract ones.

The contents are divided into the following general areas:

-- Numbers
-- Geometry
-- Algebra
-- Discrete Mathematics
-- Analysis
-- Logic
-- Metamathematics
-- Probability and statistics
-- Mathematical physics
-- Games and recreation

The Intro says that the book is "aimed at.... anyone with a curiosity about mathematics, from the novice to the informed student or enthusiast. Whatever the reader's current knowledge, I'm sure that there will be material here to enlighten and engage."

I think the author succeeds....
(It's a hardback, and I believe worth the $25 price, but of course can be gotten cheaper.)

In their tributes to the Father of Fractals, Benoit Mandelbroit (recently deceased), several sites have done posts on "infinite coastlines." One of the related notions that I've always found mind-blowing is that of a closed border or perimeter having INFINITE length, yet enclosing a region of FINITE area (very counter-intuitive!). The Koch Snowflake is probably the most oft-used example of this (though there are any number of other possible cases).
Most of you are likely well-familiar with it, but if not, or if you need a refresher, a couple of the many sites expounding on it:

Further memorializing the recently-deceased, Ivars Peterson reports on yet another oddball Mobius-band trick passed along to him by Martin Gardner some years ago:

...and could only have one book to read I think I know what it would be. . . .

Yes, I'll squeeze in one more bit about Martin Gardner before his day of honor tomorrow (to any who are tired of hearing about Martin Gardner by now, I apologize, but HEY! it's MY blog so deal with it ;-))
When Gardner's "The Colossal Book of Mathematics" came out in 2001, I looked at its size, price, and the many chapters covering topics I wasn't particularly interested in, and ignored it (I already had plenty of Gardner books on my shelf). It was only years later that I checked it out from a public library and discovered not only how many chapters were of great interest to me, but also how many of the topics I wouldn't normally have found interesting were made so by Gardner's deft and insightful writing.
So again I highly recommend this volume to anyone lacking it on their shelves. If you're ever stranded on a desert island it would offer you weeks/months of mental entertainment (...and, as an alternative, just in case I got tired of math, I might take along Gardner's essay-anthology, "The Night Is Large," as well!).

Meanwhile, 'Mathematics Rising' blog has just covered the 3rd of my 'Fab Four,' Bernhard Riemann, here:

Here he references one study I find a bit hard to believe:

"The sound a number makes can influence our decisions about it. In a recent study, one group was shown an ad for an ice-cream scoop that was priced at $7.66, while another was shown an ad for a $7.22 scoop. The lower price is the better deal, of course, but the higher price (with its silky s’s) makes a smaller sound than the lower price (with its rattling t’s).
And because small sounds usually name small things, shoppers who were offered the scoop at the higher but whispery price of $7.66 were more likely to buy it than those offered the noisier price of $7.22 — but only if they’d been asked to say the price aloud."

A few posts back I linked to an entry at another blog on one of my designated 'Fab Four' (Carl Gauss), and now RJ Lipton has put up a nice post on his blog on another of those Fab Four, David Hilbert and his insights:

And as we approach the Oct. 21 'Celebration of Mind' in honor of Martin Gardner, yet another piece on him, this time from the latest edition of American Scientist magazine:

His name will forever leave its stamp on mathematics. In memory, I'll just re-post the 3rd post I ever ran on this blog, one of the many tributes to his Mandelbrot Set that can be found on the Web:

YouTube video of Mandelbrot Set with Jonathan Coulton's lyrics below:

lyrics:

Pathological monsters! cried the terrified mathematician
Every one of them is a splinter in my eye
I hate the Peano Space and the Koch Curve
I fear the Cantor Ternary Set And the Sierpinski Gasket makes me want to cry
And a million miles away a butterfly flapped its wings
On a cold November day a man named Benoit Mandelbrot was born

His disdain for pure mathematics and his unique geometrical insights
Left him well equipped to face those demons down
He saw that infinite complexity could be described by simple rules
He used his giant brain to turn the game around
And he looked below the storm and saw a vision in his head
A bulbous pointy form
He picked his pencil up and he wrote his secret down

Take a point called Z in the complex plane
Let Z1 be Z squared plus C
And Z2 is Z1 squared plus C
And Z3 is Z2 squared plus C and so on
If the series of Z's should always stay
Close to Z and never trend away
That point is in the Mandelbrot Set

Mandelbrot Set you're a Rorschach Test on fire
You're a day-glo pterodactyl
You're a heart-shaped box of springs and wire
You're one badass f**king fractal
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way

Mandelbrot's in heaven, at least he will be when he's dead
Right now he's still alive and teaching math at Yale
He gave us order out of chaos, he gave us hope where there was none
And his geometry succeeds where others fail
If you ever lose your way, a butterfly will flap its wings
From a million miles away, a little miracle will come to take you home

Just take a point called Z in the complex plane
Let Z1 be Z squared plus C
And Z2 is Z1 squared plus C
And Z3 is Z2 squared plus C and so on
If the series of Z's should always stay
Close to Z and never trend away
That point is in the Mandelbrot Set
Mandelbrot Set you're a Rorschach Test on fire
You're a day-glo pterodactyl
You're a heart-shaped box of springs and wire
You're one badass f**king fractal
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way
Go on change the world in a tiny way
Come on change the world in a tiny way

If you're headed off to college soon (or already there), you're probably too young to remember the movie "The Graduate" but nonetheless I have one word for you... "mathematics." Read about its importance and attractiveness as a career:

The latest edition of "Mathematics Magazine" from MAA lists the top-scoring students in the last USA Mathematical and Junior Mathematical Olympiads, and the names are as follows:

Timothy Chu, Calvin Deng, Michael Druggan, Brian Hamrick, Travis Hance, Xiaoyu He, Mitchell Lee, In Sung Na, Evan O'Dorney, Toan Phan, Hunter Spink, Allen Yuan, Yury Aglyamov, Ravi Bajaj, Evan Chen, Zijing Gao, Gill Goldshlager, Youkow Homma, Jesse Kim, Sadik Shahidain, Alexander Smith, Susan Di Yun Sun, Jiaqi Xie, Jeffrey Yan, Kevin Zhou

One can't help but be struck by the degree to which names of Oriental and Asian ancestry seem to predominate this list of USA Olympians. Even in my own college math courses on the west coast 40 years ago I saw such a predilection; certain foreign nationalities seemed to have a 'knack' for math and analytics that Americans often struggled with. One wonders how deep/real these differences are, and if they are culturally-based or even possibly have a genetic component... or, I'd be curious how many of these students early-on learn foreign ancestral languages in addition to English, and if early exposure to certain languages predisposes one toward a mathematical aptitude or analytical skills in general (in a Whorfian sort of way).

I suspect somewhere out there this has all been looked at, or at least argued over, but don't know if there is a resolution to the notion (and of course the above names represent too small a sample size to draw any real conclusions from, so I'm just wildly wondering out-loud here...). American teaching methods for mathematics have often come under fire in recent times (leaving many young people phobic of the subject), and yet certain students always seem to excel and remain enthused, regardless of method.

In an interview, Gardner once said his two favorite books (that he authored) were, "The Whys of a Philosophical Scrivener" (a favorite of mine as well, but one of his lesser-known and non-mathematical works) and his novel "The Flight of Peter Fromm" (which I've never read) --- seemingly odd choices from his vast outpouring, but then Gardner was unpredictable in so many ways. He critically (and hilariously) reviewed/debunked the 'Whys...' book for the NY Times under the alias George Groth; just one of many keenly-clever playful stunts he pulled off over the years.

I'm currently re-reading his "The Colossal Book of Mathematics," the wonderful compendium of his best "Scientific American" columns over the years (with great addenda added)... a must-have volume for any math-fan!
Gardner was a hugely humble individual who might likely be amazed at the outpouring of fondness/admiration expressed upon his death. Luckily he has left us with a lifetime-or-two of reading material. Even putting aside his mathematical work, I regard him as one of the best essayists of all time. Even when disagreeing with his takes on certain matters he was a joy to read (and that is a test of a great writer --- when you treasure reading them even when you disagree with them).
And to think, he barely studied math (academically) beyond high school! What a mind!! (worth a yearly celebration).

Martin... THANKS again for the memories, the musings, and the mathematics!

I briefly mentioned Shing-Tung Yau's new book, "The Shape of Inner Space" in previous posts (a book about the geometry underlying string theory), but have now read a review copy sent along by the publisher, and can say a little more about it.

In the last couple of decades many books about cosmology have been published (by physicists or science journalists) for a general audience. And many of those volumes have been quite good and accessible to non-professionals, even when the more technical material is not very comprehensible to the average-Joe. Yau's book is somewhat different, on several counts.

"A compact Kahler manifold with a vanishing first Chern class will admit a metric that is Ricci flat."

If the above sentence leaves you in a fog (or worse), I'm not sure this will be a book for you. Yau's book is filled with such language (every time I thought I was entering a more layman-friendly section, it would be short-lived, before I was once again in way over my head). If you are not very familiar with string theory, M-theory, Calabi-Yau manifolds, black holes, branes, and the like, you may want to pass on this volume; having a casual interest in physics/cosmology won't get you through it.
The book involves a lot of heavy-duty mathematics (moreso than other cosmology-type books for the masses; indeed some of it reads more like a textbook than a trade book). I actually enjoy the challenge of reading certain science material that is beyond my comprehension, but I suspect I'm an anomaly, and most folks don't have the patience for reading through material they simply don't understand. In short, I think there is probably a very limited audience for this offering, though that particular audience may very much relish it (...and I stand in awe of them!).

A little background: For any who don't know, Yau is a mathematician (not physicist) and Fields Medalist, with a focus on very advanced/abstract geometry, who's theories (especially proof of 'Calabi-Yau manifolds') came to underlie the mathematics of string theory. For those who don't follow such things, string theory is much more controversial now then when it was first introduced and there seemed to be an almost faddish bandwagon in its direction. Yau's book appears at a time when interest in string theory may even be waning, or at least taking a lot of heat. Yau is quite cognizant of the difficult road ahead for string theory, and how dominant views could change; at one point he writes:

"I personally think Calabi-Yau manifolds are the most elegant formulation [of the underlying geometry of the universe], as well as the most beautiful manifolds constructed so far among all the string vacua. But if the science leads us to some other kind of geometry, I'll willingly follow....
"Despite my affection for Calabi-Yau manifolds --- a fondness that has not diminished over the past thirty-some years --- I'm trying to maintain an open mind on the subject, keeping to the spirit of Mark Gross's earlier remark: 'We just want to know the answer.' If it turns out that non-Kahler manifolds are ultimately of greater value to string theory than Calabi-Yau manifolds, I'm OK with that. For these less-studied manifolds hold peculiar charms of their own. And I expect that upon further digging, I'll come to appreciate them even more."

Indeed one of the charms of this book is that while so many popular cosmology books beat the drum of the author's particular hardened point-of-view, Yau, as a mathematician, recognizes that he is somewhat apart from these physicist wrestling matches, and can step back, still offering his own personal leanings, while remaining more freely open to new conceptualizations than some other debaters seem to be.
The two chapters I most enjoyed (comprehended) came toward the end of the volume, "Truth, Beauty, and Mathematics," and "The End of Geometry?," where he waxes somewhat philosophical, even poetic, about the nature of mathematics/geometry and its interplay with physics, and also speculates about a future entailing what he terms 'quantum geometry'... but by then I was pretty tuckered out from the 280 pages that preceded those chapters!

If you are considering purchasing this book I would recommend that you read Peter Woit's review, AND the comments that follow it, here:

(You might also want to read the Wikipedia entry for "Calabi-Yau manifolds" to get a sense of whether or not you can follow this material.)

And I would also recommend that anyone choosing this volume initially read the 12-or-so pages of glossary at the back of the book just to familiarize yourself with many of the more heavily used terms ahead-of-time (unfortunately, many lesser-used, but difficult, terms are not included in the glossary).

One last ironic note: this book is published by "Basic Books"... one thing it is NOT though, is "basic!"
I don't doubt that it is an excellent exposition of its subject, but it is a challenging read to-be-sure. (In fairness to Basic let me say that I'm also currently reading another of their prior cosmology offerings, Frank Wilczek's "The Lightness of Being," from 2008, and enjoying it considerably more than the Yau volume.)

There is a lot more on Calabi-Yau manifolds around the internet, as well as many more reviews of this particular book available.

And last month I mentioned Carl Gauss as one of my "Fab Four" mathematicians of all time. If you need to know more about him, Steven Colyer recently posted a nice mini-biography at his blog:

Recursivity revisited (or taken to the ultimate limit)...

"If my mental processes are determined wholly by the motion of atoms in my brain, I have no reason to believe that my beliefs are true... and hence I have no reason for supposing my brain to be composed of atoms." --- J.B.S. Haldane, "Possible Worlds" (1927)

Turns out the "Gathering For Gardner" folks are planning a worldwide "Celebration of Mind" in honor of Martin Gardner for Thur. October 21 (his birthday)... parties wherever individuals organize one. Read about it here (maybe host a get-together yourself!):

Many (most?) readers here are likely on Twitter. If you think the new Twitter design (if you're using it yet) looks pleasing, the below blogger says there's a mathematical reason for that: the design is based on the Golden Ratio:

Just noticed that the current issue of "Skeptical Inquirer" (Sept./Oct.) has a large section of tributes to Martin Gardner (by many who worked with him over the years in the skeptics' movement). It also includes the final (non-math) column he wrote for the magazine shortly before he passed away. Worth a gander if you have access to it (the issue isn't online as yet).

By way of review, just a list of some of the books I've mentioned favorably in the last month or so, in no particular order; some older, some new (some I've read myself, some I've only read reviews of):

"The Lifebox, the Seashell, and the Soul" by Rudy Rucker

"Meta Math" by Gregory Chaitin

"Mathematical Fallacies and Paradoxes" by Bryan Bunch

"The Calculus Lifesaver" by Adrian Banner

"The P = NP Question and GÃ¶del’s Lost Letter" by RJ Lipton

"The Shape of Inner Space" by Shing-Tung Yau

"Proofiness: The Dark Arts of Mathematical Deception" by Charles Seife

"Everything and More" by David Foster Wallace

"Group Theory In the Bedroom, and Other Mathematical Diversions" by Brian Hayes

"Quantum Man: Richard Feynman's Life in Science" (forthcoming) by Lawrence Krauss

"Loving and Hating Mathematics: Challenging the Myths of Mathematical Life"(forthcoming) by Reuben Hersh and Vera John-Steiner

...and I'll throw in one additional book that I haven't seen myself, but Sol over at WildAboutMath is strongly recommending:

"The Mystery of the Prime Numbers" by Matthew Watkins

Nice article in current edition of "The Atlantic" magazine on the 1-year-old community-forum Website, "Math Overflow," where serious mathematicians go to collaborate on solutions to all manner of math problems:

"Boasting 2,700 active users ranging from especially bright undergrads to Fields medalists, the basic function of the site is to answer the highly technical questions that crop up in math research."

"...organizationally, Math Overflow stands apart from its predecessors. Math Overflow is a community-moderated forum; users vote on the most accurate answers to the questions posed and gain reputation points based on participation, the most active of whom are granted various moderation privileges. The best answers are voted to the top of the page, while the worst ones are voted to the bottom."

"Math Oveflow is almost an anti-social network, focused solely on productively addressing the problems posed by its users. Heavily moderated, the guidelines for asking questions are designed to discourage unnecessary chatter and keep the community's focus on a question at hand...
"We've tried to make the forum as 'professional' as possible," said Scott Morrison..."

"Math Overflow has been a something of a revolution for how collaborative math is carried out on the Web..."