"Discover, Reflect, Understand" Math

i.e... Think!  (this comes from math teacher Rick White)




And, in a different form, more "advice to a young mathematician" here from the Princeton U. Press's "Princeton Companion to Mathematics":

http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf

American Classic (no math)

I'll have a little math post up later this afternoon... but for the moment, just a bit in memory of Pete Seeger (1919 - Jan.27, 2014):

JMM and MoMath

Brian Hayes of "bit-player" offers a wonderful account of his experience at the recent Joint Math Meetings in Baltimore:

http://bit-player.org/2014/notes-from-the-jmm

… really, a must-read; especially interesting stuff on Yitang Zhang, of twin-prime fame (and apparently now tackling the Goldbach conjecture), including this quote from one of his former Purdue colleagues T.T. Moh:

"When I looked into his [Zhang's] eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the north pole, a mountaineer who determined to scale Mt. Everest, and a traveler who would brave thunders and lightnings to reach his destination."

Meanwhile, Peter Woit reports on a "cagematch" between Ed Frenkel (Platonist) and Jim Holt (non-Platonist) held at the Museum of Mathematics in New York recently; he calls it "a no-holds-barred discussion of Platonism and mathematics in front of a standing-room-only crowd"... sounds like it was quite entertaining:

http://www.math.columbia.edu/~woit/wordpress/?p=6609

Don't know if MoMath will eventually upload the session to their YouTube channel, but if you wish to watch for it, or explore some of their other events, the channel is here:

http://www.youtube.com/museumofmathematics

This Is the Title of This Blogpost

I do love self-reference puzzles and here are a couple of doozies (I think I may have used one of these in the past?)  Knock yourself out:

http://faculty.uml.edu/jpropp/srat-Q.txt

http://www.mathpuzzle.com/20questions.htm

Mathematics Past Its Prime (so to speak)

This year's annual Edge  question was, "What scientific idea is ready for retirement?"

The (~175) replies are listed here:

http://www.edge.org/contributors/what-scientific-idea-is-ready-for-retirement

There are a great many wonderful responses, but I'll just cite a few which impinge directly on mathematics (but do take a gander at the entire thought-provoking list):

http://www.edge.org/response-detail/25414 (Charles Seife)

http://www.edge.org/response-detail/25401 (Nassim Taleb)

http://www.edge.org/response-detail/25349 (Emanuel Derman)

http://www.edge.org/response-detail/25345 (Carlo Rovelli)

http://www.edge.org/response-detail/25492  (Bart Kosko)

http://www.edge.org/response-detail/25344  (Max Tegmark)

Interestingly, a majority of these have issues with some aspect of that whipping-boy, statistics.

A Weekend Potpourri… (including Tegmark, Riemann, JMM14)

1) Not being an educator, I often ignore tweets appearing in my math feeds which seem geared strictly for teachers, but a recent one was getting RE-tweeted so often I finally caved and had to look at the link being passed along to see what it was. Indeed it was a delightful, simple idea most any primary school could employ… check it out if you've not seen it:

https://twitter.com/wterral/status/423755360512790528/photo/1

2) Below, a quick note on the packed talk covering recent twin-prime gap work at the Joint Math Meetings in Baltimore, MD.:

http://blogs.ams.org/jmm2014/2014/01/17/mind-the-gap/

3) I'm currently about half-way through Max Tegmark's new book, "Our Mathematical Universe."  Probably won't do a full review of it here since a) it's far more physics than mathematics (despite the title) and b) there are already many reviews of it around, so no need to add to the cacophony of publicity (good and bad) it's receiving.  Having said that, I will say I'm immensely enjoying it as a popular science volume -- in fact, it's one of the BEST cosmology reads I've yet come across (and I've read many)… which is not an endorsement of its views but simply of its readability… as cosmology books go, I'd call it a lively romp! For a more critical look though, see multiverse-phobic ;-) Peter Woit's review here, and be sure to peruse the comments:

http://www.math.columbia.edu/~woit/wordpress/?p=6551

A New Scientist review below (and Dr. Tegmark has been all over the Web promoting the volume as well, just google the book's title):

http://tinyurl.com/jwdfjbv

and more quick takes here:
http://space.mit.edu/home/tegmark/mathematical.html

4) Finally, a fantastic overview (in under 20 mins.) of the Riemann Hypothesis from that gregarious explainer James Grime (if anyone can make the Riemann Hypothesis fun, James can!):



That should fill your weekend for awhile.

Polygons, Circles and Limits, Oh My… an interesting errata

I was re-reading parts of the classic, and widely acclaimed, "Mathematics and the Imagination" by Edward Kasner and James Newman when I came across an old math chestnut that struck me as just a bit odd -- I'll explain why in a moment, and perhaps someone can further clarify the situation, but first, here's the chestnut, in case you're not familiar with it, regarding polygons, circles, and limits:

1)  Consider a circle of radius "1". Draw an equilateral triangle inside it. Then draw the largest possible circle inside that triangle, and then a square inside that circle, followed by another circle, followed by a pentagon… (keep going with circles and regular n-gons). As one approaches an infinite n-gon, what will be the limit of the radius of the consequent circle that fits inside?

2) Now, reverse the situation: start with a circle of radius 1. Circumscribe it with an equilateral triangle. Circumscribe that triangle with a new circle followed by a circumscribed square… then a circle… followed by a pentagon, a circle, etc. As you approach an infinite n-gon, what is the limit of the radius of the circumscribing circle?
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.answer below
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If you answer too quickly without thinking it through, the tendency for many folks is to respond that in case #1 the circle approaches a radius of zero, and in case #2 the radius approaches infinity. ...But that would be wrong (not even very close).

As the number of sides of the polygon increase, the size (perimeter) of the polygon (going in either direction, larger or smaller) fairly quickly stops increasing or decreasing appreciably, and a limit is rapidly reached. But this is where the errata enter.  Kasner and Newman have computed the limit of the radius in the first case to be approximately 1/12 (of the initial radius), and in the second case, the inverse, or about 12 (times the radius you begin with).

BUT, when I read those values somehow they didn't sound quite right to my memory, so I checked further, and sure enough all other citations of this problem I found listed approximately .115 (instead of Kasner/Newman's .083 or 1/12) as the so-called "polygon-inscribing constant" (limit) and 8.70, instead of 12, as the approximate "polygon-circumscribing constant" value -- quite a difference! See John Derbyshire's treatment of the problem for one example of the calculation involved (that all others seem to agree with):

http://www.johnderbyshire.com/Opinions/Diaries/Puzzles/2004-08.html

In examining further, I noticed Kasner/Newman used "n - 2" as the final (limiting) denominator of the multiplying sequence they employed, while everyone else simply uses "n". I presume this is the source of the difference/error, but WHY they used n - 2, I don't know (or if there has ever been any dispute over the correct approach?). The K/N book was originally published in 1940 and the edition I have is from 1989, so I'm wondering if this has ever been corrected in later editions???
At any rate, if anyone is familiar with any further interesting history to this problem or errata please let us know. It's occasionally comforting to see errors in mathematics that get passed on along the way, demonstrating that math is indeed, a very human endeavor! ;-)

Food For Thought, or Over-concern?

A long recent post from "Math Misery" blog on the forthcoming "Man Who Knew Infinity" movie being made on the life of Ramanujan:

http://tinyurl.com/kbolmpp

I assume most of us very much look forward to that film endeavor on an incredible mathematical legacy, but Manan Shah takes a different perspective and worries over how mathematicians might be depicted or stereotyped in the film. His long-read ends with this worrying summation:

"How will mathematicians come out at the end of this movie? More appreciated and admired? Or further pushed to the edges of society understood only as 'smart and odd and somehow necessary for society’s benefit, but their absence wouldn’t be missed because society doesn’t understand how much it relies on mathematics and mathematicians'?

"There are many ways to make this movie an absolute insult to mathematicians and there are many ways to really show one of the most amazing stories of a man whose contributions may never had been able to grace humanity had it not been for another man seeing past the biases of the time and reading the original letter filled with brilliant mathematics the way any person should — with an open and unassuming mind."

The Centennial Underway...

A wonderful talk about some fellow named "Martin Gardner" given by Colm Mulcahy for a skeptics group in December has been posted to YouTube. Hour-long, and worth it for Gardner fans:




(This year marks the centennial of Gardner's birth, with news and events throughout the year.)

Mathblogging.org ...plus Futility Closet

"From research to recreational, from teaching to technology, from visual to virtual, hundreds of blogs and sites regularly write about mathematics in all its facets. For the longest time, there was no good way for readers to find the authors they enjoy and for authors to be found. We want to change that."

That's from the description of the mathblogging.org site, which recently had its 3-year anniversary!

I suspect most math bloggers, but perhaps not all math readers, are aware of the aggregator site which has "collected over 700 [math-related] blogs and other news sources in one place," and aims "to be the best place to discover mathematical writing on the web."

Mathblogging.org was a sort of specialty outgrowth of the more general scienceblogging.org and ScienceSeeker.org sites which preceded it.

Additionally, they also have a Facebook page here:

https://www.facebook.com/Mathblogging

and, their main Twitter feed is: @MathbloggingAll

[I had emailed one of their editors a few questions for inclusion here, but haven't gotten those responses back :-( -- if they show up later, I'll insert them here.]

For any math-lovers not already familiar with them, Math Stack Exchange and MathOverflow (primarily for professionals) are other noteworthy math-related sites (involved, not in blogging, but in questions-and-answers).


On a sidenote: I'm so pleased to see Greg Ross, proprietor of "Futility Closet," break out on his own as he describes in a new (24-min.) interview with Boing Boing:

http://boingboing.net/2014/01/10/incredibly-interesting-authors-4.html

When I discovered Futility Closet many years ago I thought I'd stumbled upon a delicious, rare treat, few knew about... over the years I discovered more and more people who shared the exact same feeling, and Greg's site grew quickly by sheer word-of-mouth (or email). When I interviewed Greg for Math-Frolic over a year ago I asked him if there might ever be a Futility Closet book and now, YES there is!:

http://www.futilitycloset.com/book/

Sincere, well-deserved congratulations to Greg on his new endeavors, and much future success!

"the wonders of the world of algebra"

Sir Roger... and algebra:

Currently lacking time for longer posts, so just another quotation today, this time from the eminent Roger Penrose remarking on his introduction to algebra as a youngster (taken from his Foreward to Mircea Pitici's "The Best Writing on Mathematics 2013"):

"My earliest encounter with algebra came about also at an early age, when, having long been intrigued by the identity 2 + 2 = 2 X 2, I had hit upon 1.5 + 3 = 1.5 X 3. Wondering whether there might be other examples, and using some geometrical consideration concerning squares and rectangles, or something -- I had never done any algebra -- I hit upon some rather too-elaborate formula for what I had guessed might be a general expression for the solution to this problem. Upon my showing this to my older brother Oliver, he immediately showed me how my formula could be reduced to 1/a + 1/b = 1, and he explained to me how this formula indeed provided the general solution to a + b = a X b.  I was amazed by this power of simple algebra to transform and simplify  expressions, and this basic demonstration opened my eyes to the wonders of the world of algebra."

Quote... Unquote

Just a quotation today, from P.R. Halmos, from his 1968 essay "Mathematics as a Creative Art":

"…a brief and perhaps apocryphal story about David Hilbert… When he was preparing a public address, Hilbert was asked to include a reference to the conflict between pure and applied mathematics, in the hope that if anyone could take a step toward resolving it, he could. Obediently, he is said to have begun his address by saying 'I was asked to speak about the conflict between pure and applied mathematics. I am glad to do so, because it is, indeed, a lot of nonsense -- there should be no conflict -- there is no conflict -- in fact the two have nothing whatsoever to do with one another!' "

 ;-)

Devlin Retrospective

In his latest Devlin's Angle post, Keith Devlin looks back over 23 years of writing math columns (his "mathemaliterary journey" as he calls it) that divide into 3 main themes: 1) "What is multiplication?" ...the theme that created a "firestorm," 2) "Mathematical Thinking," and 3) MOOCs:

http://devlinsangle.blogspot.com/2014/01/23-and-me-play-it-again-sam.html

with enough links to fill your weekend reading time… (perhaps 2 weekends).

It Was A Very Good Year...

...in math blogging, anyway  (...sayonara 2013).

Wasn't going to do any sort of year-end blog retrospective this year, but having scanned several at other math blogs, am feeling a little more inspired now… at around 240 posts(!) last year, Math-Frolic though, would require too much effort to sort through for favorites. So instead I'll just highlight some favorites (from the mere 33 total) over at MathTango, in case you missed any of these...

I love doing the interviews and learning more about people who, before the internet, I'd never imagine crossing paths/words with. Some of my favorite interviews aren't with 'big' names, but rather with folks I know very little about, and thus most of what I hear from them is new to me. Thus, it won't be quite so surprising that my May 1 interview with Vickie Kearn of Princeton University Press is a favorite -- she doesn't blog, write math books, or teach… rather, as an editor at PUP she operates behind the math scenes shepherding the books we love to fruition. It was fun and fascinating to learn more about how that whole process works at likely the best publisher of popular math around.

I also have to admit (excu-u-u-use the self-promotion) that I had a lot of fun interviewing myself(!) back on Sept. 22.

All of that said, the interview which actually drew the most hits from the Web was, not altogether surprisingly, my second (June 21) interview with Keith Devlin, which involved NO math, but spelled out Keith's thoughts about the Ed Snowden/NSA controversy, which has been newsworthy ever since.

I've occasionally used MathTango as a place to write "essays" about various mathy matters. My favorite in that category came Sept 11 ("Mathematics… Not Immune") and was centered around the topic of "skepticism" -- it was in part a followup to an Aug. 24 post ("Numbers, Mnemonism, Savantism, Oh My") related to Daniel Tammet's last book.

Finally, I love finding popular math books that I can enthusiastically recommend to readers. Hands-down my favorite, heartfelt review ("Undiluted Martin Gardner") last Oct. 1, was of Martin Gardner's autobiography (and the week prior I wrote a shorter, broader-brush review of the same volume).

Two other mini-reviews I especially enjoyed writing were, "From Whence…?", a review of Jim Holt's "Why Does the World Exist?" (Apr. 21) and  "In Love… With Math," my Oct. 21 take on Ed Frenkel's "Love and Math," a widely-acclaimed book I relished, despite my inability to fathom large chunks of it.

Having highlighted those reviews, I will mention however that the book I MOST enjoyed reading this year, and which I wish had more momentum behind it, remains "The Outer Limits of Reason" by Noson Yanofsky… phenomenal volume for a wide range of folks!

If you missed any of those posts I do hope you'll go back and take a gander... or alternatively, any oldsters out there can just sit back and reminisce with this classic piece of nostalgia: