EVERY One of Them...

 For any who missed it, the following attention-getting tweet or email made the rounds recently:

"ALL CREDIT CARD PIN CODES IN THE WORLD LEAKED"
leading in turn to this webpage:

http://pastebin.com/2qbRKh3R

Sure enough, I found every one of my PINs there! ;-)

MathLove Redux

(Lovebirds via Peter Bekesi/WikimediaCommons)

Time to try a li'l experiment in audience participation (...which may totally flop)!

Not everybody loves math, but everybody loves love, so perhaps it's no surprise that the third most popular post I've ever done at this blog (in terms of  of 'hits' received) was a short one for Valentine's Day last Feb., entitled "MathLove." I'm perfectly willing to try to find more of what the audience likes.

That February post dealt with lovebird bloggers Jennifer Ouellette (English-major-turned-science-enthusiast) and Caltech physicist/husband Sean Carroll. If you have a similar story from your own life, somehow involving mathematics and love, that you don't mind sharing with the world send it along to me (SheckyR@gmail.com) for possible inclusion on the blog; or if you want to bypass me (and it's not too long) just go ahead and post it in the comments section below! [P.S. - Blogspot.com has a limit of 4096 characters for a comment].
Don't y'all be shy now...!

Devlin Online

If you're not already following along with Keith Devlin's "MOOCTalk" blog I recommmend you do so.  We're lucky to have someone like Keith, who is simultaneously a great math teacher, math communicator, education-devotee, and active blogger, chronicling his first-time experience with a 'Massively Open Online (math) Course." The course (via Stanford) is an introduction to mathematical thinking. (Outline HERE and text for course, here, from Amazon.)

The 'experiment' is just recently underway, and here is what one (of the 35,000 enrollees!) has to say from the early sessions:

http://mathfour.com/logic/is-math-a-language

Keith realizes well that there are pluses and minuses to online instruction, and watching his experience unfold in more-or-less real-time (and his insights therefrom), ought be fascinating to follow.

Primes On Parade...

The below visual simulator/producer of prime numbers (which also indicates the factors for any non-prime) has been getting passed around the Web in the last week or so:

http://numbersimulation.com/

It was initially posted on Reddit by its creator "QuantumTunneling" (contact address on the above page is greatgrahambini@gmail.com ):

http://www.reddit.com/r/math/comments/zl6j2/i_made_a_number_simulation_which_gives_a/

There are now enough 'visuals' for prime numbers that I didn't pay it much attention at first, but the more I've looked at it the more intriguing it becomes (though still not clear what, if any, deeper significance it has for the pattern or production of prime numbers).

If anyone out there knows more about it or can comment more astutely on it feel free to do so (I've seen various comments around the Web about it, but nothing terribly significant or substantive, or from an expert number theorist). Is it potentially anything more (or less) than a nice visual???

Lockhart's Way...

"My idea with this book is that we will design patterns. We'll make patterns of shape and motion, and then we will try to understand our patterns and measure them. And we will see beautiful things!
But I won't lie to you: this is going to be very hard work. Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don't know of any human activity as demanding of one's imagination, intuition, and ingenuity. But I do it anyway. I do it because I love it and I can't help it. Once you've been to the jungle, you can never really leave. It haunts your waking dreams." 

-- from the Introduction to "Measurement" by Paul Lockhart

How could anyone read the above words and not want to proceed to read the book that follows!?

I'm only 3/4 of the way through Paul Lockhart's new volume but will go ahead with an overview blurb on it, because it's obvious to me that I like this idiosyncratic offering and want to recommend it! The book is composed of just two main parts: Part 1 on "Size and Shape," and Part 2 on "Time and Space." Geometers especially will love Part 1 (a sort of geometry primer), though Part 2 (which I'm only partially into) is likely the richer, more fascinating, and slower, more arduous read (essentially an introduction to calculus).

Lockhart's writing style is conversational; refreshingly so, compared to the usual prescriptive tone of much math-writing. He stresses intuition over, or at least equal to, logic. One feels at times as if he is in the same room, ever-standing over your shoulder, talking to you, or, perhaps, like a kindly grandfather, holding your hand as he takes you on a stroll pointing out things along the way that he finds exciting. The book's tone very slightly reminds me of David Berlinski's "The Advent of the Algorithm," another writer with a unique and passionate style.

One of the things I particularly like about the book is that Lockhart is bluntly honest with his audience right from the start (as indicated in the quote above). So many popular books these days imply that they will make math fun and easy for you:  "Learn Calculus In Your Sleep" or "Quantum Mechanics in 3 Easy Steps" (ok, so I made those titles up, but you get the idea). But this offering doesn't pretend to be a "Math For Dummies" book. For most of us, math (at some level) is hard, and Lockhart acknowledges that; some folks who are very bright in other areas, have real mental blockages for mathematical thinking. Once again from Lockhart's intro:

"…expect it to be slow going. I have no desire to baby you or to protect you from the truth, and I'm not going to apologize for how hard it is. Let it take hours or even days for a new idea to sink in -- it may have originally taken centuries!
"I'm going to assume that you love beautiful things and are curious to learn about them. The only things you will need on this journey are common sense and simple human curiosity." 

Thanks grandpa ;-)
So I'm sure there are math-challenged or -phobic individuals out there who will simply find Lockhart's effort just as dry and indecipherable as any other math volume. But for those with an inclination toward the subject matter this volume will likely be a gem.

Unfortunately though, the title doesn't convey that gem-like quality. I suspect "Measurement" was thought to be an elegantly simple title, but I fear it will sound boring and even misleading to many prospective readers, for whom the word conjures up tedious, rote procedures. This book is full of 'elucidation' or 'illumination' (through math), and even play. It is no casual or beach read of course, but nor is it a cold textbook or instruction manual either; and certainly it's a valuable book for teachers to have on hand (so many wonderful examples/ideas herein). The title is not inappropriate, but it may not be the attention-getter the book deserves.

One of my favorite quotes from the volume comes when Lockhart is explaining the calculation of the area of a circle by utilizing an infinitely-sided inscribed polygon:

"In other words, an infinite sequence of lies with a pattern can tell you the truth. It is arguable that this is the single greatest idea the human race has ever had." [bold added]

Gotta love that enthusiasm, and it beams forth from the book repeatedly.

I do have one major quibble with the volume though:

Lockhart regularly tosses out various thought questions or problems for the reader to figure out on their own to more fully fill out the ideas being discussed. Some are easier than others, but they are good and instructive, and it is ashame (even annoying and unsatisfying) that he nowhere offers the answers to these lobbed exercises, so readers can check themselves or see the intended answer if need be. Given Lockhart's joy at elucidating logical steps for the reader I can't imagine why he drops the ball on these interspersed problems, and leaves the reader potentially hanging somewhat -- a simple appendix or addendum at the end could've covered many of them.

Toward the conclusion Lockhart notes, interestingly, that there has been precious little in the book about "reality;" rather, he is discussing mathematics in the abstract, as part of an imaginary world, or a world inside our own heads, while 'reality' is little more than the 'possibly illusory sensory input' from the world around us.
He also writes at one point, "…we love patterns. Mathematics is a meeting place for language, patterns, curiosity, and joy. And it has given me a lifetime of free entertainment."
To which I say, 'Thanks for sharing, Paul!!'

Despite the simple title, this is not a simple or altogether easy book… but it is an easy one to give a thumbs-up to for NON-math-phobes! 

Lockhart's own YouTube promo for the book here:

Slinky Night Fever?

For some reason, just felt compelled to share this (there's got to be some math in here! ...Cycloids gone wild?):



(...awww, go ahead, get up and dance ;-)

p.s.: for your further entertainment and perusal the 90th blog 'Carnival of Mathematics' is now up HERE, and includes this interesting real-life application of geometric series.

The Six Chair Puzzle

Since I've just mentioned Martin Gardner (previous post) might as well put up one of his old, simple 'Gotcha' puzzles again (I've re-written it):

6 students make a reservation at their favorite pizza palace, but just as they are about to leave a 7th student decides to join them. When they all arrive, the hostess realizes there are 7 students, yet she has only set up a table with 6 seats. Still, no problem she figures cleverly… She puts the first student in chair #1 and has his girlfriend sit on his lap for the time being, then she seats the third student in chair #2, the fourth student in chair #3, the fifth student in chair #4, and the sixth student in chair #5. Waaa-laaahh! Chair #6 still remains empty, and she asks the first student's girlfriend to now abandon his lap and take the last free seat. EVERYbody's ready for pizza!!… or, NOT!

Turning Mathematicians Into Children ;-)

The 3rd annual "Gathering For Gardner" Celebration of Mind, or G4G for short (in honor of recreational mathematician Martin Gardner) is a little over a month away and beginning to build steam. Local gatherings will be held in multiple cities around the globe.
If you're not familiar with this October celebration read more about it here:

http://www.g4g-com.org/

(…and if you're not familiar with Martin Gardner, one of the very best math communicators and essayists ever, well, then, be quicketh to your library or Kindle!)

If you're on Twitter the twitter-handle for the gathering is @G4G_CoM

And here is an intro to it from the ever-creative Scott Kim:



...Wherever you are on October 21, let your inner (mathematical) child soar! ;-)

Huge News... That Few Can Fathom

Shinichi Mochizuki's claim of proving the 'abc conjecture' is receiving widespread notice in the scientific press, although it may take years to confirm the highly-complex, 500-page-long finding, which according to some is currently only comprehensible to Mochizuki himself (who invented new math in the process of reaching his results)!
The conjecture is viewed as being central to number theory and the underlying relationships of prime numbers; described by some "as a sort of grand unified theory of whole numbers."
For those who wish to follow along a bit, a few more of the many Web entries that have been reporting on the claim:

http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture

http://www.scientificamerican.com/article.cfm?id=proof-claimed-for-deep-connection-between-prime-numbers

http://bit-player.org/2012/the-abc-game

I suspect Amir Aczel ;-) or somebody may already be at work writing a book for the rest of us to understand!

Strogatz on Singularities

It's a good day when you wake up to find Steven Strogatz writing in the NY Times about mathematics. Today was a good day:

http://opinionator.blogs.nytimes.com/2012/09/10/singular-sensations/

(the 1st of a new 6-part series from Strogatz)

Theorem-lovers: Take Your Pick

(via Tosha at Wikimedia Commons)

                   

Richard Elwes (@RichardElwes) recently tweeted a link to his latest article on "Pick's theorem," a beautifully simple, comprehensible, almost quirky piece of geometry, dealing with the area of a polygon in a lattice grid:

http://education.lms.ac.uk/2012/09/in-praise-of-pick%E2%80%99s-theorem/

In a sentiment that is echoed by many others, Cliff Pickover says in his "The Math Book," "Pick's theorem is delightful for its simplicity, and it can be experimented with using a pencil and graph paper."

Another treatment of the theorem here:

http://simomaths.wordpress.com/2011/12/08/picks-theorem-and-some-interesting-applications/

Interestingly, Pick's theorem works in the geometric plane, but NOT in 3-dimensional space.

Austrian mathematician Georg Pick originally proposed the theorem in 1899, but it wasn't popularized until 1969, long after Pick himself had died in a Nazi concentration camp in 1942.

Go Little Worm Go

Ahhh, the 'harmonic' workings of large numbers... here's a recent paradoxical braintwister from "Futility Closet":

http://www.futilitycloset.com/2012/09/01/high-hopes/

(The problem can be found at Wikipedia for those who want a fuller explanation of the counter-intuitive result: http://en.wikipedia.org/wiki/Ant_on_a_rubber_rope ...the key is to understand the full "elastic" nature of the band).

The 'Futility' problem reminds me a tad of a more complicated Raymond Smullyan puzzler that I love and have posted here before:

http://math-frolic.blogspot.com/2011/01/seemingly-impossible-task-that-isnt.html

--> Addendum note: in a nice little bit of coincidence, I just discovered that Jason Rosenhouse posted a short while back that he will be editing a tribute book to Raymond Smullyan due out sometime next year from Dover Publications. Great news!

The Joy of Strogatz

From Clifford Pickover, to Steven Strogatz, another great math popularizer. The below wonderful article reports on Strogatz (who gained fame and fans with a 15-part series of very popular NY Times columns a couple years back), and his new book "The Joy of X" (which is in part a compendium of those Times' columns):

http://tinyurl.com/c4w4t53

The second part of this 'Cornell Alumni Magazine' piece is an excerpt on arithmetic, from Strogatz's new book:

http://tinyurl.com/ctp65dv

Strogatz views math as "an intoxicating way to look at the world," and the article bears further good news:

"This fall, he'll [Strogatz] write more online essays for the Times, this time an eight-week series using multimedia to explore mathematical concepts."

Indubitably, something to look forward to….

Pickoverisms...

In his Twitter feed, Clifford Pickover (@Pickover) recently linked to this very good, and visual, presentation of Benford's Law:

tinyurl.com/d445tjw

...and here are a few other miscellaneous Pickover tweets I've jotted down over the last few months:

23 is the only prime number p such that p! is p digits long.

Aside from 17, humans will never find a prime that is the average of two consecutive Fibonacci numbers.

 19 is the only known prime of the form n^n - 8. The next prime of this form (if it exists) must be more than 34,000 digits.

Prince Rupert's cube and 1.06066 at Wikipedia: tinyurl.com/8vjep4k

Answer this question with "yes" or "no". Will your next word be "no"?

Will Another Conjecture Bite-the-dust?

Peter Woit reports the claim that the "abc conjecture" has been proven:

http://www.math.columbia.edu/~woit/wordpress/?p=5104

The abc conjecture is a number theory conjecture from 1985 which has been called "the most important unsolved problem in Diophantine analysis."

Peter links to this more detailed report and discussion of the claim:

http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/

and here's a more layman-friendly interpretation of it from a few years back:

http://bit-player.org/2007/easy-as-abc

The Inimitable Dr. Wildberger ?

Well, this is sort of interesting (to me at least)…

Quite awhile back I came across a link to Aussie Dr. Norman J. Wildberger's math videos on YouTube and looked briefly at a couple of them. They seemed interesting, and with better production values than most math videos, especially Khan Academy-type clips, so I went ahead and put a link to them in my right-hand column thinking at some point I'd get back for a closer look.
I even went so far as to list them among 10 favorite math sites in a post I did on such, even though I hadn't explored them very fully.
Well, lo-and-behold the other night I was reading some commentary on a forum and saw Dr. Wildberger's name briefly arise more-or-less in the context of crackpot math, so figured I ought do some more checking.

In googling around the Web I found a fascinating array of opinions on Dr. Wildberger (who IS a fully-credentialed, professional mathematician and university professor). He was written off as a crackpot by several, and as simply misguided by many more, but also praised by a few (even if they don't completely agree with him); and some even calling his work "important," "exciting," "groundbreaking," and the like. It's not often in math that one sees such a wide divergence of opinions on a body of work.

Anyway, someone who doesn't mince words much about "crackpots" is Mark Chu-Carroll of "Good Math, Bad Math" blog, so I clicked over there and searched his site for "Wildberger" to see what I might find… bingo! sure enough Mark had done a post on him back in 2007:

http://scientopia.org/blogs/goodmath/2007/10/15/dirty-rotten-infinite-sets-and-the-foundations-of-math/#more-529

It is a very long post followed by a very long list of interesting and heavy comments (it concentrates a lot on set theory, and contains more than one can digest in a single sitting! -- I wouldn't even attempt a summary), but what is fascinating is to see Mark simultaneously roundly denouncing much of Wildberger's approach, but also, noting a few good, interesting, or agreeable points here and there; indeed he admits that Wildberger is worth a look (somewhat backhandedly Mark writes that Wildberger "...isn't the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery" ;-).

Wildberger strongly believes that the foundations of math and math teaching are both problematic. I think I was initially attracted to him in part because he claims to be re-building mathematics on a more "intuitive" (easier-to-understand) basis. Through high school I thought algebra, geometry, and even early calculus were somewhat intuitive, but that trigonometry was NOT intuitive at all, so someone who claims to make trig intuitive (as Wildberger claims) catches my attention.
Anyway, I'm in no position to judge Dr. Wildberger's novel work (and still hope to find time to view more of his videos), but simply use him as an example that even in a field as cut-and-dry as math is perceived to be, there is room for a variety of opinion (although the negative opinion seems to far outweigh the positive in this case).

If you wish to explore more...:

Professor Wildberger's personal website is here:

http://web.maths.unsw.edu.au/~norman/

His YouTube videos here:

http://www.youtube.com/user/njwildberger/videos

A Wikipedia page entry for his notion of "Rational Trigonometry":

http://en.wikipedia.org/wiki/Rational_trigonometry

and his book on same, "Divine Proportions,"  is here:

http://www.amazon.com/Divine-Proportions-Rational-Trigonometry-Universal/dp/097574920X

Grasping Statistics

Noting that "A practical understanding of probability and statistics at an advanced, at least college, level is increasingly important in the modern world," John McGowan offers a great little primer on such matters here:

http://math-blog.com/2012/09/02/how-to-hang-yourself-with-statistics/

He briefly brings drug-testing, global-warming, derivative-based securities, and autism into the discussion, and notes that "By far the greatest and most common problem with using probability and statistics in the real world lies in the definition of terms, categories, and measured values," finally closing out thusly:

"In public policy debates, scientific controversies, and other real-world applications of probability and statistics issues about how the data were collected, how the terms and values are measured and defined, and what the categories used actually mean often take center stage and are the subject both of bitter controversy and simple confusion. It often requires extensive research to resolve these issues; often they are not resolved, certainly to the satisfaction of all.

Conclusion

"A good understanding of probability and statistics is increasingly necessary in the modern world. There are many ways to misuse probability and statistics, both intentionally and by accident. One should almost never take a statistic at face value, especially when powerful vested interests are at stake. The best course of action is to examine the data and the analysis of the data carefully. Unfortunately, this is often time consuming, but there is no substitute for important issues."

 ...well, there is a substitute... it's called ignorance or gullibility!

Starts With a Triangle

Ethan Siegel of "Starts With a Bang" blog regularly writes excellently on physics topics, but recently did a fine post pertaining more specifically to a geometry puzzle that often winds its way around the Web (...counting the number of triangles within a triangle):

http://scienceblogs.com/startswithabang/2012/07/28/weekend-diversion-triangles-a-puzzle-and-beauty/

a YouTube explanation for the same puzzle is available here:

http://www.youtube.com/watch?v=FxByF38grkQ&feature=player_embedded

Though he doesn't discuss it, Ethan's post actually starts off with a picture of the Sierpinski pyramid, a fractal-based object (which approaches a surface area and volume of 0) that is likely even more fascinating than the puzzle he is elucidating.

Boltzmann, Cantor, Gödel, Turing

From the BBC "Dangerous Knowledge" series:






If you've not seen these wonderful videos on 4 greats, try to set aside enough time at some point to enjoy them (each ~45 mins.).

What's The Number?

I've been on a self-referential kick lately, and though I've used this puzzle before, just stumbled upon another nice presentation of it, so will cite it again:

http://tinyurl.com/c84l9ns

I'll put the answer in comments section in a few days (there's only one correct answer, in base 10).