Just For Fun

Visual proof that the area of a circle equals π r² :

Good Summary of P vs. NP

Very nice summary and update on P vs. NP here:

http://cacm.acm.org/magazines/2009/9/38904-the-status-of-the-p-versus-np-problem/fulltext

TedEd...

Do you enjoy TedTalks?… Well, maybe your pre-teen will find them inspiring as well.  "Math Insider" thinks he's found 9 TedTalks that may spur your young person's interest/appreciation in mathematics:

http://www.mathsinsider.com/ted-ed/

Couple a Books

I haven't read these yet, but already believe they're worth noting...:

Mathematician Gregory Chaitin is out with a new book (barely over 100 pgs., and looks good): "Proving Darwin: Making Biology Mathematical" -- a book that treats the DNA code as a universal programming language, while looking at the mathematical underpinnings of biology and evolution; definitely, adding it to my reading queue.

a blurb about it here:

http://physicsdatabase.com/2012/05/03/proving-darwin-making-biology-mathematical/
(includes a Chaitin lecture video)



Secondly, not exactly a math book… but from one of the best math explicators out there, and with a great title and story, worth mentioning here, the last book from Steven Strogatz: "The Calculus of Friendship."

It's the touching story of Strogatz' long-time relationship with one of his math teacher-mentors, and I can't help but think many a mathematician will relate to some aspects of that relationship. A couple of Web reviews of the book here:

http://www.americanscientist.org/bookshelf/pub/a-growing-affinity

http://www.bookslut.com/nonfiction/2009_09_015101.php

And a couple of clips of Strogatz talking about the volume here:

http://www.youtube.com/watch?v=9piYoYqIf3I

http://www.radiolab.org/2009/nov/30/calculove/ 

And Strogatz has a new book scheduled for release this coming fall:
 
"The Joy of x: A Guided Tour of Math From One to Infinity"

...gots to be good!

Gödel Incompleteness 101

(Kurt Gödel via Wikimedia Commons)

From a prior RadioLab show, with Steven Strogatz, this podcast segment recounts Gödel 'Incompleteness' for us layfolk (starts with a minute+ of advertising):

Friday Puzzler

Absent-minded Ziggy has forgotten the combination to his wall-safe that contains the family jewels… he remembers the five different numbers involved, but can't quite recall the order they go in. He tries 3 different combinations, none of which work:

a) 7-2-1-8-3
b) 2-7-3-1-8
c) 1-3-8-2-7

IF it is given that in the first and second guesses (a and b) only one number is in the correct order (in each attempt), and in the third guess three numbers are in the correct places, then what must be the proper combination to open the safe?
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ANSWER:

1-7-8-2-3

String Theory... of sorts

A BBC Horizon episode apparently investigated the length of a common ordinary piece of string:

http://www.ams.org/news/math-in-the-media/mathdigest-index#201205-string




(It also relates back to discussion of infinite fractal coastlines.)

Probably Probable...

Mr. Honner uses Google searches (and their autocomplete function) as an everyday
example of applied 'conditional probability':

http://mrhonner.com/2012/05/17/google-and-conditional-probability/

Revenge of the Stat Majors...

Remember the old days when math was the nerdiest of subjects, and statistics was the most boring area of that nerdy field... no more! Article from NY Times details that statistics is now a place to be IF you wish to be in demand!:

http://bits.blogs.nytimes.com/2012/01/26/what-are-the-odds-that-stats-would-get-this-popular/

(...may require free registration to read)

The article notes that statistics is "a service field to other disciplines," much needed in today's world of massive digital data, and computers capable of processing it.

From the piece:

"Arcane statistical analysis, the business of making sense of our growing data mountains, has become high tech’s hottest calling. There are billions of bytes generated daily, not just from the Internet but also from sciences like genetics and astronomy. Companies like Google and Facebook, as well as product marketers, risk analysts, spies, natural philosophers and gamblers are all scouring the info, desperate to find a new angle on what makes us and the world tick. Computing has become cheap and available enough to process any number of formulas.
What no one has are enough people to figure out the valuable patterns that lie inside the data."

Number Line... Real or Invented

(via Wikimedia Commons)

Study concludes that the number line is not, as usually presumed, hard-wired within human intuition, but rather culturally learned and reinforced:

http://www.sciencedaily.com/releases/2012/04/120425192742.htm

full, original PLoS article here:

http://tinyurl.com/7dcucue
 
(interestingly, this all relates back to the sort of material covered in the David Berlinski book I reviewed last week)

Some quotes from the Science Daily piece:

"Influential scholars have advanced the thesis that many of the building blocks of mathematics are 'hard-wired' in the human mind through millions of years of evolution. And a number of different sources of evidence do suggest that humans naturally associate numbers with space"…
"Our study shows, for the first time, that the number-line concept is not a 'universal intuition' but a particular cultural tool that requires training and education to master. Also, we document that precise number concepts can exist independently of linear or other metric-driven spatial representations."

"Mathematics all over the world -- from Europe to Asia to the Americas -- is largely taught dogmatically, as objective fact, black and white, right/wrong, but our work shows that there are meaningful human ideas in math, ingenious solutions and designs that have been mediated by writing and notational devices, like the number line… Mathematics is neither hardwired, nor 'out there.'"

"These findings suggest that how we think about abstract concepts is even more flexible than previously thought and is profoundly affected by language, culture and environment."
"Our familiar notions on 'fundamental' concepts such as time and number are so deeply ingrained that they feel natural to us, as though they couldn't be any other way."

Mathigon.org

Another new educational math site has entered the arena; pretty skeletal at this point but probably worth keeping an eye on:

http://www.mathigon.org/

Friday Puzzle

Just a bit of wordplay today (I've slightly re-worded this, to make it a little harder, from a Twitter offering appearing earlier in week.):

Name a country that is an anagram of a synonym of a homophone of a prime number?
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ANSWER:
"Laos" from "also" (synonym of "too," homophone of prime no. "two")

Brief Berlinski Book Blurb….

As simple as one, two, three....

Of the math books I've been perusing/reading lately, the only one grabbing and holding my attention much (and oddly a volume I wasn't expecting to like), is David Berlinski's 2011 volume, "One, Two, Three." Indeed, I've enjoyed the book quite a bit, but must throw in my frequent precaution that it won't suit everyone's taste… or, to put it more bluntly, I suspect MOST readers (even math readers) may NOT relish it.
Leonard Mlodinow, in reviewing the volume, called it, "candy for the intellectually curious"… but I'd feel more comfortable deeming it 'candy for the epistemologically curious' -- some 'intellectuals' (and mathematicians for that matter) don't have the patience for philosophy that this volume requires. The book deals with matters at such an elementary level, where ideas tend to be ingrained and taken-for-granted, that many readers simply may fail to appreciate the significance of what is expounded.

Berlinski calls his subject matter "absolutely elementary mathematics" (which is also the sub-title of the book) or AEM for short -- it's not clear to me if that is his own concocted term or comes from somewhere else (and one Web commenter mistakenly bought the volume thinking it would be a nice introduction to math for her youngster... this it definitely is NOT!). At any rate, the subject matter includes the very most rudimentary foundations of math, logic, arithmetic. You either like this sort of thing… or, you don't. The author romps through a playground of ideas regarding numbers, algebra, axioms, sets, integers, fractions, rings, polynomials, proof, and clearly enjoys himself while doing so (even if it isn't always such a merry-go-round for the reader!).  Parts of it will read quite pedantically to many readers, but this kind of elementary and obvious-sounding subject matter can't help but be pedantic at times… and it is FAR less so, in Berlinski's deft hands/voice, than it would be within some other textbook relating the same notions.

Berlinski's style is terse and economic, and yet with such careful, and even interesting, word choice that I'd be tempted to call it breezy; at times surprisingly engaging; even witty and humorous (though still with interspersed and inevitable dryness at points). His rhetoric can often be fun, even when it is challenging.
There are also interesting brief sidebars about various historical mathematicians throughout the volume, though it wasn't always clear to me just why some of these were included… yet they did make a nice interlude between some of the more tedious sections. The penetrating focus though is always on pulling back the curtain on the fundamental logical and axiomatic thought processes/assumptions that underlie mathematics. I do recommend the book, but not to the mathematically-nervous nor philosophically-naive, nor to readers who read for pleasure and not to be challenged. Whereas a Keith Devlin or Ian Stewart book might connect with a large audience of the mathematically curious, this volume may only resonate with a narrow slice of the popular-math crowd.

I'll admit that I find it hard to objectively read Berlinski's math writings because I find some of his other philosophical/science/political writings rather distasteful (…and yet having said that, there are elements of his "radical skepticism," as some call it, that I'm in sympathy with). He remains a highly controversial figure. Worth noting though, I did also enjoy one of his earlier math works, "Infinite Ascent," a nice historical look at major ideas in mathematics.

The Wikipedia entry on Berlinski is here:

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

and here are a few more Web entries that lend some sense of the controversy surrounding the man:

http://tinyurl.com/7sdcuj9

http://www.observer.com/1998/06/is-the-big-bang-just-a-big-hoax-david-berlinski-challenges-everyone/

http://recursed.blogspot.com/2008/04/david-berlinski-king-of-poseurs.html

(…of course googling his name will lead you to many more references on him, including clips on YouTube)

Leaf Vascularization… and Math

via Wikimedia Commons

Scientists study leaf patterns to better elucidate the relationship between geometric structure and biological function in nature:

http://tinyurl.com/cung4je

from the article:

" 'This research is a unique interdisciplinary partnership in which physics is used to address biological problems, and it is our belief that the mathematical and physical sciences will play a major role in biomedical research in this century,' says Krastan Blagoev, director for the Physics of Living Systems program in NSF's Mathematical and Physical Sciences Directorate, which funded the research.

"Magnasco says this research is a jumping off point for understanding other systems that branch and rejoin, including everything from river systems to neural networks and even malignant tumors."

Deja Vu Gödel

I've linked to this before… and will again… 'cuz it's one of my favorite citations. A phenomenally short, succinct, inspired, mind-twisting, 1994 monosyllabic explanation of Gödel's Second Incompleteness Theorem from George Boolos. Gotta luv it!:

http://tinyurl.com/cgexvsy

or, available here as pdf:

http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

GoGetPapers.com

Nice resource portal to "lectures," "tutorials," and "essays" on all manner of mathematics here:

http://www.gogetpapers.com/Faculty/Mathematics

h/t to @MathematicsProf

(I've added a permanent link to the site in right-hand column under instructional sites)

Friday Puzzle

another old classic for a pre-weekend puzzle:

Which fits better or tighter (meaning it leaves less unused space, percentage-wise): a tight-fitting square peg in a round hole or a tight-fitting round peg in a square hole?

for the solution see here:

http://plus.maths.org/content/peg-and-hole-puzzle-solution

...Of Random Samples

Several years ago a journal paper was published entitled "The Weirdest People In the World" detailing how UNrepresentative the sample groups often employed in statistical (particularly psychological and behavioral) studies were. If I had to boil it all down (and I agree with the impetus of the paper) it simply says that good research studies, with generalizable results, ought employ random samples… and, guess what… there's no such thing as a truly random sample…

Anyway, if you enjoyed William Briggs' thoughts in a prior recent post, here's his take on this topic:

 http://wmbriggs.com/blog/?p=5566

A couple of quick excerpts:

"Over-confidence and over-certainty abounds in many areas, the closer the subject is to mankind the greater the false surety. This is why this new paper is such a delight: it is a rare admission that all might not be as solid as hoped."

"the authors list some recommendations, such as that “Journal editors and reviewers should press authors to both explicitly discuss and defend the generalizability of their findings. Claims and confidence regarding generalizability must scale with the strength of the empirical defense.” They also say to use people from more areas of the world."

More Education Snippets

While searching for something else recently, I stumbled across (isn't the internet great!) yet another brief essay regarding math education that piqued my interest:

http://thealternativepress.com/articles/learning-off-the-clock-in-search-of-a-timeless-u

The ultimate focus seems to be on Mathnasium Learning Centers, and so while I'm linking to the piece, it is not intended as an endorsement of Mathnasium, which I have no connection to or experience with (except having driven by their facilities on occasion). I did quickly scan a number of reviews of them on the Web which mostly seemed quite positive (I suspect some of my readers are much more acquainted with them than I am though).

The piece starts off thusly:

"It is tempting to compare our brains to computers, but when it comes to speed, a neuron takes its time, sending signals at a maximum of 180 miles (280 km) per hour, the top speed of a Formula One racecar, or a quarter of the speed of sound. In comparison, an electrical circuit approaches the speed of light, which is 671 million miles (1.08 billion km) per hour. Clearly, speed is not our strong suit when we are pitted against a circuit board. Our advantage is understanding – not mindlessly crunching ones and zeros. Yet for some reason, math educators continue to emphasize speed at the expense of humanity's greater gifts of interpretation and problem solving."

On a sidenote, Keith Devlin anticipates his upcoming (fall) online math course, and how others may help with it, here:

http://mooctalk.org/2012/05/08/help-wanted/

As someone who used to limit his classes to 20 students, Devlin realizes how daunting it will be to potentially reach 10's of 1000's. He calls for the involvement of other mathematicians to assist him in the effort:

"I’m going to make my course just five weeks long, starting in early October. By incorporating participation in my Stanford course as part of your students’ learning experience, everyone could benefit. For one thing, your students are likely to be inspired by being part of an educational revolution that for millions of less privileged people around the globe can quite literally be life changing.
"Because they will be supported by being part of a physical learning community, with the personal support of you, their instructor, your students will be highly empowered, privileged members of that online community. They can take advantage of your support so that they can help others. And as we all know, there is no more powerful way to learn than to try to teach others...

"...if instructors and their students across the US join me, then maybe we can collectively achieve something remarkable...

"Those of us in education know how it can change lives... Please join me this fall as we learn how to teach the world."

It is exciting to see a 'luminary' of sorts with the recognition and following of Keith Devlin so enthusiastic to experiment with this brave new world of digital math education to the masses. I'm sure the initial glitches and weaknesses of such attempts will be many... but also sure that, with time and effort, they're surmountable.

Tweet Tweet

just a few recent Twitter tweets...:

One from @BenVitale:

"There are only 31 numbers that cannot be expressed as the sum of distinct squares. And 128 is the largest one. Amazing!"

from @pickover this:

"Cool. Only one Pythagorean triangle exists whose area is expressed in repeating-digits: (693, 1924, 2045) with area 666666."

also @pickover offers this link to an old math joke:

http://www.lesjones.com/2011/09/16/math-joke/

from @DivByZero a quote:

“If ‘publish or perish’ were really true, Leonhard Euler would still be alive.”—Eric Bach

...and lastly, from @ProfKeithDevlin a note pertinent to TODAY:

"Tomorrow (Tuesday) at 5:00PM PST, Steve Hargadon is interviewing me live (w. audience Q&A) on "The Future of Education" http://www.futureofeducation.com/ "

Statistics, Schmatistics...

William Briggs on statistics and p-values:

http://wmbriggs.com/blog/?p=5583

From the piece:

"The WSJ suggests that statistics can prove opposite results simultaneously when models are used on observational studies. This is so. But it is also true that statistics can prove a hypothesis true and false with a “randomized” controlled trial, the kind of experiment we repeatedly hear is the “gold standard” of science. Randomization is a red herring: what really counts is control (see this, this, and this)."

and it ends thusly:

 " It’s too costly and time consuming to do statistics the right way. Just look how long it takes and how it expensive it is to run any physics experiment (about genuinely unknown areas)! If all of science did their work as physicists must do theirs, then we would see about a 99 percent drop in papers published. Sociology would slow to a crawl. Tenure decisions would be held in semi-permanent abeyance. Grants would taper to a trickle. Assistant Deans, whose livelihoods depend on overhead, would have their jobs at risk. It would be pandemonium. Brrr. The whole thing is too painful to consider."

 Now if only Mr. Briggs would tell us how he really feels... ;-)

Proof... in Mathematics

A layman-friendly essay on mathematical proof… and... tennis:

http://www.livemint.com/2012/04/12212107/Something-about-proof.html

Friday Puzzle

I've taken this Friday puzzle directly from "Futility Closet" earlier in the week. It assumes familiarity with the game "rock, paper, scissors":

"Andy and Bob play rock-paper-scissors 10 times. Andy plays 3 rocks, 6 scissors, and 1 paper (in some order), and Bob plays 2 rocks, 4 scissors, and 4 paper. There are no ties. Who wins?"

...This reminds me that you can play computerized 'rock-paper-scissors' online at this NY Times site (...but please don't do it while driving ;-):

http://www.nytimes.com/interactive/science/rock-paper-scissors.html

Puzzle Answer below:
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Andy wins (by quite a bit); if you need an explanation you can visit the linked site.

The Online Onslaught

via Wikimedia Commons

In an earlier post I off-handedly mentioned my belief that sometime in the future students would no longer matriculate at individual universities, but rather pick-and-choose coursework from an array of nationwide (indeed worldwide) digital offerings from the best teachers possible. I don't know how the exact details of grades, credits, degrees etc. will work out, but I'm sure it's do-able.
Anyway, my idea was for the distant future… but funny how rapidly the future is hurtling towards us. The following NY Times piece reports on a new collaboration between Harvard and M.I.T. to offer free non-credit courses over the Web, and of course a good many other colleges are doing the same as well (and other colleges already offer for-credit/degree courses). Last month the Times reported on another outstanding consortium of schools offering a similar online venture called "Coursera" (the Harvard/MIT project, btw, is called "edX"). I think it is clear the direction this is all going. The only question is how fast?...

http://tinyurl.com/87eoujw

There are no doubt benefits to be derived from the social and other interactions that are aspects of participation on an actual physical campus... but for generations being raised on Facebook, Twitter, and similar digital social platforms, even those benefits are probably fading into obsolescence. As that great teacher from the 60's told us ;-), the times they are a changin'!

Lastly, Keith Devlin also touches upon this whole subject at some length (and with his usual thoughtfulness) in his latest "Devlin's Angle" column here:

http://devlinsangle.blogspot.com/2012/05/math-mooc-coming-this-fall.html

The first lines from the piece:

"Higher education as we know it just ended. Exactly what will take its place is not at all clear. All that can be said with certainty is that within a few short years the higher education landscape will look very different."

 (Devlin will be doing his own 5-week online course from Stanford, starting in October.)

ADDENDUM: Devlin has now started a separate blog just to focus on this subject:

http://mooctalk.org/

Polymath Redux

Speaking of crowd-sourcing (as yesterday's post did)... I've reported on Tim Gowers' collaborative Polymath Project in the past, but it's been awhile. Here though, a critical review/update on the Polymath concept from a "bystander":

http://boolesrings.org/krautzberger/2012/04/30/waiting-for-the-polymath-revolution-thoughts-from-a-bystander/

I can't help but think the author is overly critical in seeming to view the Polymath effort as not revolutionary enough, and not including enough of the mathematics community. I view it (like I view Khan Academy) as being young and still a work-in-progress. I'm not certain it's even meant to be "revolutionary," so much as simply productive. And the author seems to  want to spread the wealth of that productivity to the broader math community; I just don't know how practical that will be given the Polymath focus on highly complex problems. The author actually summarizes well though the basic potential of the Project:

"Mathematics has the greatest potential for 'doing research online'. There’s no physical entity needed and our primary standard of scientific communication is the written word. There’s nothing in mathematical research that cannot be digitalized. We will never face the problem that somebody on the other side on the net would have to actually look at our specimen, our antibody staining, our test subjects. The web works perfectly for us.
Hence, mathematicians could be at the forefront of experimenting with new research activities that use the connectivity the web can offer in new and imaginative ways."

In the end though the author pleads, "...please, for a change, let’s not ask Tim Gowers to do everything for us!"

Anyway, an interesting discussion...

Crowdsourcing to Solve the Riemann Hypothesis

h/t to C.Pickover for this piece about an Indian mathematician's "ZetaTrek Project," looking at "one-dimensional quasi-crystals" to potentially crowd-source a solution to the Riemann Hypothesis:

http://blog.zombal.com/post/a-quest-to-solve-one-of-maths-great-puzzles

also here:

http://fadereu.posterous.com/knk103-the-crystals-of-mt-zeta

(I have no ability to judge how productive this approach may... or may not be???)