The Brain Is a Wondrous Mystery

via Wikimedia

Yet another fascinating story of a math prodigy of sorts… a young man whose math prowess only came about following a severe bodily attack:

http://tinyurl.com/866msww

From the article:

"Jason Padgett, 41, sees complex mathematical formulas everywhere he looks and turns them into stunning, intricate diagrams he can draw by hand. He’s the only person in the world known to have this incredible skill, which he obtained by sheer accident just a decade ago.
“I’m obsessed with numbers, geometry specifically,” Padgett said. “I literally dream about it. There’s not a moment that I can’t see it, and it just doesn’t turn off.”

"Padgett doesn’t have a PhD, a college degree or even a background in math. His talent was born out of a true medical mystery that scientists around the world are still trying to unravel.
"...One night he was walking out of a karaoke club in Tacoma when he was brutally attacked by muggers who beat and kicked him in the head repeatedly….

“All I saw was a bright flash of light and the next thing I knew I was on my knees on the ground and I thought, ‘I’m gonna get killed,’” he said.

"At the time, doctors said he had a concussion, but within a day or two, Padgett began to notice something remarkable. This college dropout who couldn’t draw became obsessed with drawing intricate diagrams, but didn’t know what they were.

“I see bits and pieces of the Pythagorean theorem everywhere,” he said. “Every single little curve, every single spiral, every tree is part of that equation.”

"The diagrams he draws are called fractals…."

Financial Trading 101... NOT!

"The year after Myron Scholes won the Nobel prize, his hedge fund crashed."

and,

"Black-Scholes changed the culture of Wall Street, from a place where people traded based on common sense, experience and intuition, to a place where the computer said yes or no."

-- from yet another article on how the infamous Black-Scholes formula brought near-ruin to the financial system:

http://www.bbc.co.uk/news/magazine-17866646

The article is by Tim Harford, but he ends quoting Ian Stewart:

"... for Ian Stewart, the story of Black-Scholes - and of Long-Term Capital Management - is a kind of morality tale. "It's very tempting to see the financial crisis and various things which led up to it as sort of the classic Greek tragedy of hubris begets nemesis," he says.

"You try to fly, you fly too close to the sun, the wax holding your wings on melts and you fall down to the ground. My personal view is that it's not just tempting to do that but there is actually a certain amount of truth in that way of thinking. I think the bankers' hubris did indeed beget nemesis. But the big problem is that it wasn't the bankers on whom the nemesis descended - it was the rest of us."

 What I can't help but wonder, is whether we've truly learned anything through all this... or, are we essentially in the midst of repeating the whole process all over again????

Friday Puzzle

h/t to Peter Ash for this one:

Tennis balls are routinely sold in cylindrical cans with 3 stacked balls. Given such a can, which will be greater, the height of the can or its circumference?

Answer will be given in the comments….

Math Thinking... and Society

Even though I'm not actively involved in math education, and there are many math bloggers who better address that whole area, I keep finding myself drawn back to thinking about such matters by various postings...
Alexander Bogomolny at "CTK Insights" has a particularly interesting, or even provocative, recent post (entitled, "Regarding the Mess We Are In"), relating to math education, here:

http://www.mathteacherctk.com/blog/2012/04/regarding-the-mess-we-are-in/

Therefrom:

"It is often asserted that, mathematics being a deductive science, the study of mathematics is bound to have a positive effect on students' thinking ability. The evidence that this is so is mostly anecdotal. The evidence that there are other and more effective ways to improve students' thinking (and along the way their math scores) is traditionally and consistently being ignored….
"I may be mistaken, but it seems to me that the idea that study of mathematics leads to a betterment of the general thinking ability contains if not a plain logical flaw then at least an overlooked ambiguity. It's implicit in that belief that improved general thinking would bring about positive effects like avoidance of economic downturns, and in lives of individuals would lead to reaching better, more  advantageous decisions. Would not that mean that (at least in the limit - so to speak - when all think masterfully), all would be expected to arrive at the same conclusions? If so, then the argument is patently based on a faulty assumption. As a matter of fact, mathematicians - those ultimate, professional thinkers - would not agree as a group (i.e. arrive logically at the same conclusion) on almost any trifle or a matter of importance.
"There are Republican and Democrat mathematicians. There are among them liberals and conservatives, good investors and bad investors, happily married and multiply divorced…."

I suspect most of us who love math, do indeed instinctively feel that the logical, precise thinking math entails would, if all citizens attained it, translate to a better society… but, as Bogomolny suggests, there may be nothing more than anecdote or intuition to support that notion. Still one can't help but believe that raising the national level of math literacy (or decreasing what John Allen Paulos calls "innumeracy" is an overall positive). And while it's true that mathematicians themselves may run the gamut of politics, religions, lifestyles, etc. I still can't help but think that, mathematicians (and even more generally, "scientists") likely fall more significantly into the "liberal" and "Democrat" categories than the general populace as a whole. And surely somewhere, there are some surveys out there, that may indicate such (or show it false)? Can anyone point me in the direction of such studies…?

Beauty & Mystery... and Mathematics

Nice recent essay from the always-interesting Clifford Pickover:

http://asmarterplanet.com/blog/2012/04/the-beauty-and-mystery-of-mathematics.html

excerpt:

"Today, we use computers to help us reason beyond the limitations of our own intuition. Experiments with computers are leading mathematicians to discoveries and insights never dreamed of before the ubiquity of computers...
"... I believe that studying science and mathematics through the telescope of history has profound value for students and anyone curious about the evolution of thought and the limits of mind.  When we study the history of mathematics, we see the challenges of both amateur and professional mathematicians who persevered; we see abacuses morphing into slide rules, and into calculators and computers."

Twist on an Old Game

Fascinating game theory ("Prisoner's Dilemma") analysis of a British game show episode here:

http://tinyurl.com/7l5tgmm

(not your usual game plan... but it worked)

Math Learning... Let the Fun Begin

An older, long, but still interesting piece on math learning with emphasis on the "unschooling" and "Sudbury" movements, and describing "playful math," "instrumental math," "didactic math," and "college admissions math:"

http://www.psychologytoday.com/blog/freedom-learn/201004/kids-learn-math-easily-when-they-control-their-own-learning

(I probably don't agree with everything in the article, but do find it an interesting read.)

The author concludes as follows:

"And so, dear parents, please stop worrying about your kids' learning of math. If they are free to play, they are likely to play with math and learn to enjoy its patterns. If they live real lives that involve calculations, they will learn, in their own unique ways, precisely the calculations that they need to live those lives....

"And so, dear educators, please step out of your boxes and take a look at these remarkable educational movements--the unschooling and Sudbury movements -- and study them to see, from a different point of view, how education can work in such a painless and joyful manner when kids are free and in charge of their own learning. Nobody, at least no student, benefits from the thousands of hours of forced math "study" that we put kids through in our schools. The same amount can be learned in a small fraction of that time by kids who are free."

Bet'ya Didn't Know...

hmmm… according to a Clifford Pickover tweet (@pickover): "...7353 is the largest number n that humans will ever find so that both n and n^3 have only odd digits".

That, and many other numerical delights can be found here:

http://www2.stetson.edu/~efriedma/numbers.html

Friday Puzzle

Just a simple old Raymond Smullyan-type puzzle today...

In a land where all citizens are either "liars" who always lie, or "truthtellers" who always tell the truth, Larry and Tom speak as follows:

Larry: "Both Tom and I are liars."

Tom: "Only one of us is a liar."

What are Larry and Tom?

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ANSWER:
Tom is a truthteller; Larry is a liar. But... what if Tom or Larry had said, "I am a liar"??? (think about it)

Retract THIS!

This is just toooo rich not to pass along (a math paper retraction from Elsevier as reported by "Retraction Watch"):

http://retractionwatch.wordpress.com/2012/04/17/math-paper-retracted-because-it-contains-no-scientific-content/

...just to whet your appetite, here's the entire abstract of the above 2010 paper:

"In this study, a computer application was used to solve a mathematical problem."

And the posting-author asks:

"How on Earth does this stuff get past editors, peer reviewers, and publication staffs? And how did it remain in print for two years?"

…be sure to read the comments section as well (they're as entertaining as the main piece).

The Language of Statistics

A simple primer on statistics here:

http://www.abc.net.au/science/articles/2012/04/11/3474740.htm?WT.svl=healthscience1

P vs. NP... on the Big Screen?

Assuming you're familiar with the "travelling salesman" problem and related P vs. NP debate, then this trailer for a forthcoming (mid-June) movie may be of interest?? -- I thought it was some sort of parody when I first viewed it, but I guess it's a for real thriller! (from "Fretboard Pictures" ??? -- no idea how wide a distribution it will have -- if anyone can fill in more details, would be curious to learn more):



(h/t to @AndrewEckford)

Education Rant...

 I'm not in the math education loop, but as someone who was a guinea pig (victim?) of the "new math" movement of the 1960s I'm still interested in the debate that swirls around the topic of effective math teaching, especially with so many young people turned off by math at an early age. In that regard 'Wild About Math' blog recently ran a podcast with the founders of "Imagine Education" and their story-telling approach to math education. It's an attempt to make math less of an abstract or cerebral exercise, and more of a tool with direct application to daily life; an approach many favor.

I'm not convinced there is any one "best" approach to teaching math, at least not for all students (some students seem to have a natural interest in, and knack for math almost regardless of approach, while others may be dragged along kicking and screaming no matter what the format). Still, I very much enjoy seeing all these creative, digital approaches as options in the marketplace of math education. One hopes the cream (of techniques) will rise to the top. My one concern is that a method which coaxes in the greatest number of individuals to some basic level of math literacy, may not be the same as an approach that best encourages those already exhibiting an aptitude for math speedily along their way. Having so many options available for home or independent self-paced study is a wonderful thing though.

In that regard, I've long been a fan of Khan Academy as an innovator in this whole arena. After so much attention early on, Khan, is now (as often happens) being viewed more critically by many, almost in a backlash fashion. I think too often critics look at Khan Academy as some finished product full of flaws, even though it seems better viewed as a start and a work-in-progress -- the Khan Academy of 10 years from now might differ significantly from the current version. I happen to believe it's headed in the right direction regardless of imperfections. Moreover, I can't help but think some of the harshest critics of the Academy are simply in fear over their own livelihoods… the advent of Khan-like offerings could in the long run reduce significantly the number of not-only math teachers needed around the country, but instructors of a great many other subjects as well.

Even at the college-level I envision a distant future where students no longer matriculate at a lone university, but choose from a smorgasbord of offerings from the very best instructors in the country over the Net: a math course from a professor at MIT, a physics course from Stanford, an English course from Yale, etc. etc. Indeed, from a strictly economic view it is hugely inefficient that every college (or even high school) must so often duplicate the course offerings of every other one down the block. Just as brick-and-mortar retail establishments have suffered mightily with the advent of the Web, I suspect many brick-and-mortar education establishments too will fade with time. The day is approaching where a person sitting in a room in Kalamazoo, Michigan will have access to the same education as someone sitting in Cambridge, Massachusetts.

Where I live, 3 prominent Universities within 30 minutes of each other of course all have full-scale English Depts., Biology Depts., Math Depts., etc. etc. -- simply put, I don't think that will be economically sustainable in the future (and yes, it will be at the cost of a great many teaching jobs). I know this is not a popular view, and again, I'm somewhat shooting-from-the-hip since I'm not in the trenches of education myself, so am interested to hear the full range of opinions that are out there.

In my own case, I've always suspected that my personal math future was torpedoed in college by a certain college math professor who was quite simply miserable, and have wondered what the future might've been if only I'd not had that one instructor (or if, in addition to him, I'd had today's digital resources). The simple fact is that the very best math teachers, like the best of anything, are few-and-far-between, and yet we're headed toward a day when most students, wherever they are, will be able, potentially, to access them.
Anyway, interesting times ahead for higher education....

Some Unusual Topics…

Of course 'racism' can appear in any number of contexts, but I'd not often seen it discussed in relation to mathematics… until here:

http://tinyurl.com/6nab8kc

And on a completely different subject, the simple wooden desk that supported so much brilliant and insightful thinking (and writing):

http://tinyurl.com/7ema8m7

Friday Puzzle

This is an old simple puzzle that comes in many versions:

Three golfers, Mr. White, Mr. Brown and Mr. Green, get together every Saturday morning for coffee before playing a round.

One such Sat. morning as they're sipping their java Mr. White remarks aloud, “Well that's weird, we’re each wearing a colored baseball cap today: one white, one brown, and one green; yet no one is wearing a cap of the color that matches his name!"
At which point, the guy to his right, wearing the green cap, responds, “Oh it’s just a silly coincidence. Finish your coffee and let's go play.”

So, what color cap is each man wearing?

Answer below:
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ANSWER: Mr. White has a brown hat, Mr. Green a white hat, and Mr. Brown a green hat.

The Odds Are....

If you're not too sick already of reading about the Mega Millions Lottery that was recently concluded then here's a nice post summarizing the mathematical odds involved, and why having 3 winners (as was the case) would not be unexpected:

http://mathandmultimedia.com/2012/04/09/lottery-winners-beat-the-odds/

Fractals and Finance

Wonderful piece on the application of Mandelbrotian fractals (and Levy distributions) to finance:

http://triplehelixblog.com/2012/04/fractal-finance-a-rogue-mathematician%E2%80%99s-search-for-answers/

an excerpt:

"It was the winter of 1961. When he made this discovery, Mandelbrot was at IBM, studying income distribution patterns between the rich and poor. The Harvard economics department invited him to speak about his work. He walked into the office of his host that day to a surprise. On the chalk-board was a figure with a convex shape that opened to the right. He immediately turned to Professor Hendrik Houthhakker and asked why his diagram was already drawn. Houthhakker was perplexed: ‘These are graphs of cotton prices.’
"The puzzling similarity in pattern between income distribution and cotton prices got Mandelbrot thinking. Was it pure coincidence that the two were spitting images of one another, or was there a deeper truth in the strange connection between the two pictures? And so it was that Mandelbrot was propelled into investigating the mysteries of finance."

and it concludes this way:

"These models, Mandelbrot’s body of work suggests, have caused us to misperceive risk in a dangerous way. His work is a potential explanation to unusual market volatility: it suggests that our notions of ‘usual’ might be incorrect. There are academics who have taken up the baton from Mandelbrot, who died last October. These scholars work to build fractal descriptions of markets, models that take into account the Levy distribution. It remains to be seen how long it will take Wall Street to begin using these Levy-based models."

RFI: Online Learning Suggestions (calculus, etc.)

I'm a big fan of the idea of online video instruction (and link to several math-teaching sites in the right-hand column)… so I was a little embarrassed when someone recently asked me which site I would recommend to get a good introductory (first-year) video course in calculus, and I couldn't name one I would favor over others (I simply don't have enough direct experience with them). So, dear readers,

1) if, from experience, you can especially recommend a specific online site for calculus instruction I'd be curious to hear which one it is.

2) for future reference, if there are specific sites anyone cares to recommend for other specific math topics/courses (say, high-school level and above) feel free to mention them as well.

3) and finally, more generally, if there are any excellent (and free) math course sites that you feel are missing from my 6-member list in the right-hand column (under "Math Instruction on the Web") please tell me which ones they are, with their URLs.

Thanks!

Martin Gardner in Video

I've linked to this in the past, but since the 10th "Gathering For Gardner" celebration recently ended, worth showing this David Suzuki tribute to Martin Gardner yet again:

Friday Puzzle

For a Friday puzzle, another adapted Futility Closet offering:

5 tires are used in the course of driving a normal car 20,000 miles. If every tire sustains exactly the same amount of mileage over that time, than how many total miles must be put on each tire?

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ANSWER: 16000 mi

Carnival of Math

A new Carnival of Math is out now, chockful of good and varied stuff:

http://travels.aperiodical.com/2012/04/carnival-of-mathematics-85.html

Enjoy...

Be Aware...

In case it missed your attention somehow, THIS is officially "Math Awareness Month"! The 'theme' this year is "Mathematics, Statistics, and the Data Deluge," with some related resources here:

http://www.mathaware.org/mam/2012/related.html

And to read up on a little of the history of Math-Awareness-Month see here:

http://www.mathaware.org/mam/2012/about.mam.html

"Polygon Circumscribing"

 From Clifford Pickover's "The Math Book":

"Draw a circle, with a radius equal to 1 inch (about 2.5 centimeters). Next, circumscribe (surround) the circle with an equilateral triangle. Next, circumscribe the triangle with another circle. Then circumscribe this second circle with a square. Continue with a third circle, circumscribing the square. Circumscribe this circle with a regular pentagon. Continue this procedure indefinitely, each time increasing the number of sides of the regular polygon by one. Every other shape used is a circle that grows continually in size as it encloses the assembly of predecessors. If you were to repeat this process, always adding larger circles at the rate of a circle a minute, how long would it take for the largest circle to have a radius equal to the radius of our solar system?"

That's the question Cliff Pickover posed in the c.1940 entry of his book. As he then notes, it might seem that the circle radii would continually grow larger and larger toward infinity... however, it ain't so! As he says, "the circles initially grow very quickly in size," but then slow down and approach a limiting value, far short of a solar system [given by the infinite product: R = 1/(cos(π/3) x cos(π/4) x cos(π/5)….]
That limiting value, as calculated in 1965, turns out to be ~8.7000, but incredibly, before that calculation was made it was believed erroneously (for around 20 prior years according to Pickover) to be about 12.

This quickie video from WolframAlpha gives a sense of how rapidly the limit is approached:

Puzzle THIS!

In the event you have a couple spare hours to burn off...:

If you didn't care for the prior Friday puzzle, then read NO further....
As long-time readers know I'm especially fond of self-referential or recursive conundrums in logic/mathematics, and in response to the "Futility Closet" puzzle from Friday Mr. Honner on Twitter linked to a far more elaborate such problem. It's TOOOO involved and complicated for one of my 'Friday Puzzles' but I link to it below for any who can't get enough of this stuff (not for the faint-of-heart though, nor the short-on-time! ;-)):

http://www.maa.org/mathhorizons/Puzzles/Feb05-SRAT/original_puzzle.htm