Tuesday, December 30, 2014

Two Cultures (EXCELLENT read)


Almost two weeks ago I posted about the obituary for Alexander Grothendieck that was rejected by the journal Nature. Many math/science sites covered that little news story, and I sort of assumed by now it was over-and-done-with.  But even in death, as in life, Grothendieck seems to spread ongoing controversy!
Today, launching off that rejected obituary, mathematician/biologist Lior Pachter of UC Berkeley has posted a remarkable, really incredible and rich post I think, about the two "cultures" of mathematics and molecular biology, which he straddles, but finds little common ground on for its participants.
It's a long, and often technical post, but I think all should have a go at reading it (it may well require more than one sitting, and don't expect to follow all parts).  As a layperson myself, I'm more interested in the broad strokes he is painting than many of the technical arguments that I can't grasp. His "list of specific differences" between mathematicians and biologists is especially interesting, and Pachter is pessimistic about the relationship between the "two cultures," writing at one point, "The relationship between biology and mathematics is on the rocks and prospects are grim," and "The extent to which the two cultures have drifted apart is astonishing." As I implied in my original post (linked to above) I'm not so sure we really have a 'two culture' problem anymore (the term, as most know, comes originally from C.P. Snow over fifty years ago), so much as a fiefdom problem, with intense specialization having subsumed pretty much every field of technical study.

Anyway, if you're a working biologist or mathematician (or really, a scientist of any stripe) READ this piece!:

https://liorpachter.wordpress.com/2014/12/30/the-two-cultures-of-mathematics-and-biology/

As I post this, there are 3 comments to Pachter's article; I suspect there will be many more over time.
Agree or disagree with him, there's LOTS to chew on.


Monday, December 29, 2014

Puzzletime...


To ease into the week, a problem I adapted from one seen over at the 7puzzle blog site:

From the numbers 1-37, find the five integers that remain when you eliminate the following:

1)  any integers containing a 1
2)  prime numbers
3)  factors of 72
4)  numbers divisible by 3 or 5

Once you have the five 'finalists,' eliminate those number pairs that add up to 60.  Then, what number is left?
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ANSWER:  22


Sunday, December 28, 2014

The Density of Numbers


Sunday reflection on rationals and irrationals....

"It is possible to show that both the rationals and the irrationals are densely distributed along the number line in the following sense: Between any two rational numbers, there lie infinitely many irrationals and, conversely, between any two irrationals are to be found infinitely many rationals. Consequently, it is easy to conclude that the real numbers must be evenly divided between the two enormous, and roughly equivalent, families of rationals and irrationals.
"As the nineteenth century progressed, mathematical discoveries came to light indicating, to the contrary, that these two classes of numbers did not carry equal weight. The discoveries often required very technical, very subtle reasoning. For instance, a function was described that was continuous (intuitively, unbroken) at each irrational point and discontinuous (broken) at each rational point; however it was also proved that no function exists that is continuous at each rational point and discontinuous at each irrational point.  Here was a striking indicator that there was not a symmetry or balance between the set of rationals and the set of irrationals.  It showed that, in some fundamental sense, the rationals and irrationals, were not interchangeable collections, but to the mathematicians of the day, it was unclear exactly what was going on.
"

-- William Dunham in "Journey Through Genius"

Tuesday, December 23, 2014

"Trip The Light"



Shalom... Namaste... Noel... Gratis... Wonder... Joy... Amity... Equanimity. . . . . .

Last posting 'til after Christmas day.  Per usual, there will be a "Sunday Reflection" here on the forthcoming Sunday, and probably a potpourri list over at MathTango on same day).
After a trying year for the world in so many ways, may everyone find reasons for cheer & celebration in the days ahead... and beyond.



(...or, for another treat, watch the fun Matt Harding version HERE.)


Monday, December 22, 2014

A Big Family (puzzle)


Another problem to kick off the week, once again adapted from Henry Dudeney:

Max, who already has some children from a prior marriage, marries the widow Wilma who also has some prior children. A dozen years later their family has a total of 12 children, including all prior children and the new ones resulting from their marriage. Each partner, Max and Wilma, have 9 children (out of the 12) that they are direct parents of.  How many children have been born to Max and Wilma together in the last 12 years?:
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[...And alternatively, if you want a little meatier puzzle, Mike Lawler just walked his kids (and readers) through one from MIT yesterday: http://tinyurl.com/njkkepd ]
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ANSWER:  6 of their own  (having 3 each from prior marriages)


Sunday, December 21, 2014

A Mathematical Tension


The unexplained mystery...

"I, for one, find Gödel's incompleteness theorems rather comforting. It means that mathematicians will never be complete. There will always be something else which is undecidable with the current axioms. Should the human species survive another few million years and continue churning out mathematics at the rate we've done for the past few thousand years, we still won't have considered it all. There will always be work for all of the future mathematicians. As always, some of that work will go on to be incredibly useful for the rest of civilization, and much of it will remain the pointless but endlessly amusing plaything of academics.
"There's an unexplained mystery behind all of this, which I've been delicately avoiding throughout the book. If maths is the consequence of games and puzzles, the result of pure intellectual thought, why does it end up being so practically useful?  I keep promoting maths as a bit of fun, yet no one can ignore that mathematical techniques are the workhorse of modern technology. In reality, mathematics is a serious industrial endeavor. There's a tension between what I claim to be the origin of maths and where it ends up being used.
"

-- Matt Parker from "Things To Make and Do In the Fourth Dimension"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Thursday, December 18, 2014

Honoring Grothendieck...


We live in a day of extraordinary and over-riding science specialization....

H/T to Jordan Ellenberg for pointing to this post about an obituary for Alexander Grothendieck that was rejected by Nature -- a fascinating read, even if Nature didn't find it so for their obit. purposes. David Mumford, one of the authors, finds it "very depressing" that a STEM publication would judge this piece unsuitable for its readers, but I'd opt for a different view... namely, that Grothendieck was simply too far advanced beyond the minds that run (or read) generalist journals like Nature and Science (which are far from the bastions they once were, before such modern-day field-of-study specialization took hold):

http://www.dam.brown.edu/people/mumford/blog/2014/Grothendieck.html

Early on, the piece reads as follows:
"His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves -- and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed. His strength and intensity were legendary. He worked long hours, transforming totally the field of algebraic geometry and its connections with algebraic number theory. He was considered by many the greatest mathematician of the 20th century."
Surely there is a far more appropriate (specialist) math journal out there that would love to run Mumford and John Tate's wonderful tribute piece for an appreciative audience....


Wednesday, December 17, 2014

A Li'l More On That Wily Matt Parker....


I already wrote a blurb at MathTango about Matt Parker's fantastic book, "Things To Make and Do In the Fourth Dimension," but now that I've finished reading it, just want to add a few quick notes:

1)  First, I'll reiterate it's a wonderful volume -- I enjoyed the second half (which touched on several of my favorite topics, and also told perhaps the most fun story of Tartaglia's rivalry with Fior over algebraic/cubic equations) even more than the first half.
2)  Do note however, that at least parts of the volume may require slightly more math sophistication, or interest, or just persistence, than some of the other volumes I included on my Holiday gift list; i.e. while Matt's book is a fun and educational read, not every chapter is an easy read. 
3)  Also, one small complaint:  the book lacks an index, which because of the sheer number and diversity of topics/information included, would've been helpful.
4)  Finally, (and the REAL reason for this additional posting), BE SURE to read the "Acknowledgements" section at the very conclusion of the book! (...a section readers often skip over). Not only is the section entertaining to read, BUT in it are buried these innocuous, cryptic lines:
"Oh yeah, and there is a competition hidden somewhere in this book. If anyone wins it, I'll think of a suitable prize. Beware of the traps."
Leave it to Matt to concoct such a ploy! And I assume by "competition," he is not referencing proving the Riemann hypothesis! ;-).  (The book poses various questions and problems at points, but I'm not sure what is being referred to as "a competition" or "the traps" -- could be fun going back through the pages trying to figure out what it's all about.)
Anyway, have at it, and, with this heads-up, may one of my readers win the prize!


Monday, December 15, 2014

Triangle Puzzler


To start the week, a quick and simple (...or, not so simple) problem:

You have a triangle. In some particular order, the three sides and height of the triangle are four consecutive integers. What is the area of the triangle?
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ANSWER:  84  (a 13-14-15 triangle with height 12)

Sunday, December 14, 2014

An Epiphany


Sunday reflection via Steven Strogatz....

"The teacher, Mr. diCurcio, said, 'I want you to figure out a rule about this pendulum.' He handed each of us a little toy pendulum with a retractable bob. You could make it a little bit longer or shorter in clicks in discrete steps. We were each handed a stopwatch and told to let the pendulum swing ten times, and then click, measure how long it takes for ten swings, and then click again, repeating the measurement after making the pendulum a little bit longer. The point was to see how the length of the pendulum determines how long it takes to make ten swings. The experiment was supposed to teach us about graph paper and how to make a relationship between one variable and another, but as I was dutifully plotting the length of time the pendulum took to swing ten times versus its length it occurred to me, after about the fourth or fifth dot, that a pattern was starting to emerge. These dots were falling on a particular curve I recognized because I'd seen it in my algebra class. It was a parabola, the same shape that water makes coming out of a fountain.
"I remember having an enveloping sensation of fear. It was not a happy feeling but an awestruck feeling. It was as if this pendulum knew algebra. What was the connection between the parabolas in algebra class and the motion of this pendulum? There it was on the graph paper. It was a moment that struck me, and was my first sense that the phrase 'law of nature' meant something. I suddenly knew what people were talking about when they said there could be order in the universe and that, more to the point, you couldn't see it unless you knew math. It was an epiphany that I've never really recovered from.
"

-- Steven Strogatz from "Who Cares About Fireflies"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]
 

Friday, December 12, 2014

A Friday Puzzle


To end the week, a simple-to-state puzzle that I've re-written/adapted from an old Henry Dudeney volume:

In the course of a year, the cats (and there are more than one) on Mr. Schlobotnik's farm killed 999,919 mice. If every cat killed exactly the same number of mice (and more than 1), then how many cats reside on the farm, given that the total number of cats is LESS than the number of mice killed per cat?
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Answer:
there were 991 cats, who each killed 1009 mice

Wednesday, December 10, 2014

NOT To Be Missed... on number theory/prime gaps


"After a while, these things taunt you".... (T. Tao)

FANTASTIC piece from Erica Klarreich and Quanta Magazine today on another obvious, but deep question from number theory (how LARGE can prime gaps be? ...sort of the reverse of the twin-prime question):

https://www.quantamagazine.org/20141210-prime-gap-grows-after-decades-long-lull/

Includes a "favorite joke" of number theorists that I'd not heard before :-) and also perhaps my favorite photo from all of mathematics: Paul Erdös and Terence Tao (as a child) together.
Seriously, with mentions of Yitang Zhang, Erdös, Tao, James Maynard, prime gaps, a crazy-ass log formula, and $10,000 prize, what is there not to love!


Tuesday, December 9, 2014

"Mathematical Mystery Tour"


The BBC has done some great hour-long mathematical presentations. Recently, Cliff Pickover tweeted one of the old Horizon episodes (I've linked to before), called "Mathematical Mystery Tour." It's interesting with its discussion of "proof" in mathematics (in light of the post I did a bit ago over at MathTango), as well as other subjects. And though it's rather dated, still a worthwhile 50 mins. if you've missed it, and have some time:



[In other news, I've now posted a blurb on Matt Parker's recent volume, "Things To Make and Do In the Fourth Dimension," over at MathTango.]


Sunday, December 7, 2014

A Book Recommended


I mentioned last week (at MathTango) that Richard Elwes' 2013 book, "Chaotic Fishponds and Mirror Universes," was one of my very favorite reads of 2014. So this morning just a blurb from its Introduction, as the Sunday reflection, in hopes of encouraging you to check it out further:

"Of all the subjects studied, debated and fought over in the course of human history, I happen to believe that the most fascinating is mathematics. That's a bold claim -- perhaps mystifying to readers who were bored or baffled by the subject at school. Well, of course fascination is in the eye of the beholder, and certainly there will be those who need some persuading. I hope this book will go some way towards doing that.
"What is irrefutable, however, is that in modern life mathematics is both important and ever-present. Even the most entrenched maths-hater has an awareness that it plays a central role in today's world, touching our lives in more ways than ever before. But that is where the details are liable to become hazy... yes, important, but where exactly is it used, and in what ways?
"In response, I present in the pages that follow a selection of 35 diverse applications of mathematics. I attempt to unravel some of the principles that underlie aspects of our daily lives, as well as those that inform today's boldest thinkers....


"I hope that, by the end of this book, readers will have a more precise sense of where mathematics fits into modern life -- and, en route that some doubters become devotees of the subject that I find endlessly, gloriously, fascinating.
"









Wednesday, December 3, 2014

Deja Vu: Revisiting The Flash Mind Reader



About 9 months ago, in a tweet, math teacher Fawn Nguyen casually mentioned "The Flash Mind Reader," a delightful Web-based puzzle, that I was unfamiliar with, though she apparently has known of it for close to a dozen years. Go here to check it out, if perchance you've not seen it:
http://www.cs.nyu.edu/~dodis/magic-ball.swf

What a great game/puzzle for younguns, but it also stumped me for awhile before I figured it out, and wrote a post alluding to it. So I was delighted to be reading "The Best Writing On Mathematics 2014" recently and come upon this great teaser once again (pgs. 171-5). It comes up in a selection aptly entitled "Wondering About Wonder In Mathematics" by Dov and Rina Zazkis.  It's one of my favorites of many great selections in this year's anthology. The authors pinpoint "surprise" as the underlying component of "wonder" in mathematics, and then list four types of "mathematical surprise":

1) perceived "magic"
2) counterintuitive results
3) variation on a known result or procedure
4) paradoxes

"The Flash Mind Reader" falls under the 'perceived magic' category, and they write this about it:
"We have used this activity [the Mind Reader] several times with both elementary school and university students. It's not uncommon for members of both groups to try to cover the webcams on their computers or face away from the screen, as if the Mind Reader was determining what number was in their head using some elaborate eye-tracking mechanism. Obviously, these actions do not prevent the Mind Reader from working.  However, these reactions serve both to illustrate some rudimentary theory testing -- 'Is this website tapping into the webcam?' -- and to demonstrate students' need to understand how this 'Mind Reader' works, which is catalyzed by their curiosity."
I'm heartened to know that university students can be as duped by this little gem as I was at first blush ;-) Of course a lot of number and card tricks are based on pure mathematics; in some ways, Flash Mind Reader takes the element of 'distraction,' which is often a component of such "magic," to another subtle level, which helps make it so effective. [In the event you don't see how the puzzle works, you'll have to buy the book, or google for the answer, I won't give it away here!]
The rest of the chapter looks at some other classic and interesting examples from mathematics, placing them in the four categories above. The Mandelbrot set, Platonic solids, the Monty Hall problem, and Simpson's paradox, are among standards mentioned in the chapter.

Anyway, I encourage folks to get a hold of this year's "Best Writing On Mathematics," as I think it the best edition yet (and unfortunately most expensive) of a series that I hope maintains interest and support. It was recently reviewed by Alexander Bogomolny:  http://tinyurl.com/p28acrz
[I included it on my recent list of books for the Holidays at MathTango.]


On a sidenote, thinking there might be some interesting back-story here, I attempted to find information about web designer Andy Naughton, who created The Flash Mind Reader, to include in this post, and was surprised that though his name and the game are found MANY times in Google searches, I couldn't actually find any background info on him... is he alive??? is he very private? Is Naughton his real name (both "Andy Naughton" and "Andy Wolfe" seem to be associated with "FlashLight Creative" -- are they 2 different people or one-and-the-same?) Is there some mystery to all this? Does anyone happen to know much about the fellow?  Just curious what the history to the Mind Reader might be, and how, if at all, its success affected Andy's life??? (...if I could locate him, and he was willing, I might be interested in doing one of my Math-Frolic interviews with him, as well). ...maybe if I just hone my own mind-reading skills I can find him.

Monday, December 1, 2014

Math, Women, Tessellation, Intuition

(image: WikimediaCommons)

A lot of discussion around the Web these days about women in STEM, and at Math-Frolic I'm even more interested in women in math, so thought it would be fun/timely to recount the unusual story of Marjorie Rice -- worth repeating, even if most are familiar with it, as a rare instance of someone becoming involved with math almost by accident.
[Most of this information was reported over a year ago in a MathMunch piece on Marjorie here:
http://mathmunch.org/2013/02/25/marjorie-rice-inspired-by-math-and-subways/  also see Ivars Peterson's 2010 piece here: http://mathtourist.blogspot.com/2010/06/tiling-with-pentagons.html ]

Marjorie discovered her senior year in high school that she found math interesting, but by then it was too late to do much with it. She went on to marry, have children, be a housewife; i.e. she took NO mathematics past high school. But after getting a subscription to Scientific American for her son, she began reading the Mathematical Games column of Martin Gardner, including a 1975 column concerning "pentagon tessellations," i.e. pentagon forms that could cover an entire plane, repeating themselves with no gaps, like a jigsaw puzzle. At one time mathematicians believed there were only five such pentagon shapes that achieved tessellation, but in 1968 three more were discovered, and a fourth new one had just been added in 1975 that Gardner was reporting on.

Marjorie was intrigued. And playing with different pentagons, with different internal angles, she finally found a fresh one that accomplished the feat of tessellation. Inventing her own unconventional notation to describe her work she wrote to Gardner showing the result. And he sent her correspondence on to another female mathematician, Doris Schattschneider, who confirmed Marjorie's success and translated her work into more standard mathematical format.  Marjorie went on to find yet three more successful pentagon tessellations, and also DISproved a conjecture made by Doris.
Successful amateurs have made significant contributions to astronomy, but in most sciences, and particularly in mathematics, it is rare for an academically-untrained amateur to accomplish something missed by professionals... but apparently Marjorie didn't know that! Her own website on her work is here:
https://sites.google.com/site/intriguingtessellations/home

She is now over ninety, and remains an inspiration, not just to women, but to amateur enthusiasts everywhere. What I love most though about the Marjorie Rice story isn't that she was a female in mathematics, nor even that she was an amateur contributing to a technical field, but rather what the story says about the role of intuition and insight in math. Despite mathematics' image of being cold, dry, and rigid, and despite its abstractness in advanced study, scrape below the surface and there remains, on occasion, a powerful substrate of intuition and mental imagery, accessible to many.

Below is a video segment (from about the 31:50 point to 35:45) talking about Marjorie's work (again h/t to MathMunch for this):





We now know of 14 tessellating pentagon forms! Are there more?


Sunday, November 30, 2014

Flashes, Pitches, Models, etc.


"We think that if, say, two variables are causally linked, then a steady input in one variable should always yield a result in the other one. Our emotional apparatus is designed for linear causality. For instance, if you study everyday, you expect to learn something in proportion to your studies. If you feel that you are not going anywhere, your emotions will cause you to become demoralized. But modern reality rarely gives us the privilege of a satisfying, linear, positive progression: you may think about a problem for a year and learn nothing; then, unless you are disheartened by the emptiness of the results and give up, something will come to you in a flash."
-- Nassim Taleb from "The Black Swan"

...and from Edward Tufte, this:

People and institutions cannot keep their own score accurately. Metrics soon become targets and then pitches, and are thereby gamed, corrupted, misreported, fudged…
"Examples: premature revenue recognition, Libor rates, beating the quarterly forecast by a single penny, terrorist attacks prevented, Weapons of Mass Destruction, number of Twitter followers, all body counts (crowd sizes, civilians blown up). Sometimes call the Principle of Lake Woebegone, where all children are above average.

...and a couple more quickies:

"If you torture the data long enough, it will confess to anything." -- Ronald Coase (British economist)

"Essentially, all models are wrong, but some are useful."  -- statistician George Box


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]


Wednesday, November 26, 2014

Happy Thanksgiving Holiday To All

(via WikimediaCommons)



*****************************


"If the only prayer that you say 

your whole life is 'thank you'

that would suffice."

~~ Meister Eckhart

*****************************

Tuesday, November 25, 2014

Tuesday Tidbits


Ed Frenkel has a stirring tribute to "visionary" Alexander Grothendieck in the NY Times today, hitting on some of the highlights of the reclusive mathematician's life, and ending as follows:
"A party of one, he was unafraid to be himself and to speak his truth. The man who had advanced mathematics in the most profound ways did not believe that math was the answer to everything. He taught us that life is more valuable than any equation."
And at MathTango yesterday I (aided by Keith Devlin) considered the nature of "proof":
http://mathtango.blogspot.com/2014/11/proofiness.html

BTW, I'll probably be posting a 2014 book wrap-up for the popular math volumes of the year sometime after Thanksgiving. It's pretty much already written, but if you have volumes you're partial to for people's Holiday lists, feel free to mention them in the comments here, or in the later post to come.
And for my own interest, curious if anyone is directly familiar with the book "Gödel's Mistake" by Ashish Dalela? If so, do you recommend it, or, not so much (for an interested layreader, not academic logician)?


Sunday, November 23, 2014

Mathematics: "A Growing Organism"... "A Connected Web"


This Sunday's Reflection:

"Mathematics is a living and growing organism; within it are intricate and delicate structures of strong aesthetic appeal.  It offers opportunities for surprise as unexpected vistas open the mind to new lines of thought...
"Mathematics was created by all manner of people. There were religious bigots and atheists, political reactionaries and wild revolutionaries, snobs and egalitarians; some were people of great charm, some odious. If there is any common denominator, it is a driving curiosity, a desire to understand, a need to build, even if the structures be abstract. Admirably suited though mathematics is to modelling the real world, it can be developed totally without dependence on anything outside itself. Parts of it are simply mind creations, owing nothing to the physical world. It is a playground for the mind...

"Regrettably, many of us have never been allowed to see what mathematics is. It has been obscured by pointless emphasis on routines rather than ideas. This failure to distinguish what is important has led many people to see mathematics as a collection of totally arbitrary rules which have to be learnt by rote, and performed with the exactness and precision of a religious rite. Ask a person if there is much to be remembered in mathematics; if they speak of an overwhelming mass of material, their education in this area has been counter-productive; not merely neutral. Mathematics, properly seen, is a connected web; grasp at one piece and all the surrounding region comes to mind.
"

-- Laurie Buxton from the Introduction to  "Mathematics For Everyone" (1984)


Thursday, November 20, 2014

Euler's Alchemy


                              e + 1 = 0 


Nice Lee Simmons piece today for Wired, rhapsodizing on "Euler's identity":

http://www.wired.com/2014/11/eulers-identity/

Love this wormhole analogy toward the end:
"But the weirdest thing about Euler’s formula—given that it relies on imaginary numbers—is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems (you know, sines, secants, and so on) into more tractable algebra—like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digitize music, and it tames all manner of wavy things in quantum mechanics, electronics, and signal processing; without it, computers might not exist."
Give it a read....


Wednesday, November 19, 2014

A Geometry Wednesday


Today, a simple, practical guide for estimating the height of a tree from Ian Stewart's recent "Professor Stewart's Casebook of Mathematical Mysteries."

Stewart describes this as "an old forester's trick (the trick is old, not the forester) for estimating the height of a tree without climbing it or using surveying equipment."

and continues:

"Stand at a reasonable distance from the tree, with your back towards it. Bend over and look back at it through your legs. If you can't see the top, move away until you can. If you can see it easily, move closer until it's just visible. At that point, your distance from the base of the tree will be roughly equal to its height."

The angle your line-of-sight is forming with the ground is now roughly 45 degrees, and thus the line-of-sight itself is the hypotenuse of an isosceles right triangle with the base-distance and tree height equal side values (thus, measure or walk off the base, and you have the height).

...As for those of us of an age where bending over and looking through our legs isn't such a practical affair, I guess we're out-of-luck :-(

Anyway, Stewart's volume is his usual mix of older and fresher, easier and more technical, math entertainment. Give it a gander.

One of my old favorites that shows up in Stewart's book is what he calls the "Square Peg Problem" (it goes by some different names) -- another one of those seemingly easy, yet exquisitely difficult-to-prove (100+ year-old) conjectures. It asks whether one or more squares can always be fitted upon every closed planar curve by selecting four points on that curve, or stated more succinctly as "Does every simple closed curve have an inscribed square?" For a fuller treatment of it see here:

http://www.webpages.uidaho.edu/~markn/squares/

and here's a 2014 article on it (pdf) from AMS:

http://www.ams.org/notices/201404/rnoti-p346.pdf

via Wikipedia




Monday, November 17, 2014

"Women In Maths"


The University of Nottingham has been doing a delightful series of short videos of "Women in Maths" (apparently inspired, in part, by Maryam Mirzakhani as the first woman to receive a Fields Medal last August). The introductory video is here:



And the other videos (focusing on individual mathematicians in ~2 minutes or less) can be viewed here:
http://tinyurl.com/ml8yayr

The on-site video description reads in part:

"Although around 40% of the UK undergraduates in mathematics are women, there is a well-documented leaking pipeline when it comes to women choosing to do a PhD and then choosing an academic career path. The proportion of women mathematicians declines rapidly the higher one looks on the academic ladder. Unfortunately, this often makes women who do choose this career path invisible, to students who are about to choose their A-levels, even to students who are already pursuing a maths degree. Therefore the women at the School of Mathematical Sciences at University of Nottingham have made some videos to: become more visible and thus hopefully inspire others; fight stereotypes; talk about what it is like to be a mathematician in academia today and why they chose academia; communicate the passion they feel for what they do and what they love about it; describe the creativity needed for research. And yes -- it can be combined with having a family!"


Sunday, November 16, 2014

Think About It...


short and sweet...

"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)


Friday, November 14, 2014

'To Thine Own Self Be True' ...A Legend Passes


"This above all -- to thine own self be true,
And it must follow, as the night the day,
Thou canst not then be false to any man.
Farewell. My blessing season this in thee!
"

 -- Polonius (in Hamlet)


As most know by now Alexander Grothendieck passed away yesterday at age 86. 
This news aggregator has a lot of good material:

https://news.ycombinator.com/item?id=8604814

And here are some of the older pieces various Twitterers linked to overnight:

http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf 

https://www.sciencenews.org/article/sensitivity-harmony-things?mode=magazine&context=585

https://www.youtube.com/watch?v=AgXsgqmCaIY&feature=youtu.be

http://inference-review.com/article/a-country-known-only-by-name

from Grothendieck himself, in Récoltes et Semailles


"...I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my 'elders' and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle -- while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

 "In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective of thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.
"


R.I.P.  and as is often said (though not that often in mathematics), "Farewell to one of the great ones"....


Thursday, November 13, 2014

Charts, Graphs, Facepalms



(from: http://www.tylervigen.com/ )


Nautilus posted a simple, worthwhile piece recently that was a lesson on how easily the public is fooled by manipulated charts:

http://nautil.us/issue/19/illusions/five-ways-to-lie-with-charts

It reminds one a bit of another site, "Spurious Correlations," that focuses on examples (like the one above) of graphs that show... guess what...: spurious correlations (well, likely spurious, anyway):

http://www.tylervigen.com/

It's all a valuable reminder that people aren't just fooled by empirical-sounding numbers, but by easily-misinterpreted visual presentations as well.

Finally, if you prefer your charts with a dose of humor, check these out:

http://www.boredpanda.com/funny-graphs-and-charts-2/ 

...and plenty more here:

http://www.allfunnycharts.com/


Wednesday, November 12, 2014

Either Have to Laugh or Cry...


After last week's elections I either need something to laugh at, or, a triple-dose of Prozac....

Soooo... from the LMAO Dept.:  yesterday Steven Strogatz passed along (on Twitter) a site and old post I'd never seen before... if bad words put you off, don't even bother looking... but if you read it (after sending any younguns out of the room) and guffaw you can thank me for passing it along (newly-minted $20 bills would be much-appreciated)... or, if you read it and hate it, you can blame that low-down, good-for-nuthin, wacko Cornell math-obsessed professor:

http://tinyurl.com/keujusd

...and I've already done the heavy-lifting for you to find the few-other math-related posts on the site:

http://tinyurl.com/nf6xlz7
http://tinyurl.com/23lkdsw
http://tinyurl.com/qy8egts
http://tinyurl.com/4k9dqmm


Monday, November 10, 2014

Bevis's Birthday


To kickstart the week, a little algebraic puzzle I adapted from this week's "Ask Marilyn" column in the Sunday Parade Magazine:

Suppose Bevis's birthday is this month (he's an adult), and on his birthday he will be the same age as the 2-digit year in which he was born (i.e. 1940, or whatever). How old will Bevis be on his upcoming birthday?
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answer:  57 years old

Sunday, November 9, 2014

Mathematicians As Mavericks (Sunday Reflection)


"Mathematicians are mavericks -- inventors and explorers of sorts; they create new things and discover novel ways of looking at old things; they believe things hard to believe, and question what seems to be obvious. Mathematicians also disrupt patterns of entrenched thinking; their work concerns vast streams of physical and mental phenomena from which they pick the proportions that make up a customized blend of abstractions, glued by tight reasoning and augmented with clues glanced from the natural universe. This amalgam differs from one mathematician to another; it is 'purer' or 'less pure,' depending on how little or how much 'application' it contains; it is also changeable, flexible, and adaptable, reflecting (or reacting to) the social intercourse of ideas that influences each of us…

"And here comes a peculiar aspect that distinguishes mathematics among other intellectual domains: Mathematicians seek validation inside their discipline and community but feel little need (if any) for validation coming from outside. This professional chasm surrounding much of the mathematics profession is inevitable up to a point because of the nature of the discipline. It is a Janus-faced curse of the ivory tower, and it is unfortunate if we ignore it."


-- Mircea Pitici from the Introduction to "The Best Writing On Mathematics 2013"

(p.s. -- I believe the 2014 edition of "The Best Writing On Mathematics" will be appearing in stores by the end of this month.)


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

 

Thursday, November 6, 2014

It's a Wonderful Subject


Another short clip today of Steven Strogatz (from about a year ago at the World Science Festival), explaining simply why he's motivated to know and teach math:



I recently finished reading Dr. Strogatz's 2009 "The Calculus of Friendship" for the second time (having loved it the first time around), and oddly, perhaps because of the holidays approaching, it caused me to think of the old Jimmy Stewart movie, "It's a Wonderful Life," a film often viewed as a tradition for this festive (and nostalgic) time of year. I've never made a point of re-watching particular movies at particular seasons, but have to admit I'm now thinking of making Dr. Strogatz's book my own personal tradition to re-read each year as the holidays approach. It's a wonderful, if subdued, tale, and one wishes that everyone, whatever field you're in, could have a "Joff" in their lives... or... be a "Joff" to others (..."Joff" being Steven's inspiring high school math teacher).



ADDENDUM Now, coincidentally, The Aperiodical blog has just put up a (15-min.) student-made documentary about doing/teaching mathematics:

http://aperiodical.com/2014/11/logically-policed/


Monday, November 3, 2014

The Monster That Lurks


Not sure which I love more about this Numberphile offering... Tim Burness's step-by-step explanation of the "Monster Group" (with it's almost 200,000 dimensions) or John Conway's sheer astonishment that it exists at all:




Sunday, November 2, 2014

Contemplating Riemann


"In [his 1859 paper], Riemann made an incidental remark -- a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years...

"...it is that incidental remark -- the Riemann Hypothesis -- that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant at work -- subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age...

"It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. ...Hunting down the solution to the Riemann Hypothesis has become an obsession for many -- the veritable 'great white whale' of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution.
"

 -- John Derbyshire, from "Prime Obsession"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]


Wednesday, October 29, 2014

Moravec's Paradox


This isn't exactly math, but it's artificial intelligence (AI), and that's close enough... especially since a few posts back I wrote about IBM's "Deep Blue" and its 1997 defeat of chess grandmaster Gary Kasparov (at the time, a long-held goal of AI). Well, Moravec's paradox is the interesting idea that advanced or high-level reasoning and logic is much more easily mimicked by a computer system than are low-level sensori-motor skills that have evolved over millions of years... it's easier for a computer to learn to play chess, than to recognize human faces. This is one of those things that is fairly obvious when you stop to think about it... but, we often don't stop to think about it!
Here's what Steven Pinker wrote in "The Language Instinct":
“The main lesson of thirty-five years of AI research is that the hard problems are easy and the easy problems are hard. The mental abilities of a four-year-old that we take for granted – recognizing a face, lifting a pencil, walking across a room, answering a question – in fact solve some of the hardest engineering problems ever conceived…. As the new generation of intelligent devices appears, it will be the stock analysts and petrochemical engineers and parole board members who are in danger of being replaced by machines. The gardeners, receptionists, and cooks are secure in their jobs for decades to come.”   
A more recent blog piece applies the paradox to Google's self-driving cars, a creation I've certainly had trouble comprehending, given the countless issues/variables involved:

http://www.eugenewei.com/blog/2014/10/13/moravecs-paradox-and-self-driving-cars

[p.s. -- actually, where are the dang flying jetpacks I grew up believing we would all have by now... forget the cars Google, I want my personal commuting jetpack!]

anyway, below, another somewhat provocative post applying Moravec's paradox to brain processing:

http://blog.jim.com/science/moravecs-paradox-rna-and-uploads/


Tuesday, October 28, 2014

Just Passing This Along


Colin Hegarty, who runs Hegartymaths has been bestowed a "Gold" tech-teaching award for his free math-tutorial site in Britain:

http://bit.ly/1322gNb

His videos are here:
https://www.youtube.com/user/HEGARTYMATHS/videos

I've not actually experienced the site or videos, so not directly endorsing it, but just recognizing that others attest to its value. It sounds a lot like (and was indeed inspired by) Khan Academy, which remains controversial in various quarters.

Anyway, check it out if you're looking for adjunct math tools.  Also, Colin tweets here:  @hegartymaths

Sunday, October 26, 2014

Taleb on Randomness


Today, a number of bits from an older Nassim Taleb volume, "Fooled By Randomness":

"Probability is not a mere computation of odds on the dice or more complicated variants; it is the acceptance of the lack of certainty in our knowledge and the development of methods for dealing with our ignorance. Outside of textbooks and casinos, probability almost never presents itself as a mathematical problem or a brain teaser. Mother Nature does not tell you how many holes there are on the roulette table, nor does she deliver problems in a textbook way (in the real world one has to guess the problem more than the solution)."

"This book is about luck disguised and perceived as nonluck (that is skills) and, more generally, randomness disguised and perceived as non-randomness (that is, determinism). It manifests itself in the shape of the lucky fool, defined as a person who benefited from a disproportionate share of luck but attributes his success to some other, generally very precise, reason."

"We are still very close to our ancestors who roamed the savannah. The formation of our beliefs is fraught with superstitions -- even today (I might say especially today). Just as one day some primitive tribesman scratched his nose, saw rain falling, and developed an elaborate method of scratching his nose to bring on the much-needed rain, we link economic prosperity to some rate cut by the Federal Reserve Board, or the success of a company with the appointment of a new president 'at the helm.'"

"Disturbingly, science has only recently been able to handle randomness (the growth in available information has been exceeded only by the expansion of noise). Probability theory is a young arrival in mathematics; probability applied to practice is almost nonexistent as a discipline"

"Indeed, probability is an introspective field of inquiry, as it affects more than one science, particularly the mother of all sciences: that of knowledge. It is impossible to assess the quality of the knowledge we are gathering without allowing a share of randomness in the manner it is obtained and cleaning the argument from the chance coincidence that could have seeped into its construction. In science, probability and information are treated in exactly the same manner. Literally every great thinker has dabbled with it, most of them obsessively.
"


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]



Wednesday, October 22, 2014

"The Man vs. The Machine"


(via MichaelMaggs/Wikimedia)

Math fans usually like chess, so I'll refer readers to FiveThirtyEight's first mini-documentary film (17 mins.), on the historic 1997 match between then-World-Champion Garry Kasparov and IBM's "Deep Blue" (actually it's the RE-match that Kasparov LOST). Some interesting history... and following its victory and acclaim, Deep Blue "retired":

http://fivethirtyeight.com/features/the-man-vs-the-machine-fivethirtyeight-films-signals/

ADDENDUM:  I've now discovered, for the more-thoroughly chess-ensconced (who have 90 minutes to devote to the Kasparov/Deep Blue battle), this older film on the same topic:





Tuesday, October 21, 2014

He'd Be Embarrassed By All the Attention...



Wasn't planning to do a post on Martin Gardner's Centennial today (...I did my little reflection post on him this past Sunday), since I've covered him plenty in the past, and knew many others would be paying tribute this week. But so many good posts have gone up, I don't want to ignore them, and thus offer a small sampling below.
It's impossible to overdose on Martin Gardner, incredible thinker/writer that he was (who hardly took a math course beyond high school!), so enjoy...  (possibly I'll add additional links in next 24 hrs., but really there are too many to choose from!):

http://www.bbc.com/news/magazine-29688355

http://www.theage.com.au/national/education/martin-gardner-and-mr-hyde-20141016-3i584.html

http://wordplay.blogs.nytimes.com/2014/10/20/mg100-2/?_php=true&_type=blogs&smid=tw-share&_r=0

http://plus.maths.org/content/five-martin-gardner-eye-openers-involving-squares-and-cubes

http://headinside.blogspot.com/2014/10/100-years-of-martin-gardner.html

Also, my year-old review of Gardner's autobiography here:
http://mathtango.blogspot.com/2013/10/undiluted-martin-gardner.html

ADDED:  http://en.chessbase.com/post/martin-gardner-s-100th-anniversary

One suspects Martin is now somewhere off demonstrating the joy of hexaflexagons to a whole new audience of enthralled angels... or, just maybe, he and Paul Erdös are sitting together, sipping coffee, and excitedly reading each other passages from "The Book." ;-)


Monday, October 20, 2014

Some Monday Stuff


So many interesting, varied things passing by my computer screen the last 48 hrs.; have to pass a few along rather than hold onto until the Friday "potpourri" collection:

First, this wonderful video on P vs. NP... about as good as any quick (11-min.) intro I've ever seen on this important subject:



On the education front, fans of Robert Talbert should read his "Medium" piece outlining the future of his "Casting Out Nines" blog.

And Grant Wiggins has an update on the education post that made the rounds at his blog last week, and turns out to have been written by his daughter! More soon to come from her:
http://tinyurl.com/k2tcv3p

And finally HERE, Tracy Zager re-visits the below Robert Kaplinsky video, covering an interesting problem/issue that actually goes back to at least 1986:




Sunday, October 19, 2014

A Martin Gardner Sunday


This coming Tuesday marks the 100th anniversary of Martin Gardner's birth, so for a Sunday reflection, some quotes about the man:
[several of these are taken from the Martin Gardner "testimonial" page: http://martin-gardner.org/Testimonials.html ]

Douglas Hofstadter, in tribute to Martin, upon his death in 2010:
"This is really a sad day…  sad because his  [Gardner's] spirit was so important to so many of us, and because he had such a profound influence on so many of us. He is totally unreproducible -- he was sui generis -- and what's so strange is that so few people today are really aware of what a giant he was in so many fields -- to name some of them, the propagation of truly deep and beautiful mathematical ideas (not just "mathematical games", far from it!), the intense battling of pseudoscience and related ideas, the invention of superb magic tricks, the love for beautiful poetry, the fascination with profound philosophical ideas (Newcomb's paradox, free will, etc. etc.), the elusive border between nonsense and sense, the idea of intellectual hoaxes done in order to make serious points... and on and on and on and on. Martin Gardner was so profoundly influential on so many top-notch thinkers in so many disciplines -- just a remarkable human being -- and at the same time he was so unbelievably modest and unassuming. Totally. So it is a very sad day to think that such a person is gone, and that so many of us owe him so much, and that so few people -- even extremely intelligent, well-informed people -- realize who he was or have even ever heard of him. Very strange. But I guess that when you are a total non-self-trumpeter like Martin, that's what you want and that's what you get."
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"Several decades passed by before I rediscovered the elegance, simplicity, and depth of his writing, and most importantly, the validity of his approach to mathematics. Then, in due course, I had the great pleasure of meeting him in his old age. He was nothing like the stern-looking man on all those book covers: in reality he was a sweet-natured, kind, wise and modest to a fault, with a twinkle in his eye, and a total joy to be with. While I can't say that Martin's columns or books steered the early course of my life, his extraordinarily diverse written legacy, his devotion to learning, his generous sharing of his toys, and his sheer decency, all conspired to reset my course in midlife.
He was also extremely egalitarian and generous with his time: he didn’t care if you were a prince or a pauper, if you had an interesting idea then he wanted to know about it, and he’d encourage you to get it in front of others. In a sense he was the original (mathematical) community organizer, at a time when it was neither profitable nor popular."
-- Colm Mulcahy  
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"Martin Gardner was an artist of mathematical writing. His work stands and will continue to stand the test of time. It is a springboard for others. I can gaze and contemplate his work over and over and see new things and create new ideas."
-- Tim Chartier
***************************
 "All of us who dare to aim our writing at 'the general reader' follow as best we can in Martin's footsteps. He is the Archimedes of mathematical writing."
        — Keith Devlin
***************************
and this:
"We glibly talk of nature's laws
but do things have a natural cause?
Black earth turned into yellow crocus
is undiluted hocus-pocus."

-- Piet Hein
used as the frontispiece to Martin's autobiography, "Undiluted Hocus Pocus"
***************************

To conclude, this wonderful, older and rare (14-min.-edited) interview with Martin was uploaded this week to YouTube... delightful:



Finally, CelebrationOfMind.org has this tribute page running this month in honor of the Gardner Centennial:
http://www.celebrationofmind.org/OCTOBER.html


Friday, October 17, 2014

"A New Universal Law"


Natalie Wolchover never fails to enthrall. Her latest piece at Quanta is on a "curiously pervasive statistical law" that connects math, physics and biology.  It's known as the Tracy-Widom distribution, after the founders who discovered that "Systems of many interacting components — be they species, integers or subatomic particles — kept producing the same statistical curve." In other words, similar to the bell curve, the Tracy-Widom distribution seems to operate universally, describing many complexly-interacting systems. Unlike the bell curve though, the tails are asymmetric in some manner relating to the universal nature of phase transitions, and phase transitions, the article notes, "are for statistical physicists 'almost like a religion.'" The article also notes that where physicists are often satisfied with "a preponderance of evidence," mathematicians want more rigorous "proof" of a relationship. Read more here:

http://www.simonsfoundation.org/quanta/20141015-at-the-far-ends-of-a-new-universal-law/


Thursday, October 16, 2014

A Mathematical Parable... and Ebola


I've shortened and simplified this post considerably. It wasn't even intended for the regular math-literate readership here who almost certainly know the classic story of grains of wheat accumulating on a 64-square chessboard; but rather for their possible math-phobic friends who need a more vivid understanding of the potential exponential nature of numbers-growth, in lieu of the Ebola story unfolding.

Ever since the virus spread (completely unlike prior decades) beyond the villages it was usually confined to, and especially since its spread beyond the shores of West Africa, some of us have had a more cautionary, skeptical view than the CDC's confident stance, because of the simple mathematics of the situation (combined with the fact that NO amount of medical protocols/regulations realistically offers 100% prevention of spread, given that humans who must carry out such protocols are imperfect, suffer lapses, make mistakes, are forgetful or tired or ill-trained, or in a hurry, etc.etc. (And that's no fault of theirs, that's just being normal humans, instead of machines). While the 70+ contacts of the Dallas Liberian victim might seem a manageable number, 300, 500, or 1000 potential contacts/exposures will not be easily manageable. (The fact that the virus doesn't spread through the air is lucky for us, but by no means precludes widespread infection.). Enough said:



[p.s., in a recent release the World Health Organization warned that before the end of this year there could be as many as 10,000 new cases of Ebola in Western Africa alone every week -- I'd be a bit surprised if that happened... but that IS the point of the above video, it could happen that fast.] Somewhere between calm and panic there is an appropriate state of alarm and alertness that the American public needs to find, to be prepared for the major disruption this epidemic, and consequent public health measures, could cause society. "Be prepared" is often a more trenchant maxim than "stay calm." Or, to put it a different way, the "precautionary principle" again takes hold (better to be overly precautious, than not precautious enough).

As an aside, in the short term, I'll say that my own confidence lies, not in our ability to necessarily control the spread of this disease, but rather in our ability to attain early diagnosis and more effective treatment for it, cutting the current 70% fatality rate significantly (but that too certainly isn't assured).


Wednesday, October 15, 2014

Grant Wiggins Presents "A Veteran Teacher Turned Coach"


 Chances are, if you read this blog, you've already seen this... because at last check well over ~300,000 500,000 had... a super piece from Grant Wiggins -- actually a posting he's taken from another educator/writer ("a veteran teacher turned coach") -- about a teacher becoming a student for a day (because we all forget too quickly what it's like!).
If somehow you've missed it, take a gander; it makes so much common sense, of the sort we often look right passed, with lots of suggestions (and lots of interesting comments as well):

http://tinyurl.com/oztopao

This morning, I noticed someone on Twitter responding to the post by mentioning that they knew a couple of classrooms where regular school seats were replaced with ball chairs -- I thought that was a fascinating idea, even if not always practical -- just an example of the simple (and perhaps healthier?) outside-the-box thinking the article encourages.
By the way, Grant promises that the writer will be doing a follow-up to the piece.

ADDENDUM: Michel Reed, the teacher who mentioned the above ball chairs example, later tweeted this photo of such a classroom :-):




Monday, October 13, 2014

A Puzzle to Kickstart the Week


A sweet, simple puzzle to kickstart your week, taken directly from a recent Brian Brushwood "Scam School" episode; and it's one of those grand facepalm-type puzzles, you'll kick yourself for, IF you don't solve it:

Multiply together a long sequence as follows:

(a-x) X (b-x) X (c-x) X (d-x)...... (y-x) X (z-x)  i.e., utilizing ALL the letters of the alphabet once

What will be the end product of this sequence multiplied out???

["Scam School" has been around a long time, but if you've missed it by any chance, you can check out it's many entertaining videos HERE.]
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Answer = 0 ...just before the final two sequence entries listed (but not shown), would be (x-x)

Sunday, October 12, 2014

The Tao of Tao ;-)


Terry Tao from his blog, "What's New":

"The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field."


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Friday, October 10, 2014

"Right Now the Math Still Favors the Virus"


This started out as a link, but turned into a bit of commentary...

Society often walks a fine line between panic and laxity over any potential crisis. Ebola is no different, and I understand why the medical community took a somewhat pollyannish view toward it in pronouncements to the public. But the numbers involved, our highly mobile society, and the fact that complex containment protocols on paper will not be completely carried out (because they never are) calls for a more sober assessment. This Washington Post article, "The Ominous Math of the Ebola Epidemic," offers a somewhat more realistic view of the numbers and potential exponential growth involved:

http://tinyurl.com/pknyxun

I'm not terribly confident of success in "containing" Ebola in the short-term, but I do suspect we will find effective treatments for managing it in those diagnosed early, and thus significantly cutting the fatality rate (the successes we've already had are quite encouraging). In trying to avoid panic the CDC and others bent over too far in the direction of preaching calm and confidence. And the problem with that scenario is the backlash it may lead to.
This country already has a perturbingy large, vocal anti-science component in it. If the medical community misplays the Ebola epidemic it will add ammunition to their arsenal: 'see, we can't trust scientists; they don't really know what they're doin'. The anti-vaccers, anti-evolutionists, climate-denialists etc. will have a field day, long-term, if, after all the calls for calm/trust, the epidemic spreads widely. I'm almost as concerned over that as I am over the medical crisis itself.
In so many ways of course we have a wonderful medical community in this country, especially when it comes to medical emergencies. One just hopes they're not already in over their head in this case. We are probably already in the stage of swinging from calm to panic (there is so little middle-ground):

http://boingboing.net/2014/10/09/black-man-sneezes-on-plane-t.html

I don't know if the medical community could've done any better in their public communications -- they were caught between a rock and hard place... walking a tightrope... over a mass public that little understands how real science operates.
Anyway, to those on the front lines, where so much courage, care, commitment, and selflessness are now required, I sit in awe of you.

ADDENDUM:  highly-respected Laurie Garrett has now posted a piece that I think pretty well nails the proper cautionary stance/tone needed in this circumstance, while addressing "five myths about Ebola" (glad to see her do it!):

http://www.newsday.com/opinion/five-myths-about-ebola-laurie-garrett-1.9489635


Thursday, October 9, 2014

"The Upside-Down World Paradox"


Given my fondness for paradox, just linking today to this quirky, fun little (non-mathematical, but logical) post about the 'upside-down world game' (an offshoot of 'the liar paradox'):

http://m-phi.blogspot.com/2014/10/the-upside-down-world-paradox.html

The blogger's seven-year-old daughter enjoys the game in the post, so if you have young kids maybe they will as well.




Wednesday, October 8, 2014

Of Friends, Face-to-Face and Virtual


Always love to see math making it into the popular or mainstream press, so nice to see this Maria Konnikova article on Dunbar's Number and social networks (and the new ramifications of digital social media) in the New Yorker:

http://tinyurl.com/mx4qgmx

As the article states, "...no one really knows how relevant the Dunbar number will remain in a world increasingly dominated by virtual interactions," or as Dunbar himself is quoted, “We haven’t yet seen an entire generation that’s grown up with things like Facebook go through adulthood yet.

There are potential neuroscience, and in turn social, implications to all this reliance on virtual interaction. It is, for now, a sort of grand, ongoing experiment, outcome unknown.

Tuesday, October 7, 2014

Happy Birthday Neil Sloane and OEIS!


Wonderful Alex Bellos piece in The Guardian today, on Neil Sloane and the OEIS (Online Encyclopedia of Integer Sequences) he founded. Fascinating reading:

http://tinyurl.com/n6wcev4

I was happy to learn of the Kolakoski sequence which combines a number sequence with self-reference (one of my favorite topics), and of which Bellos writes, "Mathematicians drool over this sequence."
 The OEIS has been around as a go-to resource for mathematicians of all stripes for 50 years now, and today includes some 250,000 number sequences, while still growing, according to the article, at a rate of about 40 new sequences each day! Some sequences can be quite creative of course, and open up interesting, difficult-to-solve questions.
Rutgers' Doron Zeilberger goes so far as to say that the OEIS has made Sloane the world's most influential mathematician!
Lots more in the article, including a video.



Sunday, October 5, 2014

Couple of Physicists' Views


Sunday thoughts on science...:

"Science in its everyday practice is much closer to art than to philosophy. When I look at Gödel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as a Chartres cathedral. The proof destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas."
-- Freeman Dyson in "Nature's Imagination"

...and from another physicist, a related view:

"…to sum up, science is not about data; it's not about the empirical content, about our vision of the world. It's about overcoming our own ideas and continually going beyond common sense. Science is a continual challenging of common sense, and the core of science is not certainty, it's continual uncertainty -- I would even say, the joy of being aware that in everything we think, there are probably still an enormous amount of prejudices and mistakes, and trying to learn to look a little bit beyond, knowing that there's always a larger point of view to be expected in the future."
-- physicist Carlo Rovelli in "The Universe" edited by John Brockman

or, just perhaps, Richard Feynman summed it up succinctly ;-):

"Physics is like sex: Sure, it may give some practical results but that’s not why we do it.”


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]


Thursday, October 2, 2014

How Would American Kids Do?


An interesting pair of successive tweets from Alexander Bogomolny this morning, showing a Hong Kong 1st-grade "admissions" test question :








Wednesday, October 1, 2014

Bayes-mania


Seeing an awful lot written on Bayesian ideas in the last year (and week!).
Jason Rosenhouse uses the Monty Hall problem and a NY Times article as a launching point for a discussion of the subtlety of Bayes here:

http://scienceblogs.com/evolutionblog/2014/10/01/a-poor-description-of-the-monty-hall-problem/

Rosenhouse takes the Times' article to task, and ends simply with:
"Applying statistics correctly is hard, even for people with professional training in the subject. But the problems are found in the complexity of real-life situations, and not in the underlying philosophical approaches to probability and statistics."
(Rosenhouse's 2009 book on the The Monty Hall problem is great, by the way, if you've never read it -- yes, an entire volume on that one problem)

Meanwhile, The Guardian also presented Bayesian statistics this week: http://tinyurl.com/ojjazs4

And in other statistics briefs, Jeff Leek tries to come to the defense of much-maligned p-values here:
http://simplystatistics.org/2014/09/30/you-think-p-values-are-bad-i-say-show-me-the-data/ 

Tuesday, September 30, 2014

True Grit


Another TEDTalk today. This is actually from 2013, but I just saw it getting passed along #MathChat/#EdChat circles today. It's about success in math and beyond. If you missed it, set aside 6 mins. for it:




Monday, September 29, 2014

Mathematics a-a-and Sex


Don't know how I missed this very fun TEDTalk from a few months back, but h/t to Cliff Pickover for tweeting it this weekend:



The speaker, mathematician Clio Cresswell, actually has a book out with the same title ("Mathematics and Sex"):
http://www.amazon.com/Mathematics-Sex-Clio-Cresswell/dp/1741141591


Sunday, September 28, 2014

Sunday Reflection on Big Data


Living in a digital web....

"…faceless Big Data's predictions are hard to opt out of.
"One Minnesota man learned that a few years ago. He stormed into a suburban Minneapolis Target department store, demanding to speak to the manager. 'My daughter got this in the mail,' the man explained. The manager took a look at what the customer had brought. It was a Target mailer, like millions sent out every year, addressed to the man's daughter. This one looked harmless enough, with pictures of adorable infants, baby furniture, and maternity wear.
" 'Are you trying to encourage her to get pregnant?' the man sputtered. His high-school age daughter was unmarried.
"The manager apologized and said he'd look into it. When he did, he learned that Target was using predictive analytics. It aggregated the information it had on its customers -- from visits to the website, purchases in bricks-and-mortar stores, calls to customer support, and the use of coupons or rebates. Software parsed this haystack of data in order to make specific, actionable forecasts of individual customers' future behavior.
"One secret initiative was to predict which customers were pregnant. Expectant mothers have to buy a lot of products that they may never have purchased before. With novelty comes indecision. This makes mothers-to-be receptive to advertising discounts, and anything else that might nudge them in the direction of buying at Target. A customer who comes to depend on Target while expecting a baby may decide to do her grocery shopping there -- for decades to come.
"Target's pregnancy predictions were much more accurate than random guessing, though of course not 100 percent certain. A few wrong guesses were acceptable. The awkward exception is when a customer gets really, really upset by a wrong prediction.
"A few days later, the manager called back the irate customer to apologize a second time.
" 'I had a talk with my daughter,' the customer said. 'It turns out there's been some activities in my house I haven't been completely aware of. She's due in August. I owe you an apology.'"
"Witness the new human condition. A department store's software can guess that a woman is having a child; her own father can't. Should we marvel at how clever our algorithms are, or at how bad we are at listening to and understanding our fellow beings?"

-- from "Rock Breaks Scissors" by William Poundstone

...and I now have an interview up with Mr. Poundstone over at MathTango

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]