Number Recognition

Ever think about just how varied individual handwriting is... how many different ways there are to write an "a," as well as every other letter... or number for that matter???

To keep it simple think about just the numbers 0 - 9, and the different ways they may be written, and yet immediately recognized. In fact, we of course have automatic equipment in the Post Office dedicated to do just that: automatically read Zip Codes, in all sorts of handwriting!

Tim Chartier addressed mechanical digit-reading in a recent post here:

http://forum.davidson.edu/mathmovement/2010/08/24/digit-recognition-with-pythagoras/

(Don't) Walk This Way

The  simple (bust-a-kneecap) geometry of high heels is elucidated over at "mathspig" blog (...the operative term here is "SPLAT"):

http://mathspig.wordpress.com/2010/08/25/killer-heels-that-kill/

Sent out especially for the blogger Isis the Scientist and her readers ;-)

here's some more doozies:

http://tinyurl.com/2eps8nz

...and here of course, some sensible footwear:

http://tinyurl.com/23gxp62

Proof? ;-)

A golden oldie...:

Suppose:

a + b = c.

This can then be equivalently written as:

(4a – 3a) + (4b – 3b) = 4c – 3c

After reorganizing (adding/subtracting from both sides):

4a + 4b – 4c = 3a + 3b – 3c

Take the constants out of the brackets:

4(a + b – c) = 3(a + b – c)

Remove the same term left and right:

4 = 3

Ta-daaaaaaa!!

Fields Medalists... A Breed Apart

Interesting article on the recently-awarded Fields Medals (given "for outstanding mathematical work completed before the age of 40"), and great mathematicians more generally:

http://www.openthemagazine.com/article/nation/masters-of-the-universe

There is, the article notes, an "identification of mathematics with prodigies and troubled minds" which "is reinforced by the few books and movies that have struck a chord with people at large."

While it's true that those at the highest rarified levels of mathematics are often exceptional or quirky individuals, nonetheless, the stereotype painted by popular media is likely overdrawn.

A Plug For Raymond Smullyan

I was recently re-reading an old work by mathematician/logician Raymond Smullyan, and it occurred to me how relevant many of his writings are today amidst the sudden interest in the P vs. NP problem.

Smullyan is retired, in his 90's now, and it also occurred to me (and I hope this doesn't sound morbid) that I don't know how much longer he'll be around enlightening us. His friend and even-more-famous colleague Martin Gardner of course passed away earlier this year in his mid-90's, after an incredibly productive life. Gardner's death was one of the inspirations for me starting this blog, and I feel I ought acknowledge Smullyan's contributions while he is still among us...

Although (like Gardner) he has written a fair amount of recreational mathematics, Smullyan is probably even better-known as a logician dealing with more abstract issues (recursion, self-reference, paradox) that underlie mathematics, and that when resolved, have significant application. His writings have always been creative, entertaining, original, and generally accessible to lay readers (though probably lacking the "zing" and universality of much of Martin Gardner's output).

His prolific book listings on Amazon here:

http://tinyurl.com/33325zk

Smullyan's pursuits also range across astronomy, music, magic, mysticism, and Taoist philosophy, and interestingly he seems to find a measure of unification in Taoism for the abstract mathematical paradoxes/puzzles he ponders (I always find interesting the division between those serious mathematicians who are deeply drawn toward mysticism and those who are not!).
If you're not familiar with Smullyan's work, I'd recommend getting to know him; especially if the whole P vs. NP hoopla intrigued you.

Wikipedia entry for Smullyan here: http://en.wikipedia.org/wiki/Raymond_Smullyan

...and one of his fans has put up a MySpace page dedicated to him as well:  
http://www.myspace.com/raymondsmullyan 

Like Gardner, he is another American gem!

ICM Review

A nice commentary on the recently-concluded "International Congress of Mathematicians" Conference (where Fields Medals were awarded) in India from Alex Bellos here:

http://alexbellos.com/?p=1337

The Congress meets once every 4 years, and is " the largest, most prestigious and most traditional event in maths;" Bellos adds "no other scientific discipline has an event as big and with as much historic resonance."

Attendees have been blogging and tweeting about the Conference since it opened; Bellos here summarizes a few highpoints (and promises more in follow-up post).

Book Review (of Chaitin & Rucker)

 Fine older review by Jaron Lanier of two books that have been out for awhile, and worth looking into if you've never read them: "Meta Math! The Quest For Omega" by Gregory Chaitin, and "The Lifebox, the Seashell, and the Soul: What Gnarly Computation Taught Me about Ultimate Reality, the Meaning of Life, and How to Be Happy" by Rudy Rucker:

http://www.americanscientist.org/bookshelf/pub/two-philosophies-of-mathematical-weirdness

Toward the end of the lengthy review, Lanier summarizes thusly:

"These two books are near opposites even though they appear to explore similar topics. Chaitin loves negative results and is thrilled by the prospect of future generations of mathematicians finding ever weirder math. The Chaitinesque intellectual future will be eternally youthful and anarchic. Neither mathematicians nor computer scientists will settle down into a single preferred pattern of thought. Rucker, in contrast, is reaching as high as he can to try to use available computer science and math metaphors to create a new, comprehensive, multidisciplinary sensibility. The Ruckerian future is one in which new guiding explanatory ideas will connect all areas of intellectual curiosity."

(Of these two works, I enjoyed Chaitin's volume more than Rucker's, but it's largely a personal preference, and I've enjoyed some of Rucker's other books more than this particular one.)

And a bit more current, Murray Bourne's latest "IntMath Newsletter" is up here (with a little something for everybody who enjoys math):

http://tinyurl.com/2euddty

Neuro-mathematics...

The human brain is probably the 'final frontier'...

Blogger Jason Goldman reviews some of what we know/believe about mathematical dysfunction of the brain here, based in part on case-studies of patients with different brain lesions and on fMRI studies:

http://tinyurl.com/2dz2m3a

This was actually the 4th in a series of related posts Jason did that can be looked up here:

http://tinyurl.com/266o57k

This Is The Title of This Morning's Post

 And, another view of recursion here:

http://xkcd.com/688/

An "Incredible Time For Mathematicians"

Hmmmm... looks like there may be a lucrative future in this thing we call mathematics:

http://www.livemint.com/2010/08/24001310/Math-becomes-fashionable-focu.html?h=A1

....if the movie "The Graduate" were produced today, I guess the one-word that might get passed along to Ben wouldn't be "plastics," but "algorithms."

And in a slightly related vein, "The Atlantic" has a current article relating to applied algorithms:

http://www.theatlantic.com/science/archive/2010/08/when-computers-predict-crime/61870/

Finally, on a side-note, the 29th "Math Teachers At Play" Blog Carnival is up for your perusal, with a variety of topics, here:

http://numberwarrior.wordpress.com/2010/08/23/math-teachers-at-play-29/

Math In the Arts...

Ivars Peterson interestingly reviews a play he recently took in in India, "A Disappearing Number," centered around mathematics and especially the life of Indian prodigy Ramanujan:

http://mathtourist.blogspot.com/2010/08/english-play.html

The play originated in 2007 (and won several "Best New Play" awards that year), and has been staged in many U.S. and worldwide venues, virtually always to rave reviews. The NY Times called it "lucid, dynamic and continuously engaging," and then continued:
"It’s not fundamentally about numbers, either, but about the search for meaning and the consoling satisfaction of finding the patterns that define and describe both the physical universe and individual human lives."

Another reviewer simply calls it "a thrilling, thrilling, thrilling play." And there are many additional reviews on the Web, including this 7-minute NPR podcast review: 

http://www.npr.org/player/v2/mediaPlayer.html?action=1&t=1&islist=false&id=128513493&m=128513513

It's nice to know that the wonder/beauty of mathematics has been translated to the acting stage so effectively!
And this October the play will be broadcast live across the U.S. (and the world) in select theaters. If you are near one... and you are a reader of this blog... you ought probably look for it!

The Math of DNA Matches

CSI, DNA, and what else... MATH!

...or

Why a 'DNA match' in forensics may NOT be all that it seems... what it does and doesn't mean mathematically speaking:

http://plus.maths.org/content/os/issue55/features/dnacourt/index

Men... Take Heart...

Hmmmm... another study at the intersection of set theory and evolution ;-))) :
(There are apparently more beautiful females around than ever before for we unsightly male counterparts!)

http://www.telegraph.co.uk/news/newstopics/howaboutthat/5912250/Women-getting-more-beautiful-say-scientists.html

The report begins thusly:

"Researchers found that attractive women have more children than their less attractive counterparts and that a higher proportion of those children are female.
 Once those daughters become adult they tend to be good looking themselves and so the pattern is repeated as women over the generations become steadily more aesthetically pleasing.
As attractive couples are less likely to have a boy than a girl, men, in contrast, remain as aesthetically unappealing as their caveman ancestors, the scientists claim." [emphasis added]

100-Meter Race Problem

Just a simple word problem to start the week off; adapted from "Futility Closet" (http://www.futilitycloset.com/2010/08/02/the-handicap/):

Michael challenges his brother Ryan to a 100-meter foot race. When Ryan crosses the finish line Michael has only covered 97 meters.
Michael challenges Ryan to a re-match, but this time Ryan has to start 3 meters behind the starting line.
Assuming the brothers run at the identical speed they did in the first race, who will win?

answer down below...

Or, if you want a bigger challenge, Matt Parker asks on Twitter recently:

"Is any number palindromic in bases 2, 3 and 5? I know @stecks checked up to a million. But still, it feels like there should be one..."

...have at it??? ;-)
.
.
.
.
.
.
.
.
.

Solution (1st problem):

Ryan wins again. Ryan is able to cover 100 meters in the time it takes Zachary to run 97 meters. By starting 3 meters behind the starting line, Ryan will be dead-even with his brother at the 97-meter mark, at which point he (Ryan) will then pull ahead for the victory!

"Patterns & Beauty: The Mathematics of Music"

Nice 6-minute video on the mathematics of music:

http://videos.howstuffworks.com/hsw/11978-patterns-and-beauty-the-mathematics-of-music-video.htm

(...if you search "mathematics" on this site, "HowStuffWorks," there are 270 more offerings!)

Evolution of the 'Recycle' Logo

Ivars Peterson of "The Mathematical Tourist" reviews various designs for the ubiquitous "recycle" symbol that most of us are used to seeing, pointing out that the original logo was based on a Mobius strip design, but was followed by several 'mutant' designs that lacked the Mobius feature:

http://mathtourist.blogspot.com/2010/08/recycling-arrows.html

Fox News... Math Illiterate

How NOT to graph! (...or just an example of "lies, damned lies, and statistics)":

http://scienceblogs.com/corpuscallosum/2010/06/tricks_with_graphs_deliberate.php#more

You may also want to check out this TED Talk for a chuckle, as well:

http://www.ted.com/talks/lies_damned_lies_and_statistics_about_tedtalks.html

I'll Drink To That!

Tiny bubbles....

Back in 2002, some physicists won an "Ig Nobel Prize" ;-) for their study of the mathematics of beer bubbles:

http://tinyurl.com/2elosmd

I can hardly wait to learn this year's illustrious winners come Sept. 30 at Harvard.

Heron's Formula

 Pythagoras gets all the publicity but Heron was no slouch either... Heron was another ancient who is credited with devising a formula for computing the area of a triangle from knowing only the lengths of the 3 sides involved. The formula appears in a few different forms, 2 common ones below (a, b, and c represent the 3 sides of a triangle, and "s" is the perimeter value for same).

more on Heron's formula here:

http://en.wikipedia.org/wiki/Heron%27s_formula

An offshoot of Heron's formula is Brahmagupta's formula for the area of any 'cyclic' 

quadrilateral (one that fits inside a circle):

 

or

     _________________________________

1/4 √(a+b+c-d) (a+b-c+d) (a-b+c+d) (-a+b+c+d)

Surprise Me!

I suspect this post on 'mathematical surprises' from Dave Richeson may get passed around a lot:

http://divisbyzero.com/2010/08/18/mathematical-surprises/

He lists a number of varied mathematical discoveries over the years that surprised the math community and offered new perspectives; his commenters chime in with still more examples.
I think this gets at what mathematicians love about their subject: it is full of never-ending surprises out of the blue; unlike the stereotyped image (all too often held by lay people) of math as boring, tedious, routine, cut-and-dry....