Saturday, July 31, 2010

Bzzzzzzzzzz....

Image via Wikipedia
Math-types have previously noted the oddity not only of cicadas emerging in buzz-resounding throngs every 13 or 17 years (depending on species), but the fact that those are PRIME numbers! Old article on these prime-number outbreaks, including a proposed explanation offered by the late, great Stephen Jay Gould here:

...and another piece on the subject:

Friday, July 30, 2010

Words and Numbers...

The Berry Paradox comes in a few different forms. Here is one of the simplest examples of it for easy comprehension:

Name "the smallest possible integer NOT definable by fewer than twelve words".

It's easy to imagine examples (definitions) that DON'T work:

the number of toes on your right foot [only 8 words, to define the number "5"]
the number of inches in a foot [7 words to define the number "12"]
the number of seconds in a million years [8 words to define whatever the number would be]

easy enough...

The problem arises however (if it isn't already obvious), that if you did somehow find an integer that you could only define with a sentence of 12 or more words, and it WAS the smallest such definable integer, THEN, IT could be accurately designated (defined) by the original 11-word sentence above ("the smallest possible integer not definable by fewer than twelve words") --- thus a self-referential contradiction!

...Another example of where mixing language/semantics with numbers/mathematics proves vexing, throwing light on illogical ambiguity or paradox within language. Much of the history of underlying problems with mathematical consistency entails issues of self-reference.

More on the Berry paradox via Wikipedia:

Thursday, July 29, 2010

Update...

"Pat's Blog" took the algebra problem I posed this morning and expanded on it (infinite nested radicals) for further discussion/learning here:

All of which reminded me in turn of an older post by "Division by Zero" blog on "continued fractions" (not exactly the same thing, but similar) here:

http://divisbyzero.com/2008/12/09/a-new-continued-fraction-for-pi/

And as another tie-in, I'll just note in passing that some of the famous equations discovered by Ramanujan (who I'd just referenced on Monday), fall into the above categories.

10 Dimensions

Superstring theory from physics requires there be at least 10 dimensions to the Universe. It's hard enough for most of us to comprehend 5 or 6 dimensions, let alone 10. This YouTube video tries to confer a sense of how 10 dimensions can exist:

Algebra Problem

Solve for x:

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And since the infinite series on right = 3, we can substitute to yield 9 = x + 3, therefore x = 6

Wednesday, July 28, 2010

"Calculus Diaries"

Keen science blogger/writer and latecomer-to-mathematics Jennifer Ouellette has a new book coming out Aug. 31, "The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse" (...say that 3 times quickly!), which could be very entertaining, as well as instructive for any who struggled through high-school or college calculus:

http://www.jenniferouellette-writes.com/calcdiaries.html

http://www.underthemicroscope.com/blog/cocktail-party-calculus

(available for pre-order)

"Pattern Decorrelation"

THERE'S a term I was previously unfamiliar with... It refers, in this instance, to certain complex mechanisms in the brain which have been mathematically modeled to account for workings of the olfactory system. Read about it in this Web article entitled, "Explaining Scent Mathematically," referencing some recent corroborative work of Swiss neurobiologists and mathematicians, which helps explain how olfactory neural circuits are structured so as to discriminate different odors (decorrelation also helps explain the neural circuitry of the visual system as well):

http://www.physorg.com/news199367766.html

Wikipedia entry on "decorrelation" here: http://en.wikipedia.org/wiki/Decorrelation

Tuesday, July 27, 2010

Fast-growing

A month ago I referenced the Facebook mathematics interest group with over 56,000 members. Today, it stands at 74,000 members!:

YouTube Intro to Cantorian infinity for the layman:

Monday, July 26, 2010

Got math on your mind?... A plethora of Tweeters with math-related interests, are listed by Listorious here:

http://listorious.com/search/people?q=math&rpp=25&page=1

And TLists tallies Tweeters' "math lists" here:

http://www.tlists.com/lists/search/?query=math

The Ramanujan Journal

Image via Wikipedia

You can't very well write any sort of self-respecting math blog without at some point posting something about the incredible Indian mystical mathematician Ramanujan.  I won't quite do that now, but will note a journal I only recently came across that is specifically dedicated to Ramanujan's contributions to math, "The Ramanujan Journal," linked to below (and you can follow links on upper right of page to sample free content therefrom):

http://www.springer.com/mathematics/numbers/journal/11139

(There are any number of good books and Web pages out there devoted to Ramanujan as well.)

Saturday, July 24, 2010

Let's Go to the Movies

An Amazon list of movies about mathematics here:

Friday, July 23, 2010

There's Recursion and Then There's REEEcursion

Recursion in language and logic is a favorite topic of mine, but I'd never really thought about recursion in geography! This math blogger has:

http://divisbyzero.com/2010/07/19/nested-nations/

Keith Devlin Changes His Mind

I clicked to this "Edge" page to read a blurb by Keith Devlin on changing his mind about mathematical Platonism:

http://www.edge.org/q2008/q08_4.html

....but then realized the entire section (from 2008) was interesting, on science-types answering the question, "What have you changed your mind about?" It starts here:

http://www.edge.org/q2008/q08_1.html

and the 150+ contributors are indexed here for easier sorting through:

http://www.edge.org/q2008/q08_index.html#contributors

ADDENDUM: the latest book co-authored by Devlin, "The Computer As Crucible" (on experimental mathematics), was just reviewed here:

http://tinyurl.com/23yg94f

Thursday, July 22, 2010

A Few Number Games/Puzzles

Ken-Ken on the Web here:

http://www.kenken.com/

Shikaku (and some other games) here:

http://www.puzzle-shikaku.com/

Wednesday, July 21, 2010

Just a Logic Problem

3 sealed boxes face you on a table. You are told one contains gold while the other two are empty. Each box has printed on it a written clue regarding its contents, HOWEVER only one statement is accurate, while the other two are false. Which box has the gold?

1) No gold in here     2) No gold in here   3)  The gold is in box 2

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answer: Box 1 contains the gold

Infinity and More (or Less)....

David Foster Wallace was an amazing, prolific, creative, award-winning novelist ("Infinite Jest" his greatest popular success), who died all too early at his own hands battling depression. He also wrote some non-fiction offerings, one of which, oddly-enough, was a treatise on the history of math and the concept of infinity, for W.W.Norton's "Great Discoveries" book series.

I was looking forward to reading this 2003 take of a bright, engaging, non-mathematician writer/novelist titled "Everything And More: A Compact History of Infinity," but then came across highly critical reviews of the volume on the Web by mathematicians I much respect (the book is checkered with flaws and inaccuracies). This cooled but didn't expunge my interest in the volume. And now having read it I am amazed that this English-major/novelist, could even have tackled (...let alone in such an offbeat style) the heavy math-cerebral subject-matter surveyed in this dense 300-page book (which the author, accustomed to writing 1000-page tomes, calls a "booklet") --- so, even with mistakes-and-all, I can't help but (with a grain of salt) recommend it, as unlike any other math book you are likely to encounter; written in an informal and conversational tone about ideas that are utterly UN-informal and UN-conversational (...and the multitudinous footnotes are virtually as fascinating as the main text)! Here, a few less-negative reviews:

http://www.powells.com/review/2003_12_01.html

http://www.popmatters.com/books/reviews/e/everything-and-more.shtml

http://www.nytimes.com/2003/11/16/books/review/16PAPINET.html

Definitely not everyone's cup-o-tea, but still I think an intriguing, complex, quirky read by a brilliant, troubled mind from the humanities, covering material that has troubled a great many astute minds in the world of mathematics.

Tuesday, July 20, 2010

SAT Test

Does the SAT Math test favor males? Some thoughts:

http://tinyurl.com/32lmcsc

And those preparing to take the math SAT at some point, may wish to follow these blogs:

http://satmathblog.com/

http://www.sat-tutors-blog.com/

....and some practice available here:

http://www.majortests.com/sat/math-skills.php

Soap Bubble Math

A few links to the mathematics and science of soap bubbles:

http://www.mathman.biz/html/bubbles.html

http://math.berkeley.edu/~hutching/pub/bubbles.html

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

....and some old television video of bubble entertainer Tom Noddy in action:

Monday, July 19, 2010

GĂ¶del's Theorem: A Short Version!

Interesting analogy from the wonderful Raymond Smullyan to explain GĂ¶del's theorem:

http://blog.plover.com/math/Gdl-Smullyan.html

(corrected URL ^)

Random Samples and 'WEIRD' People

35+ years ago in college it troubled me that among the first lessons learned in basic statistics was the necessity for employing "random samples" in research studies, even while in reality samples I saw in academic literature were virtually never truly random (indeed, the very concept of a really "random sample" is questionable).
35 years later, nothing has changed, but a journal paper by some behavioral psychologists, now making the rounds, entitled "The Weirdest People in the World," is highlighting this concern over how representative research conclusions from limited samples can be for humans in general.

abstract here:

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1601785

fuller citation here:

Henrich, J., Heine, S. & Norenzayan, A. (2010) "The Weirdest People in the World?" Behavioral and Brain Sciences. [PDF] [Audio File Part I] [Audio File Part II] [Coverage in Science]

Neuroanthropology blog has addressed the paper in an extensive blog post:

http://tinyurl.com/2bsazch

"WEIRD" people, BTW, are those from "Western, Educated, Industrialized, Rich and Democratic societies."

I'm just glad to see the issue getting the attention it deserves (and I'd add that while the criticisms raised are being principally pointed at behavioral and psychology studies, even biological/physiological research in humans is not entirely immune from such questions/concerns).

Sunday, July 18, 2010

Hmmmm....

@standupmaths on Twitter recently tweeted the following:

"Why is every prime squared (p > 3) always one more than a multiple of 24? [ 5x5-1=24,7x7-1=48...]"

I think I've seen this demonstrated true for all Mersenne primes, but am less clear if it is proven (or even provable) for ALL primes? Can anyone confirm...???

Saturday, July 17, 2010

Proof By Visualization

(1 + 2 + 3 +.... + n)2  =  13  +  23  + 33  +.... + n3

visualization of the proof here:

http://tinyurl.com/2955mm

Friday, July 16, 2010

Arithmetic Puzzle

Richard Wiseman's latest 'Friday puzzle':

http://tinyurl.com/36eaf3f

Great Internet Mersenne Prime Search (GIMPS)

For anyone who doesn't already know of it, the GIMPS project links up computers worldwide to search for the next Mersenne Prime number (of the form 2^n - 1) --- anyone with a personal computer can participate, and even win cash rewards (and all the work is done on your computer in the background, while you wile away the hours surfing the Web and eating pizza!).

If interested check it out here:

http://www.mersenne.org/

Some history/background on Mersenne primes:

http://primes.utm.edu/mersenne/index.html

or the Wikipedia account:

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

...and more updated info on primes here:

http://primes.utm.edu/largest.html

Finally, the largest current Mersenne prime (at the moment):  243112609 -1

Thursday, July 15, 2010

"The Mathematical Universe" of Max Tegmark

MIT Cosmologist Max Tegmark's notion of 'The Mathematical Universe':

http://www.americanchronicle.com/articles/view/24386

(...a Platonist view of mathematics)

Wednesday, July 14, 2010

Math Teachers Playing...

Latest "Math Teachers At Play" blog-carnival is posted here:

http://tinyurl.com/2dsyzfl

Somethin' for everyone...

Collatz Conjecture

A conjecture that is easy to state, easy to understand... but, difficult to prove:

http://jpcameron.com/blog/?p=47

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

Tuesday, July 13, 2010

Taking Math to the Streets

In merry old Britain, a new form of street entertainment has number-loving fans involved in "math busking" to engage interested passersby in the joys and wonder of mathematics:

http://tinyurl.com/37tntmh

Could America be next?....

Origami and Math

The amazing Robert Lang merges math and the art of origami in this 16-minute 2008 TED Talk:

Monday, July 12, 2010

Beautifully Simple Algebra

Quoted directly from Alfred Posamentier's "Mathematical Amazements and Surprises":

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"You are seated at a table in a dark room. On the table there are twelve pennies, five of which are heads up and seven of which are  tails up. (You know where the coins are, so you can move or flip any coin, but because it is dark you will not know if the coin you are touching was originally heads up or tails up.) You are to separate the coins into two piles (possibly flipping some of them) so that when the lights are turned on there will be an equal number of heads in each pile."

"Your first reaction is 'you must be kidding!' How can anyone do this task without seeing which coins are heads or tails up? This is where a most clever (yet incredibly simple) use of algebra will be the key to the solution."

Posamentier Continues:

"Let's 'cut to the chase' (You might actually want to try it with 12 coins.) Separate the coins into two piles, of 5 and 7 coins each. Then flip over the coins in the smaller pile. Now both piles will have the same number of heads! That's all! You will think this is magic. How did this happen. Well, this is where algebra helps us understand what was actually done."
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Explanation: ...At the start, there are 5 heads showing among 12 coins. After separating into piles, let's say the 7-pile now has "h" heads. The 5-pile then has "5 - h" heads, and "5 - (5 - h)" tails (or, just "h" tails). Once you flip the entire smaller pile, all the tails ("h" of them) become heads, and all the heads become tails. Thus you are left with "h" heads in the "5-pile," the same as the number in the "7-pile." Whaaa-laaahhhh! Math is a beautiful thang!!

Sunday, July 11, 2010

They Predicted It!

Some British mathematicians used Graph Theory to predict ahead of time that Spain would beat the Netherlands for the soccer World Cup:

http://www.qmul.ac.uk/media/news/items/se/31612.html

Saturday, July 10, 2010

Applied Mathematics

Mathematics + astrophysics + supercomputers, being used to study/model early Universe:

http://www.physorg.com/news197816941.html

Friday, July 9, 2010

Sing-A-Long...

Found on the internet (lyrics by Georg Cantor ; - ) ....

The world's longest song (sung to tune of "99 bottles of beer on the wall"):

"Aleph-nought bottles of beer on the wall,
Aleph-nought bottles of beer,
You take one down, and pass it around,
Aleph-nought bottles of beer on the wall...."

(...Repeat till you pass out)

[ often heard being sung by guests at the Hilbert Hotel! ;-)) ]

New Offering From Marcus du Sautoy

The books keep comin'....

British mathematician Marcus du Sautoy has a new book out:

"The Number Mysteries: A Mathematical Odyssey Through Everyday Life"

Amazon calls it the "ultimate handikit to mathematics."
And Alan Davies reviews it as "Mind-bending, fascinating and useful too. Maths didn't used to be this much fun."

Thursday, July 8, 2010

Thomson's Lamp

"Thomson's Lamp" is an example of a "supertask," a category of paradox that involves an infinitely-divisible task. One form of the paradox runs as follows: You have a lamp that can be turned on and off with a toggle switch. At the start the lamp is turned on for exactly one minute, at which point it is turned off for exactly .5 mins., and then turned on for .25 mins., and then off for .125 mins.... and so on.

The question is, at the two-minute mark is the lamp on or off?  Common-sensically, and practically, one might expect there is a simple, correct mathematically-calculable solution to the question --- after-all, at the two minute mark the lamp MUST be EITHER on or off! But in fact, we are dealing with an infinite sequence (1 + 1/2 + 1/4 +1/8 +1/16 +....), and as such there is NO one single right answer --- different mathematical arguments/solutions can be logically made, and even semantically the problem is unsettled. In part, the answer depends on how fast one assumes the (undetailed) turning on and turning off action itself takes --- is it 'instantaneous' (requiring no amount of time), or does it take some finite amount of time (say perhaps, with the speed of light as a limiting factor)? In short, Thomson's Lamp is a fun thought exercise (involving infinity) that oddly evades any proven solution.

"A New Kind of Science"

Some reviews of Steven Wolfram's 2002 tome "A New Kind of Science":

http://www.maa.org/reviews/wolfram.html

http://www.nybooks.com/articles/archives/2002/oct/24/is-the-universe-a-computer/

http://books.slashdot.org/books/02/05/21/146210.shtml?tid=134

http://www.nytimes.com/2002/05/11/arts/11WOLF.html?pagewanted=1

Wednesday, July 7, 2010

Intuition, Math, and the '2 Children' Problem

Extensive discussion of (including many comments on) the 'boy born on Tuesday' riddle here:

http://tinyurl.com/283cawp

"Rejecta Mathematica"

For all you rejects out there, "Rejecta Mathematica" is the title of an apparently real (somewhat experimental), open-access, online math journal that intends to publish articles (with some merit) rejected by other journals. Information on it here:

http://math.rejecta.org/vol1-num1

...but not clear to me if there ever was or will be a second issue!!??? :-(

And on a more strictly for-fun, humorous note, you may also want to check out the "Journal of Unpublishable Mathematics" here (including a pdf of Vol. 1, 2010) ;-)) :

http://www.mathematicians.org.uk/jum/

Tuesday, July 6, 2010

Wikipedia Math

This mathematics "portal" site on Wikipedia links to the various other math-related pages available on Wikipedia:

http://en.wikipedia.org/wiki/Portal:Mathematics

Which Is Witch

The interesting "Salem Witch" logic puzzle discussed here:

http://tinyurl.com/29e6e4b

(I don't think the problem is stated quite as clearly as needs be to avoid some confusion, but still worth a read.)

Monday, July 5, 2010

A Museum Piece

Harking back to my childhood today....

When I was a youngster visiting a science museum in my home state what most fascinated me was not the dinosaur displays, or fossils, or insect collections, nor more whizbang exhibits, but a simple large display known as a "Galton Box" (after the 1889 inventor of the first one), or known by some as a "quincunx," (...okay, so I was an odd kid).

Most of you are likely familiar with these enclosed contraptions in which balls drop from a central point at the top onto a symmetrical pegboard where they bounce around until finally falling into columns at the bottom... the majority of balls dropping, by sheer chance, somewhere in the middle columns, and fewer balls bouncing around in a manner depositing them to the outer end columns.

On the glass pane enclosing the balls and peg-grid would be drawn the 'normal' or 'Gaussian' distribution (or 'Bell curve') so central to mathematics/probability, and lo-and-behold, once all the hundreds of balls had been released they would, in the columns below, take on the shape of that normal curve, via of course the 'laws' of sheer chance, not due to any mechanical manipulation. Even as a child, not really understanding much about normal distributions, nor math/probability more generally, somehow that demonstration was very powerful to me; like a magic, unseen hand guiding the fate of those individual spheres ---  each one taking a rather random, unpredictable journey, yet the end result being highly predictable and little-changing. Even as a youngster I sensed there was something profound in that. Some kids today construct Galton Boxes or quincunxes for science fair projects. Hooray!, for still to this day I love these apparatuses and their magical outcomes (...the rest of you can go gawk at dinosaur models).

A very quickie YouTube demonstration with a miniature, sand-based quincunx here:

And more on the 'quincunx' here:

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

http://www.mathsisfun.com/data/quincunx.html

finally, more technical info on the normal distribution from Wikipedia here:

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

Sunday, July 4, 2010

T-shirts and More...

For all you fashionable math buffs out there:

http://www.mathematicianspictures.com/Shirts_.htm

Fun With Pythagoras

For those who just can't get enough of that Greek dude Pythagoras, a new book out from Alfred Posamentier, "The Pythagorean Theorem: The Story of its Power and Beauty" totally devoted to... guess what!:

http://tinyurl.com/34ezfoe

Saturday, July 3, 2010

Carnival of Mathematics #67

The rollicking new July "Carnival of Mathematics" is now posted at "Travels In a Mathematical World" blog here:

http://tinyurl.com/28jlm9r

Friday, July 2, 2010

RuBot Performs

Rubik's-Cube-Solving Robot:

...and on the math news front, Russian mathematician Grigori Perelman has now officially turned down the million-dollar "Millenium" prize he was awarded for proving the "Poincare Conjecture" :

http://tinyurl.com/25hk4o4

Thursday, July 1, 2010

The "Two Envelopes" Problem

Well-known math riddle elucidated HERE (Keith Devlin) and HERE (Wikipedia, with further links).