Multiplication is NOT repeated addition (multiplication as 'scaling', from Keith Devlin):
http://www.maa.org/devlin/devlin_01_11.html
Friday, December 31, 2010
Tuesday, December 28, 2010
Logic...
Nice quick introduction to several classic paradoxes here:
http://www.popgive.com/2010/06/brain-twisting-paradoxes.html
http://www.popgive.com/2010/06/brain-twisting-paradoxes.html
Monday, December 27, 2010
Reindeer Problem From Wiseman
Richard Wiseman's prior-Friday Christmas problem here:
http://tinyurl.com/2bqgj2b (has problem & answer)
If you want to read the problem without the answer being immediately given go here (and get a bonus video in addition, as well):
http://tinyurl.com/2fbgtlp
http://tinyurl.com/2bqgj2b (has problem & answer)
If you want to read the problem without the answer being immediately given go here (and get a bonus video in addition, as well):
http://tinyurl.com/2fbgtlp
The 'Surprise' of Mathematical Connections
Another interesting post from RJ Lipton here:
http://rjlipton.wordpress.com/2010/12/26/unexpected-connections-in-math
This time on the "beauty and power" of "unexpected connections" in mathematics, where findings or knowledge from areas that seem to have little overlap, are conjoined to produce yet newly useful knowledge or proof. This can happen in other areas of science, but is an especially integral underlying part of modern mathematics with its complex, almost unwieldy proofs, going forward.
The post starts off with reference to illustrious mathematician and economics Nobel-Prize winner John Nash (famous to lay folks from the book and movie "A Beautiful Mind"), before delving into some more abstract and technical discussion.
Mathematics, full of surprise and beauty, signifying much.
http://rjlipton.wordpress.com/2010/12/26/unexpected-connections-in-math
This time on the "beauty and power" of "unexpected connections" in mathematics, where findings or knowledge from areas that seem to have little overlap, are conjoined to produce yet newly useful knowledge or proof. This can happen in other areas of science, but is an especially integral underlying part of modern mathematics with its complex, almost unwieldy proofs, going forward.
The post starts off with reference to illustrious mathematician and economics Nobel-Prize winner John Nash (famous to lay folks from the book and movie "A Beautiful Mind"), before delving into some more abstract and technical discussion.
Mathematics, full of surprise and beauty, signifying much.
Sunday, December 26, 2010
Saturday, December 25, 2010
Tuesday, December 21, 2010
Just A Couple of Book Notes...
The new Reuben Hersh/Vera John-Steiner book, "Loving and Hating Mathematics: Challenging the Myths of Mathematical Life" should be available in stores very shortly:
http://www.amazon.com/Loving-Hating-Mathematics-Challenging-Mathematical/dp/0691142475
[Addendum: just received a review copy of the above volume in mail so will have more to say about it perhaps sometime after the Holidays.]
And a rough draft (pdf download) of yet another book for lay people on the Riemann Hypothesis is available here (Barry Mazur co-author):
http://wstein.org/rh/
http://www.amazon.com/Loving-Hating-Mathematics-Challenging-Mathematical/dp/0691142475
[Addendum: just received a review copy of the above volume in mail so will have more to say about it perhaps sometime after the Holidays.]
And a rough draft (pdf download) of yet another book for lay people on the Riemann Hypothesis is available here (Barry Mazur co-author):
http://wstein.org/rh/
Monday, December 20, 2010
Sunday, December 19, 2010
Fractals From NOVA
Hat tip to "Grey Matters" for directing me to this 2008 hour-long PBS 'NOVA' episode on fractals, entitled "Hunting the Hidden Dimension":
http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html
http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html
Friday, December 17, 2010
Links and Gifts
Sol's usual array of interesting math links for this week at Equalis Community Blog here:
http://www.equalis.com/members/blog_view.asp?id=565749
(if you don't find at least one link there interesting to you, better check to see if you still have a pulse ;-))
And if you haven't yet finished holiday-shopping for all your math-geeky friends, Denise had several suggestions here:
http://letsplaymath.net/2009/12/22/have-a-mathy-christmas/
http://www.equalis.com/members/blog_view.asp?id=565749
(if you don't find at least one link there interesting to you, better check to see if you still have a pulse ;-))
And if you haven't yet finished holiday-shopping for all your math-geeky friends, Denise had several suggestions here:
http://letsplaymath.net/2009/12/22/have-a-mathy-christmas/
Google's Word Corpus
growth in interest in prime numbers and number theory:
http://tinyurl.com/22juwgm
growth in interest in arithmetic, not so much:
http://tinyurl.com/25km6aq
http://tinyurl.com/22juwgm
growth in interest in arithmetic, not so much:
http://tinyurl.com/25km6aq
Thursday, December 16, 2010
Wednesday, December 15, 2010
Timekeeping For Geeks ;-)
I don't comprehend all these hourly designations, but some of you probably will (...worth a chuckle):
http://www.beaverdam.k12.wi.us/bd/content/mathclock%20withnumbers.jpg
...and here a prime number watch:
http://tinyurl.com/2uop37u
And finally a watch based on, of all things, the Sierpinski Triangle:
http://tinyurl.com/2fqolo5
(Perhaps some people just have too much time on their hands! ;-)
http://www.beaverdam.k12.wi.us/bd/content/mathclock%20withnumbers.jpg
...and here a prime number watch:
http://tinyurl.com/2uop37u
And finally a watch based on, of all things, the Sierpinski Triangle:
http://tinyurl.com/2fqolo5
(Perhaps some people just have too much time on their hands! ;-)
Tuesday, December 14, 2010
Poincare and the Continuum
From "Mathematics Rising" blog:
"...mathematics is not built entirely on logic. It lives somewhere between thought-governed ideal realities and physical realities created by the senses. As such, it may be able to provide its own unique view of cognition itself. It may be said that cognitive processes unfold themselves into mathematical insights."
Read the entire post on the essence of mathematics here:
http://mathrising.com/?p=335
"...mathematics is not built entirely on logic. It lives somewhere between thought-governed ideal realities and physical realities created by the senses. As such, it may be able to provide its own unique view of cognition itself. It may be said that cognitive processes unfold themselves into mathematical insights."
Read the entire post on the essence of mathematics here:
http://mathrising.com/?p=335
Sunday, December 12, 2010
Google Hidden Challenge
Interesting geeky story of Google planting Mensa-like problem in an ad and rewarding solver:
http://tinyurl.com/27v3mgq
http://tinyurl.com/27v3mgq
Friday, December 10, 2010
Weekend Miscellany
Just a little quickie recursive brain workout for starters:
1. The third sentence here is true.
2. This is the second sentence here.
3. The fourth sentence here is false.
4. The sixth sentence here is true.
5. The first sentence in this list is true.
6. The first and last word in this sentence is "the."
Is sentence #5 true or false???
Meanwhile, plenty of fun stuff at Sol's latest "Wild About Math" entry for the Equalis Community blog:
http://www.equalis.com/members/blog_view.asp?id=565749&post=116258
1. The third sentence here is true.
2. This is the second sentence here.
3. The fourth sentence here is false.
4. The sixth sentence here is true.
5. The first sentence in this list is true.
6. The first and last word in this sentence is "the."
Is sentence #5 true or false???
Meanwhile, plenty of fun stuff at Sol's latest "Wild About Math" entry for the Equalis Community blog:
http://www.equalis.com/members/blog_view.asp?id=565749&post=116258
Thursday, December 9, 2010
"The Tau Manifesto"
Mark Chu-Carroll at his blog, "Good Math, Bad Math," often posts examples of mathematical 'crackpottery' that come his way, but recently posted about an argument for why everyone's favorite number π, should really be substituted with another value tau, τ, the equivalent of 2π, and why this would all make sense in so many ways (tau is based on the radius of a circle, instead of the diameter-basis for pi). And yes, it's a serious argument:
http://tauday.com/
Many agree with the logic and reasoning of the assertions in favor of 'tau,' but... well... changing centuries of routine use of pi (and making obsolete all those 'I Love π ' t-shirts) , is of course another matter altogether.
http://tauday.com/
Many agree with the logic and reasoning of the assertions in favor of 'tau,' but... well... changing centuries of routine use of pi (and making obsolete all those 'I Love π ' t-shirts) , is of course another matter altogether.
Wednesday, December 8, 2010
"The World of Trotter Math"
I've recently been reading a lot of interesting posts, playing with numbers, over at "The World of Trotter Math," especially many recent ones on prime numbers:
http://trottermath.net/wordpress/
Upon researching to learn more about the site's proprietor I suddenly discovered his name was Terry Trotter and he passed away in 2004. I don't know who keeps the blog running, but thank you whoever is responsible.
A further-interesting tribute to Mr. Trotter is here:
http://www.guardiansofdarkness.com/GoD/trotter.html
http://trottermath.net/wordpress/
Upon researching to learn more about the site's proprietor I suddenly discovered his name was Terry Trotter and he passed away in 2004. I don't know who keeps the blog running, but thank you whoever is responsible.
A further-interesting tribute to Mr. Trotter is here:
http://www.guardiansofdarkness.com/GoD/trotter.html
Sunday, December 5, 2010
Mathematicians, Media, and Sex Appeal
Entertaining post (including lots of links) today from Jonathan Farley of The Guardian:
http://tinyurl.com/27qmk42
http://tinyurl.com/27qmk42
Saturday, December 4, 2010
Gardner Strikes Again
For today's amusement, a paradox quoted directly from Martin Gardner's "The Jinn From Hyperspace":
"Now for a final paradox. There is a certain event that I guarantee will or will not take place during the next ten minutes. You are absolutely incapable of predicting correctly whether it will or won't occur. I don't mean that it's unlikely you can predict it. I mean it is logically impossible to predict it!
"You don't believe it? Then do the following. If you think the event will occur write "Yes" inside the blank rectangle below. If you think it won't happen, write "No" inside the rectangle.
"If you predicted correctly, I'll send you a million dollars.
The event is: You will write "No" inside the rectangle."
As Gardner might say, "Gotcha!"
This is just one of many paradoxes entangled with self-reference and classic problems of causation/prediction when dealing with human language and logic.
"Now for a final paradox. There is a certain event that I guarantee will or will not take place during the next ten minutes. You are absolutely incapable of predicting correctly whether it will or won't occur. I don't mean that it's unlikely you can predict it. I mean it is logically impossible to predict it!
"You don't believe it? Then do the following. If you think the event will occur write "Yes" inside the blank rectangle below. If you think it won't happen, write "No" inside the rectangle.
"If you predicted correctly, I'll send you a million dollars.
The event is: You will write "No" inside the rectangle."
As Gardner might say, "Gotcha!"
This is just one of many paradoxes entangled with self-reference and classic problems of causation/prediction when dealing with human language and logic.
Thursday, December 2, 2010
Boys Will Be Boys...
...and also creative mathematicians:
nice storyline from "CTK Insights" on 3 young people who bucked the more mundane (and expected) answer on an IQ test to start with Pascal's Triangle and end up with "The Rascal Triangle":
http://tinyurl.com/29opquv
(a clear case of thinking-outside-the-triangle)
nice storyline from "CTK Insights" on 3 young people who bucked the more mundane (and expected) answer on an IQ test to start with Pascal's Triangle and end up with "The Rascal Triangle":
http://tinyurl.com/29opquv
(a clear case of thinking-outside-the-triangle)
Wednesday, December 1, 2010
"Sexy Primes"
six is so sexy... "Trotter Math" introduces us here (in a re-post of info from over 10 years ago!) to "sexy primes":
http://trottermath.net/wordpress/sexy-primes/
http://trottermath.net/wordpress/sexy-primes/
Monday, November 29, 2010
n + sqrt(n) = .....
James Tanton recently tweeted that for any positive integer n, the equation n + sqrt(n) rounded to the nearest integer, will never result in the square of an integer; i.e. 4 + 2 = 6 (not a square), 9 + 3 = 12, 13 + √13 = 16.605 or 17, etc.
...and Gary Davis took up the challenge to demonstrate the truth of the statement here (using proof by contradiction):
http://tinyurl.com/2bv76se
Now if I can just figure out a way to work this into my next cocktail party conversation ;-)
...and Gary Davis took up the challenge to demonstrate the truth of the statement here (using proof by contradiction):
http://tinyurl.com/2bv76se
Now if I can just figure out a way to work this into my next cocktail party conversation ;-)
Saturday, November 27, 2010
Self-recursion
Just recently stumbled upon this brief page on "self-recursion" over at Wolfram MathWorld:
http://mathworld.wolfram.com/Self-Recursion.html
One example of a self-recursive statement therefrom to whet your interest:
"This sentence contains ten words, eighteen syllables, and sixty-four letters."
...and, on a more humorous note:
"What is the volume of a pizza of thickness 'a' and radius 'z'?" Answer: pi z z a.
http://mathworld.wolfram.com/Self-Recursion.html
One example of a self-recursive statement therefrom to whet your interest:
"This sentence contains ten words, eighteen syllables, and sixty-four letters."
...and, on a more humorous note:
"What is the volume of a pizza of thickness 'a' and radius 'z'?" Answer: pi z z a.
Friday, November 26, 2010
Latest From Equalis Community Blog
"Wild About Math" has a new blog entry up at the Equalis Community blog here:
http://www.equalis.com/members/blog_view.asp?id=565749
Always includes some interesting stuff... and in this instance, introduced me to another blog I was unaware of, "Grey Matters," which I've added to the blogroll at right. Also, blogger Sol mentions he'll be doing reviews of several of James Tanton's books in the future; something to look forward to. There is a link to the latest "Math Teachers At Play" carnival, and other amusements. Good stuff for a long weekend....
http://www.equalis.com/members/blog_view.asp?id=565749
Always includes some interesting stuff... and in this instance, introduced me to another blog I was unaware of, "Grey Matters," which I've added to the blogroll at right. Also, blogger Sol mentions he'll be doing reviews of several of James Tanton's books in the future; something to look forward to. There is a link to the latest "Math Teachers At Play" carnival, and other amusements. Good stuff for a long weekend....
Wednesday, November 24, 2010
Base 12
A sort of fun post from Alex Bellos today on using 12 instead of 10 as the basis of our mathematical system, as pushed by the "Dozenal Society of America" (who knew there was such an organization!). Most folks are aware of the binary system and other bases lower than 10, but employing bases greater than 10 gets less attention. There have always been many reasons why base 12 (also known as "duodecimal" or "dozenal") would make a lot of utilitarian sense, even if as a practical matter, it will never happen:
http://alexbellos.com/?p=1462
Wikipedia on base 12 here: http://en.wikipedia.org/wiki/Duodecimal
And the Dozenal Society's website here: http://www.dozenal.org/index.php?u=31
(Somehow though, asking for a baker's dozen donuts and only expecting 11, just doesn't seem as appealing... ;-))
http://alexbellos.com/?p=1462
Wikipedia on base 12 here: http://en.wikipedia.org/wiki/Duodecimal
And the Dozenal Society's website here: http://www.dozenal.org/index.php?u=31
(Somehow though, asking for a baker's dozen donuts and only expecting 11, just doesn't seem as appealing... ;-))
Tuesday, November 23, 2010
Saturday, November 20, 2010
A Royal Wedding... and Mathematics
Great piece by Matt Parker explaining P vs. NP in layman terms:
http://tinyurl.com/2d83t7v
In fact, his introductory lines are the simplist statement of P vs. NP I've ever come across:
"Can you solve a problem as fast as someone can check your answer? Can you show that this is possible for any problem at all? Then $1m (£600,000) is all yours."
http://tinyurl.com/2d83t7v
In fact, his introductory lines are the simplist statement of P vs. NP I've ever come across:
"Can you solve a problem as fast as someone can check your answer? Can you show that this is possible for any problem at all? Then $1m (£600,000) is all yours."
Friday, November 19, 2010
Friday Pizza Puzzle
Four students order two 10-inch pizzas from two different pizzerias, to divide equally among themselves. When the pizzas arrive one is in the shape of a 10-inch (per side) square, while the other is in the shape of a 10-inch diameter circle. They plan to divide each pizza into 4 equal pieces, with each student receiving 1 piece from each pizza.
How many sq. inches of pizza will each student end up with?
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. answer below
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answer: 25 (pi + 4) sq. in.
4
How many sq. inches of pizza will each student end up with?
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. answer below
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answer: 25 (pi + 4) sq. in.
4
Thursday, November 18, 2010
Garrett Lisi's "Geometric Theory of Everything"
I've been intrigued in the past by the outside-the-box geometry/cosmology thinking of physicist Garrett Lisi (but then I'm not schooled enough in his subject matter to even know if he's only outside-the-box or, off-the-wall!). Peter Woit has a post today related to Lisi's latest theorizing:
http://www.math.columbia.edu/~woit/wordpress/?p=3292
... and here's an older TED talk Lisi gave:
http://tinyurl.com/yla4y52
http://www.math.columbia.edu/~woit/wordpress/?p=3292
... and here's an older TED talk Lisi gave:
http://tinyurl.com/yla4y52
Wednesday, November 17, 2010
Palindromically Speaking
An easy puzzle for today:
Palindromic numbers are those that read the same backwards as forwards; for example, 101, 3663, 40904.
Given the palindromic number a = 138831, come up with a palindromic number b composed of the same six digits (as a) but in another order, such that the sum of a + b also equals a palindromic number.
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answer below
...but first check out this (NON-palindromic) quirk from "Futility Closet":
http://www.futilitycloset.com/2010/11/15/home-again/
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.answer: 831138
Palindromic numbers are those that read the same backwards as forwards; for example, 101, 3663, 40904.
Given the palindromic number a = 138831, come up with a palindromic number b composed of the same six digits (as a) but in another order, such that the sum of a + b also equals a palindromic number.
.
answer below
...but first check out this (NON-palindromic) quirk from "Futility Closet":
http://www.futilitycloset.com/2010/11/15/home-again/
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.answer: 831138
Tuesday, November 16, 2010
Get Busy!
Release of Wolfram Mathematica 8 announced:
http://blog.wolfram.com/2010/11/15/mathematica-8/
...and this follow-up post:
http://tinyurl.com/3xjxp5p
http://blog.wolfram.com/2010/11/15/mathematica-8/
...and this follow-up post:
http://tinyurl.com/3xjxp5p
Monday, November 15, 2010
Friday, November 12, 2010
A Non-Math Friday...
Just a couple of notes tangential to math today:
First off, Peter Woit has a quick (and positive) review of the new book "Massive" by Ian Sample (on particle physics, and principally the Higgs Boson), up at his blog; worth a look:
http://www.math.columbia.edu/~woit/wordpress/?p=3266
On a lighter note, I'm now reading/enjoying Martin Gardner's final book, the "Colossal Book of Wordplay," a slightly odd volume. First it is NOT "colossal," and second, although he certainly dabbled in wordplay previously, it is not what Gardner is famous for.
This smallish volume is barely 150 pages with large print and smallish pages... I'm not sure if the "colossal" in the title is merely a takeoff on his previous "Colossal Book of Mathematics," or whether the title itself is intended as some sort of ironic 'wordplay' on this quite slim volume (or possibly the volume was originally to include yet more material that was never completed due to Gardner's death earlier this year?).
Nonetheless, it is a fun, jaunty book covering a wide range of wordplay with frequent intervening bits of quirky entertainment. Possibly it could be organized a little better, at times seeming to me slightly disjointed or 'thrown together,' and I wish Gardner had gone into greater depth at times (as Douglas Hofstadter has previously done on some of this material), but for nine bucks it's worth the price of admission, if you have an interest in the quirks of language (as a lot of mathematicians, and other analytical sorts, do). My expectations for Gardner are so high that this book probably falls short of them, but were it any lesser author, I'd easily give it a thumbs-up.
The content ranges from almost juvenile or goofy entries to well-known standards, to amusements that almost any reader will find new to them. There are puzzles and palindromes, poems and anagrams, brain teasers, word games, riddles, and everything in-between. If nothing on one page strikes your fancy, something on the next page likely will.
If you love words, get this book. Think of it as Gardner's final stocking stuffer gift to us. (I haven't seen it yet in a single bookstore, and had to order it online, so not sure how widely distributed it is.)
Finally, and slightly more math related, Alex Bellos covers a recent Rubik's Cube competition here:
http://alexbellos.com/?p=1429
Wednesday, November 10, 2010
Some Miscellany For Wednesday
Another look at the foundations of mathematics in upcoming book:
http://tinyurl.com/33ylyvp
If you can't get enough of Pi:
http://facts.randomhistory.com/2009/07/03_pi.html
James Tanton's newly-active Twitter feed here:
http://twitter.com/#!/jamestanton
http://tinyurl.com/33ylyvp
If you can't get enough of Pi:
http://facts.randomhistory.com/2009/07/03_pi.html
James Tanton's newly-active Twitter feed here:
http://twitter.com/#!/jamestanton
Tuesday, November 9, 2010
RSA Encryption
A fairly straightforward explanation of RSA encryption here:
http://inversezen.com/2010/11/the-rsa-algorithm/
http://inversezen.com/2010/11/the-rsa-algorithm/
Sunday, November 7, 2010
Air Molecules, Neuron-firing, Carl Sagan
Mathematically-speaking, you've been breathing some mighty famous air:
http://io9.com/5635391/youve-probably-shared-the-same-air-with-galileo
(not sure if the actual mathematics of this holds up to closer scrutiny...?)
And from the same blog, a story that's been making the rounds about an electronic contraption for brain stimulation that might aid mathematical thinking:
http://tinyurl.com/3695xty
Not precisely math-related, but on the heels of a day celebrating Martin Gardner, I feel I'd be remiss if I didn't note that a day celebrating Carl Sagan is coming up:
http://www.centerforinquiry.net/carlsaganday
http://io9.com/5635391/youve-probably-shared-the-same-air-with-galileo
(not sure if the actual mathematics of this holds up to closer scrutiny...?)
And from the same blog, a story that's been making the rounds about an electronic contraption for brain stimulation that might aid mathematical thinking:
http://tinyurl.com/3695xty
Not precisely math-related, but on the heels of a day celebrating Martin Gardner, I feel I'd be remiss if I didn't note that a day celebrating Carl Sagan is coming up:
http://www.centerforinquiry.net/carlsaganday
Friday, November 5, 2010
A Look At Pascal's Triangle
In his book "Wonder of Numbers" Clifford Pickover names the following article as having the "all-time strangest title" of any published mathematical paper:
Granville, A. (1992) "Zaphod Beeblebox's brain and the fifty-ninth row of Pascal's Triangle" American Mathematical Monthly April, 99(4): 318-331.
The paper (pdf) can be found here:
http://www.gianpierobiancoli.it/wp-content/uploads/2009/10/beeb.pdf
And if you don't know who the character Zaphod Beeblebrox is (from "Hitchhiker's Guide to the Galaxy") you can check him out here:
http://en.wikipedia.org/wiki/Zaphod_Beeblebrox
In other matters, a quickie intro to the Riemann Hypothesis from Matt Parker here:
http://tinyurl.com/2arhu4r
[includes the mention that "All prime numbers (greater than five) squared are one more than a multiple of 24."]
And revisiting Mr. Fermat:
x^n + y^n = z^n ...NO solutions for n ≥ 3
x^n + y^n = z^(n-1) ...INFINITELY many solutions for all n ≥ 3
proof:
http://tinyurl.com/2d5qucy
Granville, A. (1992) "Zaphod Beeblebox's brain and the fifty-ninth row of Pascal's Triangle" American Mathematical Monthly April, 99(4): 318-331.
The paper (pdf) can be found here:
http://www.gianpierobiancoli.it/wp-content/uploads/2009/10/beeb.pdf
And if you don't know who the character Zaphod Beeblebrox is (from "Hitchhiker's Guide to the Galaxy") you can check him out here:
http://en.wikipedia.org/wiki/Zaphod_Beeblebrox
In other matters, a quickie intro to the Riemann Hypothesis from Matt Parker here:
http://tinyurl.com/2arhu4r
[includes the mention that "All prime numbers (greater than five) squared are one more than a multiple of 24."]
And revisiting Mr. Fermat:
x^n + y^n = z^n ...NO solutions for n ≥ 3
x^n + y^n = z^(n-1) ...INFINITELY many solutions for all n ≥ 3
proof:
http://tinyurl.com/2d5qucy
Wednesday, November 3, 2010
Biology, Complexity, Dynamics... and Math
"Biology is not yet a predictive science, there are essentially no fundamental laws [as with physics]... biology, in terms of maturity, is at the stage that physics was 300 years ago..."
Interesting post over at plus.maths.org on the work of Thomas Fink et.al., essentially trying to mathematically model biological systems.
a bit more therefrom:
"Everyone says the standard model for evolution is mutation, selection and inheritance. Put those ingredients together in a box and you get evolution. But the reality is, when we put those things into models of evolution, or set up appropriate systems of artificial life, we just don't get life-like evolution — we don't find the evolution of complex, surprising things. Some fundamental is missing. What gives a system the capacity to evolve? What makes a system evolvable?"
...and later:
"The problem is, to be able to know what is interesting, one needs to know what is boring."
Interesting post over at plus.maths.org on the work of Thomas Fink et.al., essentially trying to mathematically model biological systems.
a bit more therefrom:
"Everyone says the standard model for evolution is mutation, selection and inheritance. Put those ingredients together in a box and you get evolution. But the reality is, when we put those things into models of evolution, or set up appropriate systems of artificial life, we just don't get life-like evolution — we don't find the evolution of complex, surprising things. Some fundamental is missing. What gives a system the capacity to evolve? What makes a system evolvable?"
...and later:
"The problem is, to be able to know what is interesting, one needs to know what is boring."
Tuesday, November 2, 2010
Stuff For a Tuesday
I previously posted about the intriguing and unsolved "Collatz Problem" (whether or not certain created sequences always must end in the same pattern, regardless of starting point). It is also known by the name "Hailstone Numbers" and Ben Vitale recently wrote about them here:
http://benvitale-funwithnum3ers.blogspot.com/2010/10/list-of-squares.html?spref=tw
Clifford Pickover addresses the same subject here:
http://sprott.physics.wisc.edu/pickover/hailstone.html
Meanwhile, in a different vein, I just recently discovered this fairly young blog devoted entirely to prime numbers:
http://primepatterns.wordpress.com/
Finally, I've never been much of a Sudoku fan, but I do very much enjoy Ken-Ken (are there others like me out there, and if so, why is that???)... In any event, this poster's been thinking more about Ken-Ken than I ever did:
http://bit-player.org/2010/kenken-friendly-numbers
http://benvitale-funwithnum3ers.blogspot.com/2010/10/list-of-squares.html?spref=tw
Clifford Pickover addresses the same subject here:
http://sprott.physics.wisc.edu/pickover/hailstone.html
Meanwhile, in a different vein, I just recently discovered this fairly young blog devoted entirely to prime numbers:
http://primepatterns.wordpress.com/
Finally, I've never been much of a Sudoku fan, but I do very much enjoy Ken-Ken (are there others like me out there, and if so, why is that???)... In any event, this poster's been thinking more about Ken-Ken than I ever did:
http://bit-player.org/2010/kenken-friendly-numbers
Sunday, October 31, 2010
Dr. James Tanton
Kudos again to Sol, this time for introducing me to Dr. James Tanton, a creative mathematician with his own YouTube channel of interesting videos here (okay, I'm a sucker for an Aussie accent):
http://www.youtube.com/user/DrJamesTanton
His homepage website is here:
http://www.jamestanton.com/
...and "Math Mama" reviewed some of his work here:
http://mathmamawrites.blogspot.com/search?q=%22james+tanton%22
In other news, another recent "tweet" (from Twitter) that caught my eye:
"It is known that e is irrational and that pi is irrational, but it is not known if their sum is irrational."
I'm wondering (maybe someone out there knows the answer) is it EVER the case that 2 irrational, transcendental numbers are known to sum to a rational???
http://www.youtube.com/user/DrJamesTanton
His homepage website is here:
http://www.jamestanton.com/
...and "Math Mama" reviewed some of his work here:
http://mathmamawrites.blogspot.com/search?q=%22james+tanton%22
In other news, another recent "tweet" (from Twitter) that caught my eye:
"It is known that e is irrational and that pi is irrational, but it is not known if their sum is irrational."
I'm wondering (maybe someone out there knows the answer) is it EVER the case that 2 irrational, transcendental numbers are known to sum to a rational???
Thursday, October 28, 2010
Penney's Paradox
Want to win a few coins... The "Penney Paradox" is a very intriguing though less-discussed paradox than some of its more famous counterparts (it's named after its discoverer Walter Penney, though it is also often discussed using a penny as the working example).
If one flips a fair coin 3 separate times, there are 8 equally probable (heads/tails) triplet-results: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. In this game a first player selects one of these triplets, and then a second player chooses a different one. The coin is then flipped repeatedly until one of the selected triplets appears as a run and the player having chosen it wins the game (and coin). For example, if the chosen triplets are HTH and THT and the flips go THHHTH, the last three flips mean that HTH has won... the first triplet appearing matching a player's choice, wins.
One might first think that any one triplet is just as likely to occur as any other. However, upon reflection it will probably be clear that given a series of 4-or-more flips there are more ways for a triplet like say HTH to occur than the triplets TTT or HHH to appear. But what is far more intriguing is that NO MATTER what triplet the first player chooses, there are triplets that player #2 can select giving him/her a probabalistic edge of winning.
"Futility Closet" site mentioned this a few weeks back (and how player 2 can make his/her choice), but without elaborating much on how the mathematics of it works.
"plus.math.org" covers the math here:
http://plus.maths.org/issue55/features/nishiyama/
Or you can check out a briefer treatment on Wikipedia here:
http://en.wikipedia.org/wiki/Penney%27s_game
I should also mention that IF you do have Martin Gardner's "Colossal Book of Mathematics" on-hand he covers the subject well in his chapter 23 on "nontransitive paradoxes" (it is the "nontransitivity" of the relationships involved that result in the differential probabilities for the triplets).
If one flips a fair coin 3 separate times, there are 8 equally probable (heads/tails) triplet-results: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. In this game a first player selects one of these triplets, and then a second player chooses a different one. The coin is then flipped repeatedly until one of the selected triplets appears as a run and the player having chosen it wins the game (and coin). For example, if the chosen triplets are HTH and THT and the flips go THHHTH, the last three flips mean that HTH has won... the first triplet appearing matching a player's choice, wins.
One might first think that any one triplet is just as likely to occur as any other. However, upon reflection it will probably be clear that given a series of 4-or-more flips there are more ways for a triplet like say HTH to occur than the triplets TTT or HHH to appear. But what is far more intriguing is that NO MATTER what triplet the first player chooses, there are triplets that player #2 can select giving him/her a probabalistic edge of winning.
"Futility Closet" site mentioned this a few weeks back (and how player 2 can make his/her choice), but without elaborating much on how the mathematics of it works.
"plus.math.org" covers the math here:
http://plus.maths.org/issue55/features/nishiyama/
Or you can check out a briefer treatment on Wikipedia here:
http://en.wikipedia.org/wiki/Penney%27s_game
I should also mention that IF you do have Martin Gardner's "Colossal Book of Mathematics" on-hand he covers the subject well in his chapter 23 on "nontransitive paradoxes" (it is the "nontransitivity" of the relationships involved that result in the differential probabilities for the triplets).
Tuesday, October 26, 2010
Tuesday Fun...
Just some fun stuff today:
The success of physicists and a sense of probability in poker-playing here:
http://tinyurl.com/244wukt
An entire post of math humor/goofiness here (some better than others!):
http://tetrahedral.blogspot.com/2010/10/math-jokes.html
...it includes this one, just to whet your appetitie:
And lastly just another interesting tweet from Twitter:
"Are there infinitely many prime Fibonacci numbers? Unknown, at least as of ten years ago."
The success of physicists and a sense of probability in poker-playing here:
http://tinyurl.com/244wukt
An entire post of math humor/goofiness here (some better than others!):
http://tetrahedral.blogspot.com/2010/10/math-jokes.html
...it includes this one, just to whet your appetitie:
And lastly just another interesting tweet from Twitter:
"Are there infinitely many prime Fibonacci numbers? Unknown, at least as of ten years ago."
Sunday, October 24, 2010
Mathematics 1001
It's annoying (and expensive) when, while awaiting the arrival of certain books in a bookstore, I suddenly happen upon a volume I've never heard of that looks enticing for the money I've been saving for the other specific books!...
This weekend I stumbled upon "Mathematics 1001" by Brit Richard Elwes, a new book that does a great job of covering in a nutshell a huge number of ideas/concepts/terms cutting across a broad swath of mathematics. Several such books are already available, but this looks to be the best one I've seen yet, written at a layman's level... a great little reference source and quickie introduction to various math nuggets, for your shelf. It is a sort of glossary of mathematical ideas, organized not alphabetically, but generally from simple and basic ideas that hang together to more complex and abstract ones.
The contents are divided into the following general areas:
-- Numbers
-- Geometry
-- Algebra
-- Discrete Mathematics
-- Analysis
-- Logic
-- Metamathematics
-- Probability and statistics
-- Mathematical physics
-- Games and recreation
The Intro says that the book is "aimed at.... anyone with a curiosity about mathematics, from the novice to the informed student or enthusiast. Whatever the reader's current knowledge, I'm sure that there will be material here to enlighten and engage."
I think the author succeeds....
(It's a hardback, and I believe worth the $25 price, but of course can be gotten cheaper.)
Friday, October 22, 2010
Mandelbrot, Mobius, Mirth
Image via Wikipedia
In their tributes to the Father of Fractals, Benoit Mandelbroit (recently deceased), several sites have done posts on "infinite coastlines." One of the related notions that I've always found mind-blowing is that of a closed border or perimeter having INFINITE length, yet enclosing a region of FINITE area (very counter-intuitive!). The Koch Snowflake is probably the most oft-used example of this (though there are any number of other possible cases).Most of you are likely well-familiar with it, but if not, or if you need a refresher, a couple of the many sites expounding on it:
http://www.cut-the-knot.org/WhatIs/Infinity/Length-Area.shtml
http://en.wikipedia.org/wiki/Koch_snowflake
p.s... here's another one of those video zooms of the Mandelbrot Set, magnified
http://vimeo.com/1908224
Further memorializing the recently-deceased, Ivars Peterson reports on yet another oddball Mobius-band trick passed along to him by Martin Gardner some years ago:
http://tinyurl.com/3484lyg
And to end the week with some chuckles, re-visit this old caption-writing contest from WildAboutMath blog, if you missed it the first time:
http://tinyurl.com/2atzumj
Wednesday, October 20, 2010
If I Were Stranded on a Desert Island...
...and could only have one book to read I think I know what it would be. . . .
Yes, I'll squeeze in one more bit about Martin Gardner before his day of honor tomorrow (to any who are tired of hearing about Martin Gardner by now, I apologize, but HEY! it's MY blog so deal with it ;-))
When Gardner's "The Colossal Book of Mathematics" came out in 2001, I looked at its size, price, and the many chapters covering topics I wasn't particularly interested in, and ignored it (I already had plenty of Gardner books on my shelf). It was only years later that I checked it out from a public library and discovered not only how many chapters were of great interest to me, but also how many of the topics I wouldn't normally have found interesting were made so by Gardner's deft and insightful writing.
So again I highly recommend this volume to anyone lacking it on their shelves. If you're ever stranded on a desert island it would offer you weeks/months of mental entertainment (...and, as an alternative, just in case I got tired of math, I might take along Gardner's essay-anthology, "The Night Is Large," as well!).
Meanwhile, 'Mathematics Rising' blog has just covered the 3rd of my 'Fab Four,' Bernhard Riemann, here:
http://mathrising.com/?p=270
Finally, a couple of recent math blog carnivals here for your enjoyment and delectation:
http://tinyurl.com/2bhva6c
http://tinyurl.com/2bcdscb
And I'll end today with a factoid lifted off an old Twitter posting:
A 10,000 number gap in prime numbers: 9973! + 2 through 9973! + 10006 are all composite (non-prime).
...someone please double-check those for me over the lunch hour and get back to me to confirm.
Yes, I'll squeeze in one more bit about Martin Gardner before his day of honor tomorrow (to any who are tired of hearing about Martin Gardner by now, I apologize, but HEY! it's MY blog so deal with it ;-))
When Gardner's "The Colossal Book of Mathematics" came out in 2001, I looked at its size, price, and the many chapters covering topics I wasn't particularly interested in, and ignored it (I already had plenty of Gardner books on my shelf). It was only years later that I checked it out from a public library and discovered not only how many chapters were of great interest to me, but also how many of the topics I wouldn't normally have found interesting were made so by Gardner's deft and insightful writing.
So again I highly recommend this volume to anyone lacking it on their shelves. If you're ever stranded on a desert island it would offer you weeks/months of mental entertainment (...and, as an alternative, just in case I got tired of math, I might take along Gardner's essay-anthology, "The Night Is Large," as well!).
Meanwhile, 'Mathematics Rising' blog has just covered the 3rd of my 'Fab Four,' Bernhard Riemann, here:
http://mathrising.com/?p=270
Finally, a couple of recent math blog carnivals here for your enjoyment and delectation:
http://tinyurl.com/2bhva6c
http://tinyurl.com/2bcdscb
And I'll end today with a factoid lifted off an old Twitter posting:
A 10,000 number gap in prime numbers: 9973! + 2 through 9973! + 10006 are all composite (non-prime).
...someone please double-check those for me over the lunch hour and get back to me to confirm.
Monday, October 18, 2010
Dan, David, and more Martin
Daniel Gilbert muses in the NY Times about the 'magic' of numbers:
http://tinyurl.com/25w9j6w
Here he references one study I find a bit hard to believe:
http://rjlipton.wordpress.com/2010/10/17/david-hilbert-speaks/
And as we approach the Oct. 21 'Celebration of Mind' in honor of Martin Gardner, yet another piece on him, this time from the latest edition of American Scientist magazine:
http://tinyurl.com/2fxo6k6
http://tinyurl.com/25w9j6w
Here he references one study I find a bit hard to believe:
"The sound a number makes can influence our decisions about it. In a recent study, one group was shown an ad for an ice-cream scoop that was priced at $7.66, while another was shown an ad for a $7.22 scoop. The lower price is the better deal, of course, but the higher price (with its silky s’s) makes a smaller sound than the lower price (with its rattling t’s).A few posts back I linked to an entry at another blog on one of my designated 'Fab Four' (Carl Gauss), and now RJ Lipton has put up a nice post on his blog on another of those Fab Four, David Hilbert and his insights:
And because small sounds usually name small things, shoppers who were offered the scoop at the higher but whispery price of $7.66 were more likely to buy it than those offered the noisier price of $7.22 — but only if they’d been asked to say the price aloud."
http://rjlipton.wordpress.com/2010/10/17/david-hilbert-speaks/
And as we approach the Oct. 21 'Celebration of Mind' in honor of Martin Gardner, yet another piece on him, this time from the latest edition of American Scientist magazine:
http://tinyurl.com/2fxo6k6
Saturday, October 16, 2010
Benoit Mandelbrot 1924 - 2010
Acclaimed and beloved mathematician Benoit Mandelbrot, of fractal fame, passed away at the age of 85 last Thursday. NY Times obit here:
http://www.nytimes.com/2010/10/17/us/17mandelbrot.html?_r=3&hp
His name will forever leave its stamp on mathematics. In memory, I'll just re-post the 3rd post I ever ran on this blog, one of the many tributes to his Mandelbrot Set that can be found on the Web:
YouTube video of Mandelbrot Set with Jonathan Coulton's lyrics below:
lyrics:
Pathological monsters! cried the terrified mathematician
Every one of them is a splinter in my eye
I hate the Peano Space and the Koch Curve
I fear the Cantor Ternary Set And the Sierpinski Gasket makes me want to cry
And a million miles away a butterfly flapped its wings
On a cold November day a man named Benoit Mandelbrot was born
His disdain for pure mathematics and his unique geometrical insights
Left him well equipped to face those demons down
He saw that infinite complexity could be described by simple rules
He used his giant brain to turn the game around
And he looked below the storm and saw a vision in his head
A bulbous pointy form
He picked his pencil up and he wrote his secret down
Take a point called Z in the complex plane
Let Z1 be Z squared plus C
And Z2 is Z1 squared plus C
And Z3 is Z2 squared plus C and so on
If the series of Z's should always stay
Close to Z and never trend away
That point is in the Mandelbrot Set
Mandelbrot Set you're a Rorschach Test on fire
You're a day-glo pterodactyl
You're a heart-shaped box of springs and wire
You're one badass f**king fractal
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way
Mandelbrot's in heaven, at least he will be when he's dead
Right now he's still alive and teaching math at Yale
He gave us order out of chaos, he gave us hope where there was none
And his geometry succeeds where others fail
If you ever lose your way, a butterfly will flap its wings
From a million miles away, a little miracle will come to take you home
Just take a point called Z in the complex plane
Let Z1 be Z squared plus C
And Z2 is Z1 squared plus C
And Z3 is Z2 squared plus C and so on
If the series of Z's should always stay
Close to Z and never trend away
That point is in the Mandelbrot Set
Mandelbrot Set you're a Rorschach Test on fire
You're a day-glo pterodactyl
You're a heart-shaped box of springs and wire
You're one badass f**king fractal
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way
Go on change the world in a tiny way
Come on change the world in a tiny way
http://www.nytimes.com/2010/10/17/us/17mandelbrot.html?_r=3&hp
His name will forever leave its stamp on mathematics. In memory, I'll just re-post the 3rd post I ever ran on this blog, one of the many tributes to his Mandelbrot Set that can be found on the Web:
YouTube video of Mandelbrot Set with Jonathan Coulton's lyrics below:
lyrics:
Pathological monsters! cried the terrified mathematician
Every one of them is a splinter in my eye
I hate the Peano Space and the Koch Curve
I fear the Cantor Ternary Set And the Sierpinski Gasket makes me want to cry
And a million miles away a butterfly flapped its wings
On a cold November day a man named Benoit Mandelbrot was born
His disdain for pure mathematics and his unique geometrical insights
Left him well equipped to face those demons down
He saw that infinite complexity could be described by simple rules
He used his giant brain to turn the game around
And he looked below the storm and saw a vision in his head
A bulbous pointy form
He picked his pencil up and he wrote his secret down
Take a point called Z in the complex plane
Let Z1 be Z squared plus C
And Z2 is Z1 squared plus C
And Z3 is Z2 squared plus C and so on
If the series of Z's should always stay
Close to Z and never trend away
That point is in the Mandelbrot Set
Mandelbrot Set you're a Rorschach Test on fire
You're a day-glo pterodactyl
You're a heart-shaped box of springs and wire
You're one badass f**king fractal
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way
Mandelbrot's in heaven, at least he will be when he's dead
Right now he's still alive and teaching math at Yale
He gave us order out of chaos, he gave us hope where there was none
And his geometry succeeds where others fail
If you ever lose your way, a butterfly will flap its wings
From a million miles away, a little miracle will come to take you home
Just take a point called Z in the complex plane
Let Z1 be Z squared plus C
And Z2 is Z1 squared plus C
And Z3 is Z2 squared plus C and so on
If the series of Z's should always stay
Close to Z and never trend away
That point is in the Mandelbrot Set
Mandelbrot Set you're a Rorschach Test on fire
You're a day-glo pterodactyl
You're a heart-shaped box of springs and wire
You're one badass f**king fractal
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way
And you're just in time to save the day
Sweeping all our fears away
You can change the world in a tiny way
Go on change the world in a tiny way
Come on change the world in a tiny way
Friday, October 15, 2010
Psssst... One Word For You: Mathematics
If you're headed off to college soon (or already there), you're probably too young to remember the movie "The Graduate" but nonetheless I have one word for you... "mathematics." Read about its importance and attractiveness as a career:
http://asunews.asu.edu/20101014_math_school
...and at least a slightly related post here (from Peter Woit):
http://www.math.columbia.edu/~woit/wordpress/?p=3220
http://asunews.asu.edu/20101014_math_school
...and at least a slightly related post here (from Peter Woit):
http://www.math.columbia.edu/~woit/wordpress/?p=3220
Thursday, October 14, 2010
Idle Thoughts...
Just speculatin' here....
The latest edition of "Mathematics Magazine" from MAA lists the top-scoring students in the last USA Mathematical and Junior Mathematical Olympiads, and the names are as follows:
Timothy Chu, Calvin Deng, Michael Druggan, Brian Hamrick, Travis Hance, Xiaoyu He, Mitchell Lee, In Sung Na, Evan O'Dorney, Toan Phan, Hunter Spink, Allen Yuan, Yury Aglyamov, Ravi Bajaj, Evan Chen, Zijing Gao, Gill Goldshlager, Youkow Homma, Jesse Kim, Sadik Shahidain, Alexander Smith, Susan Di Yun Sun, Jiaqi Xie, Jeffrey Yan, Kevin Zhou
One can't help but be struck by the degree to which names of Oriental and Asian ancestry seem to predominate this list of USA Olympians. Even in my own college math courses on the west coast 40 years ago I saw such a predilection; certain foreign nationalities seemed to have a 'knack' for math and analytics that Americans often struggled with. One wonders how deep/real these differences are, and if they are culturally-based or even possibly have a genetic component... or, I'd be curious how many of these students early-on learn foreign ancestral languages in addition to English, and if early exposure to certain languages predisposes one toward a mathematical aptitude or analytical skills in general (in a Whorfian sort of way).
I suspect somewhere out there this has all been looked at, or at least argued over, but don't know if there is a resolution to the notion (and of course the above names represent too small a sample size to draw any real conclusions from, so I'm just wildly wondering out-loud here...). American teaching methods for mathematics have often come under fire in recent times (leaving many young people phobic of the subject), and yet certain students always seem to excel and remain enthused, regardless of method.
The latest edition of "Mathematics Magazine" from MAA lists the top-scoring students in the last USA Mathematical and Junior Mathematical Olympiads, and the names are as follows:
Timothy Chu, Calvin Deng, Michael Druggan, Brian Hamrick, Travis Hance, Xiaoyu He, Mitchell Lee, In Sung Na, Evan O'Dorney, Toan Phan, Hunter Spink, Allen Yuan, Yury Aglyamov, Ravi Bajaj, Evan Chen, Zijing Gao, Gill Goldshlager, Youkow Homma, Jesse Kim, Sadik Shahidain, Alexander Smith, Susan Di Yun Sun, Jiaqi Xie, Jeffrey Yan, Kevin Zhou
One can't help but be struck by the degree to which names of Oriental and Asian ancestry seem to predominate this list of USA Olympians. Even in my own college math courses on the west coast 40 years ago I saw such a predilection; certain foreign nationalities seemed to have a 'knack' for math and analytics that Americans often struggled with. One wonders how deep/real these differences are, and if they are culturally-based or even possibly have a genetic component... or, I'd be curious how many of these students early-on learn foreign ancestral languages in addition to English, and if early exposure to certain languages predisposes one toward a mathematical aptitude or analytical skills in general (in a Whorfian sort of way).
I suspect somewhere out there this has all been looked at, or at least argued over, but don't know if there is a resolution to the notion (and of course the above names represent too small a sample size to draw any real conclusions from, so I'm just wildly wondering out-loud here...). American teaching methods for mathematics have often come under fire in recent times (leaving many young people phobic of the subject), and yet certain students always seem to excel and remain enthused, regardless of method.
Wednesday, October 13, 2010
Gardner Finale...
Gotta be good!:
I just discovered that Martin Gardner's last book, "The Colossal Book of Wordplay" is now out:
http://ken-jennings.com/blog/?p=2195
More fun from the man who made math and fun synonymous!
How timely too with the "Celebration of Mind" commemoration of Martin Gardner just a week away on October 21st.
In an interview, Gardner once said his two favorite books (that he authored) were, "The Whys of a Philosophical Scrivener" (a favorite of mine as well, but one of his lesser-known and non-mathematical works) and his novel "The Flight of Peter Fromm" (which I've never read) --- seemingly odd choices from his vast outpouring, but then Gardner was unpredictable in so many ways. He critically (and hilariously) reviewed/debunked the 'Whys...' book for the NY Times under the alias George Groth; just one of many keenly-clever playful stunts he pulled off over the years.
I'm currently re-reading his "The Colossal Book of Mathematics," the wonderful compendium of his best "Scientific American" columns over the years (with great addenda added)... a must-have volume for any math-fan!
Gardner was a hugely humble individual who might likely be amazed at the outpouring of fondness/admiration expressed upon his death. Luckily he has left us with a lifetime-or-two of reading material. Even putting aside his mathematical work, I regard him as one of the best essayists of all time. Even when disagreeing with his takes on certain matters he was a joy to read (and that is a test of a great writer --- when you treasure reading them even when you disagree with them).
And to think, he barely studied math (academically) beyond high school! What a mind!! (worth a yearly celebration).
Martin... THANKS again for the memories, the musings, and the mathematics!
I just discovered that Martin Gardner's last book, "The Colossal Book of Wordplay" is now out:
http://ken-jennings.com/blog/?p=2195
More fun from the man who made math and fun synonymous!
How timely too with the "Celebration of Mind" commemoration of Martin Gardner just a week away on October 21st.
In an interview, Gardner once said his two favorite books (that he authored) were, "The Whys of a Philosophical Scrivener" (a favorite of mine as well, but one of his lesser-known and non-mathematical works) and his novel "The Flight of Peter Fromm" (which I've never read) --- seemingly odd choices from his vast outpouring, but then Gardner was unpredictable in so many ways. He critically (and hilariously) reviewed/debunked the 'Whys...' book for the NY Times under the alias George Groth; just one of many keenly-clever playful stunts he pulled off over the years.
I'm currently re-reading his "The Colossal Book of Mathematics," the wonderful compendium of his best "Scientific American" columns over the years (with great addenda added)... a must-have volume for any math-fan!
Gardner was a hugely humble individual who might likely be amazed at the outpouring of fondness/admiration expressed upon his death. Luckily he has left us with a lifetime-or-two of reading material. Even putting aside his mathematical work, I regard him as one of the best essayists of all time. Even when disagreeing with his takes on certain matters he was a joy to read (and that is a test of a great writer --- when you treasure reading them even when you disagree with them).
And to think, he barely studied math (academically) beyond high school! What a mind!! (worth a yearly celebration).
Martin... THANKS again for the memories, the musings, and the mathematics!
Tuesday, October 12, 2010
NOT Your High-School Geometry...
Book Review...
I briefly mentioned Shing-Tung Yau's new book, "The Shape of Inner Space" in previous posts (a book about the geometry underlying string theory), but have now read a review copy sent along by the publisher, and can say a little more about it.
In the last couple of decades many books about cosmology have been published (by physicists or science journalists) for a general audience. And many of those volumes have been quite good and accessible to non-professionals, even when the more technical material is not very comprehensible to the average-Joe. Yau's book is somewhat different, on several counts.
"A compact Kahler manifold with a vanishing first Chern class will admit a metric that is Ricci flat."
If the above sentence leaves you in a fog (or worse), I'm not sure this will be a book for you. Yau's book is filled with such language (every time I thought I was entering a more layman-friendly section, it would be short-lived, before I was once again in way over my head). If you are not very familiar with string theory, M-theory, Calabi-Yau manifolds, black holes, branes, and the like, you may want to pass on this volume; having a casual interest in physics/cosmology won't get you through it.
The book involves a lot of heavy-duty mathematics (moreso than other cosmology-type books for the masses; indeed some of it reads more like a textbook than a trade book). I actually enjoy the challenge of reading certain science material that is beyond my comprehension, but I suspect I'm an anomaly, and most folks don't have the patience for reading through material they simply don't understand. In short, I think there is probably a very limited audience for this offering, though that particular audience may very much relish it (...and I stand in awe of them!).
A little background: For any who don't know, Yau is a mathematician (not physicist) and Fields Medalist, with a focus on very advanced/abstract geometry, who's theories (especially proof of 'Calabi-Yau manifolds') came to underlie the mathematics of string theory. For those who don't follow such things, string theory is much more controversial now then when it was first introduced and there seemed to be an almost faddish bandwagon in its direction. Yau's book appears at a time when interest in string theory may even be waning, or at least taking a lot of heat. Yau is quite cognizant of the difficult road ahead for string theory, and how dominant views could change; at one point he writes:
"I personally think Calabi-Yau manifolds are the most elegant formulation [of the underlying geometry of the universe], as well as the most beautiful manifolds constructed so far among all the string vacua. But if the science leads us to some other kind of geometry, I'll willingly follow....Indeed one of the charms of this book is that while so many popular cosmology books beat the drum of the author's particular hardened point-of-view, Yau, as a mathematician, recognizes that he is somewhat apart from these physicist wrestling matches, and can step back, still offering his own personal leanings, while remaining more freely open to new conceptualizations than some other debaters seem to be.
"Despite my affection for Calabi-Yau manifolds --- a fondness that has not diminished over the past thirty-some years --- I'm trying to maintain an open mind on the subject, keeping to the spirit of Mark Gross's earlier remark: 'We just want to know the answer.' If it turns out that non-Kahler manifolds are ultimately of greater value to string theory than Calabi-Yau manifolds, I'm OK with that. For these less-studied manifolds hold peculiar charms of their own. And I expect that upon further digging, I'll come to appreciate them even more."
The two chapters I most enjoyed (comprehended) came toward the end of the volume, "Truth, Beauty, and Mathematics," and "The End of Geometry?," where he waxes somewhat philosophical, even poetic, about the nature of mathematics/geometry and its interplay with physics, and also speculates about a future entailing what he terms 'quantum geometry'... but by then I was pretty tuckered out from the 280 pages that preceded those chapters!
If you are considering purchasing this book I would recommend that you read Peter Woit's review, AND the comments that follow it, here:
http://www.math.columbia.edu/~woit/wordpress/?p=3165
(You might also want to read the Wikipedia entry for "Calabi-Yau manifolds" to get a sense of whether or not you can follow this material.)
And I would also recommend that anyone choosing this volume initially read the 12-or-so pages of glossary at the back of the book just to familiarize yourself with many of the more heavily used terms ahead-of-time (unfortunately, many lesser-used, but difficult, terms are not included in the glossary).
One last ironic note: this book is published by "Basic Books"... one thing it is NOT though, is "basic!"
I don't doubt that it is an excellent exposition of its subject, but it is a challenging read to-be-sure. (In fairness to Basic let me say that I'm also currently reading another of their prior cosmology offerings, Frank Wilczek's "The Lightness of Being," from 2008, and enjoying it considerably more than the Yau volume.)
There is a lot more on Calabi-Yau manifolds around the internet, as well as many more reviews of this particular book available.
Monday, October 11, 2010
Monday Miscellany
Pick-A-Number From 10 to 10000...
Pretty impressive online 'guess-the-number' game here:
http://myframeshoppe.ca/math/
(H/T to Sol for leading me to it)
More on the Oct. 21 worldwide "Celebration of Mind" in memory of Martin Gardner here:
http://www.thinkfun.com/puzzlehunter/?p=308
And last month I mentioned Carl Gauss as one of my "Fab Four" mathematicians of all time. If you need to know more about him, Steven Colyer recently posted a nice mini-biography at his blog:
http://tetrahedral.blogspot.com/2010/10/carl-gauss.html
Pretty impressive online 'guess-the-number' game here:
http://myframeshoppe.ca/math/
(H/T to Sol for leading me to it)
More on the Oct. 21 worldwide "Celebration of Mind" in memory of Martin Gardner here:
http://www.thinkfun.com/puzzlehunter/?p=308
And last month I mentioned Carl Gauss as one of my "Fab Four" mathematicians of all time. If you need to know more about him, Steven Colyer recently posted a nice mini-biography at his blog:
http://tetrahedral.blogspot.com/2010/10/carl-gauss.html
Saturday, October 9, 2010
More On Prime Numbers
Heavy reading...
I don't pretend to understand this... that doesn't stop it from being fascinating:
http://mister-computer.net/primesums/Primes3D.htm
I don't pretend to understand this... that doesn't stop it from being fascinating:
http://mister-computer.net/primesums/Primes3D.htm
Friday, October 8, 2010
Duhhh...
In the food-for-thought dept. this recent 'Tweet' from John Allen Paulos:
"Told of 2 "psychics" who predict coin flips with accuracy of 60% and 20%, respectively, most think first would be more helpful."
Goat Dining...
Here's a nice (I said nice, not simple) little geometry problem to give your tired ol' brain a workout on a Friday morning:
http://www.pedagonet.com/mathgenius/test196.html
http://www.pedagonet.com/mathgenius/test196.html
Thursday, October 7, 2010
Quote... Unquote
Recursivity revisited (or taken to the ultimate limit)...
"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)
Wednesday, October 6, 2010
Not Quite...
An example of teacher MathFail:
(H/T to Mark Chu-Carroll)
(...I'll categorize this under "humor"... though, perhaps not!)
(H/T to Mark Chu-Carroll)
(...I'll categorize this under "humor"... though, perhaps not!)
Tuesday, October 5, 2010
Math Talking Heads
Here's an older clip from a BloggingheadsTV episode with John Horgan and Jim Holt discussing modern-day mathematics:
Monday, October 4, 2010
Yet More Tribute To Martin Gardner
Turns out the "Gathering For Gardner" folks are planning a worldwide "Celebration of Mind" in honor of Martin Gardner for Thur. October 21 (his birthday)... parties wherever individuals organize one. Read about it here (maybe host a get-together yourself!):
http://www.g4g-com.org/
They've also started a Twitter feed here:
http://twitter.com/#!/G4G_CoM
...or hey, get your WWMGT t-shirt here ("What Would Martin Gardner Think"):
http://www.zazzle.com/martin_gardner_fans_tshirt-235153802305461125
http://www.g4g-com.org/
They've also started a Twitter feed here:
http://twitter.com/#!/G4G_CoM
...or hey, get your WWMGT t-shirt here ("What Would Martin Gardner Think"):
http://www.zazzle.com/martin_gardner_fans_tshirt-235153802305461125
Sunday, October 3, 2010
New Twitter
Many (most?) readers here are likely on Twitter. If you think the new Twitter design (if you're using it yet) looks pleasing, the below blogger says there's a mathematical reason for that: the design is based on the Golden Ratio:
http://tinyurl.com/2w6xmcn
http://tinyurl.com/2w6xmcn
Saturday, October 2, 2010
M. Gardner Tribute
Just noticed that the current issue of "Skeptical Inquirer" (Sept./Oct.) has a large section of tributes to Martin Gardner (by many who worked with him over the years in the skeptics' movement). It also includes the final (non-math) column he wrote for the magazine shortly before he passed away. Worth a gander if you have access to it (the issue isn't online as yet).
Book Compendium
By way of review, just a list of some of the books I've mentioned favorably in the last month or so, in no particular order; some older, some new (some I've read myself, some I've only read reviews of):
"The Lifebox, the Seashell, and the Soul" by Rudy Rucker
"Meta Math" by Gregory Chaitin
"Mathematical Fallacies and Paradoxes" by Bryan Bunch
"The Calculus Lifesaver" by Adrian Banner
"The P = NP Question and Gödel’s Lost Letter" by RJ Lipton
"The Shape of Inner Space" by Shing-Tung Yau
"Proofiness: The Dark Arts of Mathematical Deception" by Charles Seife
"Everything and More" by David Foster Wallace
"Group Theory In the Bedroom, and Other Mathematical Diversions" by Brian Hayes
"Quantum Man: Richard Feynman's Life in Science" (forthcoming) by Lawrence Krauss
"Loving and Hating Mathematics: Challenging the Myths of Mathematical Life"(forthcoming) by Reuben Hersh and Vera John-Steiner
...and I'll throw in one additional book that I haven't seen myself, but Sol over at WildAboutMath is strongly recommending:
"The Mystery of the Prime Numbers" by Matthew Watkins
"The Lifebox, the Seashell, and the Soul" by Rudy Rucker
"Meta Math" by Gregory Chaitin
"Mathematical Fallacies and Paradoxes" by Bryan Bunch
"The Calculus Lifesaver" by Adrian Banner
"The P = NP Question and Gödel’s Lost Letter" by RJ Lipton
"The Shape of Inner Space" by Shing-Tung Yau
"Proofiness: The Dark Arts of Mathematical Deception" by Charles Seife
"Everything and More" by David Foster Wallace
"Group Theory In the Bedroom, and Other Mathematical Diversions" by Brian Hayes
"Quantum Man: Richard Feynman's Life in Science" (forthcoming) by Lawrence Krauss
"Loving and Hating Mathematics: Challenging the Myths of Mathematical Life"(forthcoming) by Reuben Hersh and Vera John-Steiner
...and I'll throw in one additional book that I haven't seen myself, but Sol over at WildAboutMath is strongly recommending:
"The Mystery of the Prime Numbers" by Matthew Watkins
Friday, October 1, 2010
The Atlantic Magazine Covers Math!
Nice article in current edition of "The Atlantic" magazine on the 1-year-old community-forum Website, "Math Overflow," where serious mathematicians go to collaborate on solutions to all manner of math problems:
http://www.theatlantic.com/technology/archive/2010/09/beyond-facebook-how-the-worlds-mathematicians-organize-online/63422
Some blurbs therefrom:
http://www.theatlantic.com/technology/archive/2010/09/beyond-facebook-how-the-worlds-mathematicians-organize-online/63422
Some blurbs therefrom:
"Boasting 2,700 active users ranging from especially bright undergrads to Fields medalists, the basic function of the site is to answer the highly technical questions that crop up in math research."
"...organizationally, Math Overflow stands apart from its predecessors. Math Overflow is a community-moderated forum; users vote on the most accurate answers to the questions posed and gain reputation points based on participation, the most active of whom are granted various moderation privileges. The best answers are voted to the top of the page, while the worst ones are voted to the bottom."
"Math Oveflow is almost an anti-social network, focused solely on productively addressing the problems posed by its users. Heavily moderated, the guidelines for asking questions are designed to discourage unnecessary chatter and keep the community's focus on a question at hand...
"We've tried to make the forum as 'professional' as possible," said Scott Morrison..."
"Math Overflow has been a something of a revolution for how collaborative math is carried out on the Web..."
Thursday, September 30, 2010
Wednesday, September 29, 2010
ALL Data/Knowledge on a Computer
For Real??
It's not April 1st so I guess Stephen Wolfram's stated goal to make ALL the world's data computable is a real one...:
http://tinyurl.com/39stuzc (very long post... transcript actually)
Wolfram lays forth his belief that it may be algorithmically possible to save ALL the world's data/knowledge into a working computer format for practical use:
"The idea is: take all the systematic knowledge—and data—that our civilization has accumulated, and somehow make it computable. Make it so that given any specific question one wants to ask, one can just compute the answer on the basis of that knowledge and data."
Although admitting to skepticism earlier on that this could be achieved, Wolfram (encouraged by the huge success of his Mathematica program) now seems optimistic that it is do-able.
Meanhwile, for the audiophiles, a new (hour-long) math podcast here:
http://pulse-project.org/node/230
Finally, I awoke this morning to find this problem posted on Twitter:
Between them, two numbers use all the digits (1 thru 9). What are they, such that their product is as big as possible?
hmmmm...???
It's not April 1st so I guess Stephen Wolfram's stated goal to make ALL the world's data computable is a real one...:
http://tinyurl.com/39stuzc (very long post... transcript actually)
Wolfram lays forth his belief that it may be algorithmically possible to save ALL the world's data/knowledge into a working computer format for practical use:
"The idea is: take all the systematic knowledge—and data—that our civilization has accumulated, and somehow make it computable. Make it so that given any specific question one wants to ask, one can just compute the answer on the basis of that knowledge and data."
Although admitting to skepticism earlier on that this could be achieved, Wolfram (encouraged by the huge success of his Mathematica program) now seems optimistic that it is do-able.
Meanhwile, for the audiophiles, a new (hour-long) math podcast here:
http://pulse-project.org/node/230
Finally, I awoke this morning to find this problem posted on Twitter:
Between them, two numbers use all the digits (1 thru 9). What are they, such that their product is as big as possible?
hmmmm...???
Tuesday, September 28, 2010
New Hersh Book Forthcoming
Reuben Hersh has a new book, with Vera John-Steiner, upcoming (December?) that looks to be good: "Loving and Hating Mathematics: Challenging the Myths of Mathematical Life."
Hersh is one of the premier popular explicators of mathematics of the 20th century. But interestingly, he and Martin Gardner (probably the more popular writer, but less-schooled in academic mathematics than Hersh) feuded over the years, with their opposing views of math's relationship to 'reality.'
Gardner was the adamant and traditional "Platonist" believing (as most intuitively do) that mathematics is a true reflection of the real world (outside the human mind). Seems obvious to many... but in fact a surprising number of professional mathematicians hold to a different view of mathematics, as just another creation of the human mind (not objectively discovered, but very much influenced and created by human culture, cognition, psychology, etc.). Remove humans (and their minds) from the Universe and there would be no mathematical laws, as we perceive them, operating.
I didn't realize, until reading his brief Wikipedia entry, that Hersh actually originally earned a B.A. degree in English Literature (Harvard), and worked as a machinist and writer for Scientific American, before eventually getting his Ph.D. in mathematics in 1962 (New York University).
Here is what Gardner had to say of Hersh in one of his many online interviews:
This simple 2008 piece on the Web addresses the issue:
http://www.canadafreepress.com/index.php/article/2805
And Gardner himself has a wonderful 2005 essay (actually a book review), "A Defense of Platonic Realism," reprinted as chapter 9 in his volume "The Jinn From Hyperspace," if you have access to that.
Or, an earlier Gardner essay entitled "How Not To Talk About Mathematics" in his "The Night Is Large" book (chapter 24) covers much the same ground as well (with more specific reference to Hersh). It's a fascinating debate that won't end anytime soon, and that non-mathematicians often aren't even aware of.
Hersh is one of the premier popular explicators of mathematics of the 20th century. But interestingly, he and Martin Gardner (probably the more popular writer, but less-schooled in academic mathematics than Hersh) feuded over the years, with their opposing views of math's relationship to 'reality.'
Gardner was the adamant and traditional "Platonist" believing (as most intuitively do) that mathematics is a true reflection of the real world (outside the human mind). Seems obvious to many... but in fact a surprising number of professional mathematicians hold to a different view of mathematics, as just another creation of the human mind (not objectively discovered, but very much influenced and created by human culture, cognition, psychology, etc.). Remove humans (and their minds) from the Universe and there would be no mathematical laws, as we perceive them, operating.
I didn't realize, until reading his brief Wikipedia entry, that Hersh actually originally earned a B.A. degree in English Literature (Harvard), and worked as a machinist and writer for Scientific American, before eventually getting his Ph.D. in mathematics in 1962 (New York University).
Here is what Gardner had to say of Hersh in one of his many online interviews:
"Reuben Hersh is a marvelous example of a person who thinks that mathematics is entirely a human product and has no reality outside of human culture. He has written a whole book about this called "What Is Mathematics Really?" To Reuben Hersh, mathematics is no different from art or fashions in clothes. It’s a cultural phenomenon. The postmodernists in France have essentially this point of view. And it drives me up the wall. I like to say, 'If two dinosaurs met two other dinosaurs in a clearing, there would be four of them even though the animals would be too stupid to know that.' Of course, the argument as to whether the universe exists outside of the human mind goes back to the middle ages."(I suspect Hersh might object to at least part of this characterization.) The irreconcilable differences between these two pillars of 20th century math reporting/education is fascinating, and both views have very bright, significant supporters on their sides (if anything, the Hersh view may even have made gains in recent years, though the Platonist view still predominates).
This simple 2008 piece on the Web addresses the issue:
http://www.canadafreepress.com/index.php/article/2805
And Gardner himself has a wonderful 2005 essay (actually a book review), "A Defense of Platonic Realism," reprinted as chapter 9 in his volume "The Jinn From Hyperspace," if you have access to that.
Or, an earlier Gardner essay entitled "How Not To Talk About Mathematics" in his "The Night Is Large" book (chapter 24) covers much the same ground as well (with more specific reference to Hersh). It's a fascinating debate that won't end anytime soon, and that non-mathematicians often aren't even aware of.
Monday, September 27, 2010
There Was A Young Lady With Curves...?
Not exactly Robert Frost, but just for fun today a page of math limericks:
http://www.trottermath.net/humor/limricks.html
Here's a couple of samples, just to get you in the right mood:
http://www.trottermath.net/humor/limricks.html
Here's a couple of samples, just to get you in the right mood:
"A graduate student at Trinity Computed the square of infinity. But it gave him the fidgets To put down the digits, So he dropped math and took up divinity."
"Archimedes, the well known truth-seeker, Jumping out of his bath, cried, "Eureka!" He ran half a mile, Wearing only a smile, And became the very first streaker."
Saturday, September 25, 2010
Open Science?
Without even mentioning P vs. NP (though obviously there are implications for that debate), RJ Lipton's latest post interestingly addresses the question of whether the sharing of ideas and partial results in math/science is a valuable thing (before all issues are worked out). Such 'open science' seems to me to be the wave of the future, but there are undoubtedly habits and considerations that will need to be overcome:
http://tinyurl.com/37hcy7z
(...and I love the quote he includes from Howard Aiken: “Don’t worry about people stealing an idea. If it’s original, you will have to ram it down their throats.”) ;-)
Lipton employs several famous math examples for his discussion, and ends with this question for his readers:
http://tinyurl.com/38d67g9
http://tinyurl.com/37hcy7z
(...and I love the quote he includes from Howard Aiken: “Don’t worry about people stealing an idea. If it’s original, you will have to ram it down their throats.”) ;-)
Lipton employs several famous math examples for his discussion, and ends with this question for his readers:
"Should researchers be encouraged to share ideas earlier than we do now? What mechanisms are needed to be sure that proper credit is given out? Would you publish a partial result?"
ADDENDUM: another pertinent post here:
http://tinyurl.com/38d67g9
Friday, September 24, 2010
Paul Erdos
Image via Wikipedia
A documentary portrait of the amazing Paul Erdos (1913-1996) via YouTube clips here:
http://tinyurl.com/2f2776n
To this day Erdos is the most published mathematician of all time. He was well-known as a traveling math vagabond who would simply hang out with colleagues he had all over the world, after showing up on their doorstep and announcing, "My brain is open," essentially meaning 'hey, let's do some math!' And yet despite many award-winning accomplishments in several math fields he was initially flummoxed by the well-known 'Monty Hall problem.' A fascinating character, whose likes we may never see again.
Thursday, September 23, 2010
Have I Mentioned Any Math Books Before...
Well, here's some mmmmore:
Browsing in a local bookstore I noticed the new edition of David Foster Wallace's 2003 "Everything and More" is now out. It has a newly-written Introduction, but otherwise, so far as I could tell, there were no changes/corrections to the text (???). Wallace died tragically at his own hands a couple years ago. Many mathematicians would not recommend this work, but if you're a lover of math and infinity, combined with wordplay (or what some call Wallace's "verbal pyrotechnics"), I think it a worthwhile, if eclectic, read (my previous mini-review HERE).
Wallace's volume was part of Norton's "Great Discoveries Series." Though not truly a math book, I'll just point out that another upcoming volume in that series will be a new biography of Richard Feynman from physicist Lawrence Krauss... ought to be good, keep an eye out.
Also worth noting that Stephen Wolfram has announced that his massive 2002 tome "A New Kind Of Science" is now available on the iPad:
http://tinyurl.com/33sjswd
Finally, what I did pick out (based solely on Martin Gardner's ringing endorsement on back cover) during my browse through the bookstore, was Brian Hayes' 2008 "Group Theory In the Bedroom, and Other Mathematical Diversions." Gardner wrote, "Every essay in this book is a gem of science writing at its highest level...Its scope is awesome... There isn't a dull page in the book." Into the reading queue it goes....
Browsing in a local bookstore I noticed the new edition of David Foster Wallace's 2003 "Everything and More" is now out. It has a newly-written Introduction, but otherwise, so far as I could tell, there were no changes/corrections to the text (???). Wallace died tragically at his own hands a couple years ago. Many mathematicians would not recommend this work, but if you're a lover of math and infinity, combined with wordplay (or what some call Wallace's "verbal pyrotechnics"), I think it a worthwhile, if eclectic, read (my previous mini-review HERE).
Wallace's volume was part of Norton's "Great Discoveries Series." Though not truly a math book, I'll just point out that another upcoming volume in that series will be a new biography of Richard Feynman from physicist Lawrence Krauss... ought to be good, keep an eye out.
Also worth noting that Stephen Wolfram has announced that his massive 2002 tome "A New Kind Of Science" is now available on the iPad:
http://tinyurl.com/33sjswd
Finally, what I did pick out (based solely on Martin Gardner's ringing endorsement on back cover) during my browse through the bookstore, was Brian Hayes' 2008 "Group Theory In the Bedroom, and Other Mathematical Diversions." Gardner wrote, "Every essay in this book is a gem of science writing at its highest level...Its scope is awesome... There isn't a dull page in the book." Into the reading queue it goes....
Wednesday, September 22, 2010
Factoid...
The formula n^2 - n + 41 produces prime numbers for all n's from zero to 40. At 41 it fails.
Primes and Quantum Physics
We normally think of mathematics underlying and driving ideas in physics... In this interesting 2006 article (focused around a chance 1972 meeting between Freeman Dyson and number theorist Hugh Montgomery) Marcus du Sautoy writes of how ideas from quantum physics may underlie and help deepen our understanding of prime numbers:
http://seedmagazine.com/content/article/prime_numbers_get_hitched/
Speaking of du Sautoy, you may wish to check out his website for "The Story of Maths" series he did for the BBC:
http://www.open2.net/storyofmaths/index.html
http://seedmagazine.com/content/article/prime_numbers_get_hitched/
Speaking of du Sautoy, you may wish to check out his website for "The Story of Maths" series he did for the BBC:
http://www.open2.net/storyofmaths/index.html
Tuesday, September 21, 2010
'nuther Oldie but Goodie
This is an old puzzle that comes in a variety of forms. I've adapted it here from a Martin Gardner version in his "Aha! Gotcha" volume:
6 students make reservations for a dinner at a popular pizza place. But at the last minute a 7th student decides to join them.
When the kids arrive the hostess immediately sees that there are 7 diners for her table set-up of 6. But she is a clever one. Thinking on her feet, she decides to seat the first student in chair #1 and then have his girlfriend (the second student) sit on his lap temporarily. Then the hostess can sit student #3 in the 2nd chair, student #4 in the 3rd chair, student #5 in the 4th chair, and finally student #6 goes into chair #5. Chair #6 is thus still leftover, and so of course the hostess can now move the original girlfriend to that seat. Waaah-laaaaah!!
Hope she gets a big tip... or, maybe not!?
Do you see the flaw in her method?
(For any who don't see through the flaw I'll wait 24 hrs. and explain the simple answer in the comments below... and then you can go "DOH!")
I'm sure the young lass (another child prodigy) reported on in this nice article can spot the catch in the math:
http://tinyurl.com/34arukx
6 students make reservations for a dinner at a popular pizza place. But at the last minute a 7th student decides to join them.
When the kids arrive the hostess immediately sees that there are 7 diners for her table set-up of 6. But she is a clever one. Thinking on her feet, she decides to seat the first student in chair #1 and then have his girlfriend (the second student) sit on his lap temporarily. Then the hostess can sit student #3 in the 2nd chair, student #4 in the 3rd chair, student #5 in the 4th chair, and finally student #6 goes into chair #5. Chair #6 is thus still leftover, and so of course the hostess can now move the original girlfriend to that seat. Waaah-laaaaah!!
Hope she gets a big tip... or, maybe not!?
Do you see the flaw in her method?
(For any who don't see through the flaw I'll wait 24 hrs. and explain the simple answer in the comments below... and then you can go "DOH!")
I'm sure the young lass (another child prodigy) reported on in this nice article can spot the catch in the math:
http://tinyurl.com/34arukx
Monday, September 20, 2010
Boston Museum
Nice article from Boston Globe on the "Mathematica" exhibit at their city Museum of Science, still enthralling young and old after all these years:
http://tinyurl.com/329upk5
(I was especially heartened to see the quincunx or Galton Box mentioned which I previously wrote about as one of my joys as a youngster; although it is only designated in the article as a "probability" demonstration.)
http://tinyurl.com/329upk5
(I was especially heartened to see the quincunx or Galton Box mentioned which I previously wrote about as one of my joys as a youngster; although it is only designated in the article as a "probability" demonstration.)
Sunday, September 19, 2010
Proofiness, Numerical Mendacity, and Civics
What has Stephen Colbert wrought....
NY Times favorable review (from Steven Strogatz) of Charles Seife's latest book, "Proofiness: The Dark Arts of Mathematical Deception":
http://tinyurl.com/2dphrtd
(promoted as a book about "the art of using pure mathematics for impure ends")
Strogatz concludes thusly, "For the most part, though, he [Seife] is deadly serious. A few other recent books have explored how easily we can be deceived — or deceive ourselves — with numbers. But “Proofiness” reveals the truly corrosive effects on a society awash in numerical mendacity. This is more than a math book; it’s an eye-opening civics lesson."
Saturday, September 18, 2010
Have It Your Way...
This ain't your Daddy's Burger King...
Math on the menu (billions of burger choices) at a new NY restaurant:
http://tinyurl.com/22pddta
ohhh, and do you want fries with that. . . .
Math on the menu (billions of burger choices) at a new NY restaurant:
http://tinyurl.com/22pddta
ohhh, and do you want fries with that. . . .
More Books
How timely!
RJ Lipton, who has been one of the primary communicators/bloggers of the recent P vs. NP happenings, and someone with a long-time particular interest in the issue, has newly published a book on the topic, "The P = NP Question and Gödel’s Lost Letter":
http://tinyurl.com/2v88wwd
(it's a tad expensive, and NOT bedtime reading!)
What ought make the book particularly interesting though is that Lipton is among the minority who still believe that P may indeed EQUAL NP (versus the majority belief that P ≠ NP).
Changing gears from one deep problem to another, Harvard mathematician and Fields Medalist Shing-Tung Yau has a recent well-received book out, "The Shape of Inner Space," which recounts the geometry/mathematics underlying string theory:
http://tinyurl.com/26u4o2q
RJ Lipton, who has been one of the primary communicators/bloggers of the recent P vs. NP happenings, and someone with a long-time particular interest in the issue, has newly published a book on the topic, "The P = NP Question and Gödel’s Lost Letter":
http://tinyurl.com/2v88wwd
(it's a tad expensive, and NOT bedtime reading!)
What ought make the book particularly interesting though is that Lipton is among the minority who still believe that P may indeed EQUAL NP (versus the majority belief that P ≠ NP).
Changing gears from one deep problem to another, Harvard mathematician and Fields Medalist Shing-Tung Yau has a recent well-received book out, "The Shape of Inner Space," which recounts the geometry/mathematics underlying string theory:
http://tinyurl.com/26u4o2q
ADDENDUM: Peter Woit's review of Yau's book is now up here:
http://www.math.columbia.edu/~woit/wordpress/?p=3165
http://www.math.columbia.edu/~woit/wordpress/?p=3165
Friday, September 17, 2010
The Special Number 3816547290
Read all about it (find out why it's special):
http://everything2.com/title/3816547290
...and some more on this peculiar number here (from a computational-science slant):
http://scienceblogs.com/builtonfacts/2010/08/a_conspiracy_of_digits.php
...and one more further follow-up here:
http://tinyurl.com/29rvhae
http://everything2.com/title/3816547290
...and some more on this peculiar number here (from a computational-science slant):
http://scienceblogs.com/builtonfacts/2010/08/a_conspiracy_of_digits.php
...and one more further follow-up here:
http://tinyurl.com/29rvhae
Thursday, September 16, 2010
Pi: The Number That Keeps On Giving....
Ever get the feeling that perhaps some people just have too much free time on their hands ;-)....
In news, that we've all been hungry for, it's been announced that the two-quadrillionth digit of pi has been determined... in binary... to be a zero!:
http://www.bbc.co.uk/news/technology-11313194
It's still the case that only a mere 2.7 trillion consecutive digits of pi are known, but powerful cloud computer techniques are now able to find specific digits much farther out. But more importantly... can these techniques help me find my car keys in the morning???
Seriously though, congratulations to those involved!
In news, that we've all been hungry for, it's been announced that the two-quadrillionth digit of pi has been determined... in binary... to be a zero!:
http://www.bbc.co.uk/news/technology-11313194
It's still the case that only a mere 2.7 trillion consecutive digits of pi are known, but powerful cloud computer techniques are now able to find specific digits much farther out. But more importantly... can these techniques help me find my car keys in the morning???
Seriously though, congratulations to those involved!
Just a Few Thangs
RJ Lipton offers a brief update on where things stand with Vinay Deolalikar’s PNP “proof” here:
http://tinyurl.com/2wqandk
(essentially, it's still being discussed and worked on, and we're a long way from final resolution)
The 3rd "Mathematics and Multimedia Blog Carnival" is up-and-running here:
http://math4allages.wordpress.com/2010/09/13/blog-carnival-3/
Some unresolved prime number questions here:
http://twitpic.com/2ou1x8
And lastly, another quickie puzzle (that MAA's "MinuteMath" ran recently):
A certain positive integer "x" has the property that x% of x = 4. What is "x"?
answer below:
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solution x = 20
http://tinyurl.com/2wqandk
(essentially, it's still being discussed and worked on, and we're a long way from final resolution)
The 3rd "Mathematics and Multimedia Blog Carnival" is up-and-running here:
http://math4allages.wordpress.com/2010/09/13/blog-carnival-3/
Some unresolved prime number questions here:
http://twitpic.com/2ou1x8
And lastly, another quickie puzzle (that MAA's "MinuteMath" ran recently):
A certain positive integer "x" has the property that x% of x = 4. What is "x"?
answer below:
.
.
.
.
.
.
.
.
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.
.
.
solution x = 20
Wednesday, September 15, 2010
Nerds, Geeks, Dorks, Dweebs
Why didn't they have Venn Diagrams like this back when I was in school:
http://tinyurl.com/ydvzqld
(but it doesn't include 'buffs'...)
http://tinyurl.com/ydvzqld
(but it doesn't include 'buffs'...)
The Jordan Curve Theorem
The Jordan Curve Theorem (named for a French mathematician who first proved it) states that any continuous simple closed curve in a plane, separates the plane into two disjoint regions, the inside and the outside.
...Seems intuitively pretty straightforward, or as one of the books I have on my shelf says,
"The theorem seems like a statement of the blindingly obvious. If a curve proceeds continuously without any breaks in it and returns to its starting point without crossing itself, then there will be a region outside the curve and a region inside. The two regions are separate, one is finite and the other is infinite."
While this is indeed clear for the run-of-the-mill closed curves we are accustomed to seeing, there are far more complex topological curves, including for example the Koch Snowflake, that may help one see why the theorem's proof (and it has been proved) is not at all easy (some of the proofs run to 1000's of lines).
A couple of discussions of the theorem here:
http://tinyurl.com/26tfk7v
http://www.math.ohio-state.edu/~fiedorow/math655/Jordan.html
...Seems intuitively pretty straightforward, or as one of the books I have on my shelf says,
"The theorem seems like a statement of the blindingly obvious. If a curve proceeds continuously without any breaks in it and returns to its starting point without crossing itself, then there will be a region outside the curve and a region inside. The two regions are separate, one is finite and the other is infinite."
While this is indeed clear for the run-of-the-mill closed curves we are accustomed to seeing, there are far more complex topological curves, including for example the Koch Snowflake, that may help one see why the theorem's proof (and it has been proved) is not at all easy (some of the proofs run to 1000's of lines).
A couple of discussions of the theorem here:
http://tinyurl.com/26tfk7v
http://www.math.ohio-state.edu/~fiedorow/math655/Jordan.html
Tuesday, September 14, 2010
Number Digit Puzzle
Can you find a 10 digit number such that:
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answer: 6210001000
This is called a "self-descriptive number," BTW, and it is the only one in base 10.
- the 1st digit tells how many zeros are in the number,
- the 2nd digit tells how many 1's are in the number,
- the 3rd digit tells how many 2's are in the number,
- the 4th digit tells how many 3's are in the number, etc. etc. (10th digit tells how many 9's are in no.)
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answer: 6210001000
This is called a "self-descriptive number," BTW, and it is the only one in base 10.
Einstein In Context...
A couple of folks have emailed me over time with inquiries about the (somewhat famous) quotation from Albert Einstein used in the right-hand column of this blog. For anyone interested in its fuller context, it comes from this 1921 address of Einstein's to the Prussian Academy of Sciences in Berlin, entitled "Geometry and Experience." (Einstein was essentially addressing the notion of what would later come to be known as "the unreasonable effectiveness of mathematics" (Wigner) when he made the remark):
http://tinyurl.com/2b4375e
http://tinyurl.com/2b4375e
Monday, September 13, 2010
This and That
A mathematics blogger comments on the "hoopla" around Stephen Hawking's new book here:
http://mathrising.com/?p=238
Does learning higher mathematics take away from the ability to think sharply on a problem? The question is raised here (includes a nice 'algebra' problem):
http://tinyurl.com/2v7uz7f
... and Princeton University Press is giving away a copy of the new book, "The Calculus Lifesaver," by Adrian Banner:
http://tinyurl.com/3426tfj
http://mathrising.com/?p=238
Does learning higher mathematics take away from the ability to think sharply on a problem? The question is raised here (includes a nice 'algebra' problem):
http://tinyurl.com/2v7uz7f
... and Princeton University Press is giving away a copy of the new book, "The Calculus Lifesaver," by Adrian Banner:
http://tinyurl.com/3426tfj
DOH!! Page 29...
Some more video entertainment today. A clip from back in 2008 of someone expressing the joy/passion that is evoked by a claim that the Riemann Hypothesis has been proven:
And now check out this note:
http://arxiv.org/abs/0807.0090
And now check out this note:
http://arxiv.org/abs/0807.0090
Saturday, September 11, 2010
A Book, and a Crafty Law Student
An old Dover publication I only recently stumbled upon is one of the best I've seen at succinctly covering many of the most trenchant paradoxes and self-reference issues underlying mathematics:
Bryan Bunch's "Mathematical Fallacies and Paradoxes" (1982) covers a wide selection of, guess what, fallacies and paradoxes; some fairly light, others deeper and heavier. Below I've totally re-adapted one of the many self-reference paradoxes contained in the volume:
Bryan Bunch's "Mathematical Fallacies and Paradoxes" (1982) covers a wide selection of, guess what, fallacies and paradoxes; some fairly light, others deeper and heavier. Below I've totally re-adapted one of the many self-reference paradoxes contained in the volume:
To demonstrate what a fine teacher he is, Larry the Lawyer contracts with each of his students such that they need only pay him for his individual instruction IF and ONLY IF they win their first law case. If they lose that first case they pay him no fee.The Bunch book definitely isn't for everyone (not even for all math buffs), but if you have an especial interest in the paradoxes and intrinsic issues that underlie uncertainty in mathematics, as well as in science and knowledge more generally, it's worth a look.
However, one of his students, Squiggy, upon completing the course, opts simply not to try any cases at all to avoid paying any fee. Perturbed, Larry feels compelled to sue Squiggy for payment (since avoiding trial cases in order to avoid payment was not intended as an option). Once the case comes to court Squiggy represents himself. IF he loses, then by the original contract he does NOT have to pay Larry! IF he wins the suit then the court will have ruled that he does NOT have to pay! Squiggy appears well on his way to being a superb lawyer....
Friday, September 10, 2010
"Nerd Superbowl" & The Halting Problem
Some stuff to chew on for weekend:
Nice wrap-up to the current state of the recent P vs. NP excitement from "Math Trek" here:
http://tinyurl.com/32smtbf
And below, Mark Chu-Carroll takes his readers through the 'halting problem' and its unsolvability in a recent post:
http://scientopia.info/blogs/goodmath/2010/09/08/1069/#more-1069
Nice wrap-up to the current state of the recent P vs. NP excitement from "Math Trek" here:
http://tinyurl.com/32smtbf
And below, Mark Chu-Carroll takes his readers through the 'halting problem' and its unsolvability in a recent post:
http://scientopia.info/blogs/goodmath/2010/09/08/1069/#more-1069
Thursday, September 9, 2010
The Mathematics of Sunscreen
Higher and higher SPFs....
Older NY Times article here:
http://www.nytimes.com/2009/05/14/fashion/14SKIN.html?_r=2
Older NY Times article here:
http://www.nytimes.com/2009/05/14/fashion/14SKIN.html?_r=2
Visualizing Data...
This David McCandless (journalist) 18-minute TEDTalk has been bopping around the Web of late, in which he promotes the utility of 'visualizing' numbers/data/context; probably a tad oversimplified, but still interesting:
David's blog is here: http://www.informationisbeautiful.net/
David's blog is here: http://www.informationisbeautiful.net/
Wednesday, September 8, 2010
Tuesday, September 7, 2010
Math: A Young Man's Game?
Another interesting post from Alex Bellos, this time on "maths' cult of youth" (the common notion that great mathematicians invariably do their best work in youth, and are past their prime by age 40).
http://alexbellos.com/?p=1354
Starts off focusing on now 15-year-old child prodigy Arran Fernandez, before addressing the issue more broadly.
(...Hopefully, we'll get a few more productive years out of 35-year-old Terence Tao , who some consider the world's greatest living mathematician, before he commences his downward slide. ;-))
http://alexbellos.com/?p=1354
Starts off focusing on now 15-year-old child prodigy Arran Fernandez, before addressing the issue more broadly.
(...Hopefully, we'll get a few more productive years out of 35-year-old Terence Tao , who some consider the world's greatest living mathematician, before he commences his downward slide. ;-))
My Fab Four
Speaking of Euler, as I did in the prior post....
It's of course impossible to name the four most important or influential or greatest mathematicians of all time. The arguments could go 'round-and-'round forever without resolution, and math is a surprisingly diverse, robust field where different individuals make all kinds of different contributions.
Nonetheless, one of the math T-shirts I offer at my Zazzle store is "The Fab Four" where I've designated the four I would pick out for such an honor if forced to choose. For the combined breadth, depth, and variety of their contributions they are (with their Wikipedia links):
Leonhard Euler
(1707 - 1783)
Carl Friedrich Gauss
(1777 - 1855)
David Hilbert
(1862 - 1943)
G.F. Bernhard Riemann
(1826 - 1866)
I'm leaving out both a lot of hugely important ancient and modern-day mathematicians, but these are just my subjective choices. Who might you choose for a Fab Four? Here's a list of 'greatest mathematicians' as formulated by someone else:
http://fabpedigree.com/james/greatmm.htm
It's of course impossible to name the four most important or influential or greatest mathematicians of all time. The arguments could go 'round-and-'round forever without resolution, and math is a surprisingly diverse, robust field where different individuals make all kinds of different contributions.
Nonetheless, one of the math T-shirts I offer at my Zazzle store is "The Fab Four" where I've designated the four I would pick out for such an honor if forced to choose. For the combined breadth, depth, and variety of their contributions they are (with their Wikipedia links):
Leonhard Euler
(1707 - 1783)
Carl Friedrich Gauss
(1777 - 1855)
David Hilbert
(1862 - 1943)
G.F. Bernhard Riemann
(1826 - 1866)
I'm leaving out both a lot of hugely important ancient and modern-day mathematicians, but these are just my subjective choices. Who might you choose for a Fab Four? Here's a list of 'greatest mathematicians' as formulated by someone else:
http://fabpedigree.com/james/greatmm.htm
Sunday, September 5, 2010
e^i(π) + 1 = 0
Astrophysicist Adam Frank waxes poetic (almost) over Euler's "incomparable and glorious" Identity, involving "five magic numbers," and "beauty" (...with Justin Bieber thrown in for good measure):
http://www.npr.org/blogs/13.7/2010/09/02/129610905/best-equation-ever?ft=1&f=114424647
Frank asks, "Why are so many mathematically inclined folks sent into paroxysms of delight over this string of symbols which seem like gibberish to others," and then he proceeds to answer the question for the reader.
http://www.npr.org/blogs/13.7/2010/09/02/129610905/best-equation-ever?ft=1&f=114424647
Frank asks, "Why are so many mathematically inclined folks sent into paroxysms of delight over this string of symbols which seem like gibberish to others," and then he proceeds to answer the question for the reader.
Friday, September 3, 2010
New Carnival of Mathematics
The 69th "Carnival of Mathematics" it is now available (with the usual variety of math-festive offerings) here:
http://jd2718.wordpress.com/2010/09/03/carnival-of-mathematics-69/
http://jd2718.wordpress.com/2010/09/03/carnival-of-mathematics-69/
Fermat's Last Theorem Used in Proof
For any greater than is irrational...
A demonstration of Fermat's Last Theorem being used in a very efficient ("charming") proof of the above:
http://tinyurl.com/2bof7l6
(...this is almost too simple)
A demonstration of Fermat's Last Theorem being used in a very efficient ("charming") proof of the above:
http://tinyurl.com/2bof7l6
(...this is almost too simple)
Who Knew!
I just discovered there is a YouTube channel devoted to number theory. Latest offering here:
http://tinyurl.com/26unyae
...and a few problems (from elsewhere) to play with here:
http://threesixty360.wordpress.com/2010/09/02/its-a-new-newsletter/
(I suspect 4.2.3 has some shortcut solution, though I'm not seeing it?)
http://tinyurl.com/26unyae
...and a few problems (from elsewhere) to play with here:
http://threesixty360.wordpress.com/2010/09/02/its-a-new-newsletter/
(I suspect 4.2.3 has some shortcut solution, though I'm not seeing it?)
Thursday, September 2, 2010
Cryptography Basics
Nice basic intro to cryptography (and especially RSA encryption) from Daniel Chiquito over at Equalis site (including a recommendation for Simon Singh's "The Code Book" as a good source of further learning):
http://www.equalis.com/members/blog_view.asp?id=565749&post=108495
http://www.equalis.com/members/blog_view.asp?id=565749&post=108495
Wednesday, September 1, 2010
Women In Math
Over a year ago on her blog Tanya Khovanova mentioned the numerical outcome of googling "male mathematicians" and "female mathematicians." So I just did the same and similarly found a heavily-weighted favoritism toward females:
"male mathematician" --- 3420 hits
"female mathematician" --- 8990 hits
At first glance of course this seems peculiar given the preponderance of male mathematicians in society over females, but of course upon a moment's reflection it's clear that someone doing a search would be far more likely to"search" for find "female mathematicians" than "male mathematicans" (which is, at least subconsciously, for a lot of Americans, almost a redundancy!).
At any rate, many are working hard today, to alter this gender-bias and specifically encourage females to pursue mathematics, at least as an interest, if not even a vocation. One organization active in that endeavor is the Association For Women In Mathematics:
http://www.awm-math.org/
Check 'em out, especially if you're interested in math, and you're a possessor of two 'X' chromosomes!
"male mathematician" --- 3420 hits
"female mathematician" --- 8990 hits
At first glance of course this seems peculiar given the preponderance of male mathematicians in society over females, but of course upon a moment's reflection it's clear that someone doing a search would be far more likely to
At any rate, many are working hard today, to alter this gender-bias and specifically encourage females to pursue mathematics, at least as an interest, if not even a vocation. One organization active in that endeavor is the Association For Women In Mathematics:
http://www.awm-math.org/
Check 'em out, especially if you're interested in math, and you're a possessor of two 'X' chromosomes!
Prime Numbers Always Intrigue
In 1968, a Russian mathematician conjectured that, on average, the number of prime numbers for which the sum of their digits was even was equal to the number of prime numbers for which that sum was odd.
The conjecture has now been proven true, by French mathematicians:
http://www.physorg.com/news192907929.html
Can't honestly say I understand what the significance of the finding is 8-\ but the article concludes thusly:
The conjecture has now been proven true, by French mathematicians:
http://www.physorg.com/news192907929.html
Can't honestly say I understand what the significance of the finding is 8-\ but the article concludes thusly:
"The methods employed to arrive at this result, derived from combinatorial mathematics, the analytical theory of numbers and harmonic analysis, are highly groundbreaking and should pave the way to the resolution of other difficult questions concerning the representation of certain sequences of integers.Pretty much any proof that involves prime numbers has a lot of potential consequences...
Quite apart from their theoretical interest, these questions are directly linked to the construction of sequences of pseudo-random numbers and have important applications in digital simulation and cryptography."
Tuesday, August 31, 2010
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/
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/
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