Welcome to the 37th edition of the Carnival of Mathematics!

In preparation for this edition, I actually managed to secure an exclusive interview with the number 37, and have included a small portion of our conversation below:

Logic Nest (LN): So 37, what have you been up to lately?

37: Oh, not much. I’ve always had a fairly good life given that I’m not only a prime number, but a lucky, irregular, AND unique prime. It’s summertime where I live, so mostly I hang out by the pool with my good friends 16, 21, and 28. We’re in a band together called the Padovan Sequence.

LN: Wow. That’s very interesting. I’ve heard that some people think you’re unlucky though. What do you say about that?

37: That’s totally a fabrication. Just because I’m the number 666 divided by its digits added together [37=666/(6+6+6)] doesn’t mean a thing.

LN: Understandable. I can see the confusion. I’ve heard that there’s a website out there that’s all about you, is that true?

37: Yes, and I must admit that I’m a bit embarrassed about it. Just because I pop up in all sorts of scientific, cultural, and historical situations doesn’t mean that I should have a fansite. I mean, come on now, people…

And it went on like that for a while…

Speaking of prime numbers, let’s kick the carnival off with this article submitted by Jeffrey Shallit from Recursivity about a Rutgers graduate student named Eric Rowland who has proved a new prime-generating formula that’s quite simple. There are some great comments on this post that include various programming implementations of the formula.

Over at Walking Randomly, Mike Croucher has posted his second Integral of the Week involving an exponential function and the square root of pi. The twist on this problem is that he gives you the evaluation and asks you to prove it. In addition, he’s asking readers to exclude the common evaluation method of converting the integral to polar coordinates. He’s taking solutions via the comments on the site. There are already a few proposed solutions, but take some time to think it over before jumping straight to the comments!

“A” presents an editorial on Being Bad at Math posted at It’s the Thought that Counts. This post is about the popular idea that it’s acceptable to confess a total lack of math ability, even though equivalent statements about difficulty in something like one’s native tongue would be seen as embarrassing. This post explores a cultural brushing off of mathematics, and how this idea should no longer be tolerated in the twenty-first century.

Another great lesson in math and culture comes from Barry Leiba, who points out a personal pet peeve of mine in his article That’s a mean median posted at Staring At Empty Pages, namely that people often incorrectly equate “median” with “average”, even at the New York Times. This one should get the blood of you stats people out there boiling!

Given the impending American presidential election, Barry Wright, III presents an educational post entitled Plurality Winner, Condorcet Loser? at fashionablemathematician - mathematics. The contents of the article explores various ideas that Barry is exploring from Donald Saari’s Basic Geometry of Voting, which is a text he is using “both for research purposes and to prepare to TA a class on the mathematics involved in Democracy, voting systems, and the like”. By definition, “a Condorcet winner is one which is ranked higher than every other alternative in a majority of decisions” while a “plurality winner is an alternative which receives more first-place votes than any other alternatives”. As the title implies, there is an interesting case when one can be both a plurality winner and a Condorcet loser. Good stuff.

In The Universe of Discourse : Period Three and Chaos posted at The Universe of Discourse, Mark Dominus gives us some information about Möbius functions, which are of major importance in complex analysis, where they correspond to certain transformations of the Riemann sphere. In particular, he looks at Möbius functions with real coefficients. In this post he talks about functions with a periodic point of order 3 (where f(f(f(x))) = x for some x) in connection to the Sharkovskii’s theorem. Both of these concepts are explained more fully at the link above.

Denise presents Math History on the Internet posted at Let’s play math!. She presents links for some WONDERFUL historical resources available on the web. As she says, “the story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.” There’s some really great stuff available at this link for anyone interested in picking up some mathematics history!

In his post Playing with Permutations at Reasonable Deviations, Rod Carvalho proposes a 2-player game. The goal is to find out whether a necessary condition is also sufficient. This game blends Combinatorics with Algebra, and even Algebraic Geometry. It’s an interesting game to consider and builds on a few other posts that Rod has written since January 2008.

Ron Cook from The Endeavour gives us an explanation of Random Inequalities in this three part series. Random inequalities are often used in Bayesian clinical trial methods, and should interest all the stats people who are reading. The first part introduces the reader to the concept of random inequalities, the second part shows how they are analytically evaluated, and the third shows how they are numerically evaluated when analytical evaluation is not possible.

Lastly, Ξ_Heather wants us to think about Burnt Pancakes and Godzilla at her article 4, 6, 8, 10, 12, 14,?.What comes next? posted at 360. As she explains, “the Burnt Pancake problem involves pancakes of different sizes, each with one burnt side, piled up on top of one another.” It’s great content explained in an entertaining manner. FYI, Godzilla evidently wears a chef’s hat when cooking pancakes.

Here are a few more submissions that have come in since I initially published last night:

Alvaro Fernandez presents Top 10 Brain Training Future Trends posted at Sharp Brains.This article discusses the concept of “brain training”, or how we keep our brains fit. This is particularly interesting given that mathematics is commonly perceived as a game for the young, as evidenced by this XKCD comic. Take care of your brains, people!

Are you aware that there is an Encyclopedia of Triangle Centers? David Eppstein is, and he describes another kind of triangle center, different from the ones at the Encyclopedia, here.

Catsynth asks the question, “What do you get when you mix a cat and a Fourier Transform?” in this post. Education and entertainment ensue! The lesson to be learned is simple: be careful of what mathematical transforms you perform on your pets. Obviously.

That’s all for this edition! If you’d like to post any additional articles to this edition of the Carnival, please contact me. I’ll be taking submissions through Sunday evening. Otherwise, stay tuned for the next edition which will be hosted by CatSynth.

This article intrigued me so much that I dug a bit deeper, and found that Daniel L. Everett, the Chair of Languages, Literatures, and Cultures from Illinois State University, has spent a good portion of his career working with the Pirahã people. He has collaborated in the past with Gibson on various projects in the past. Some info can be found here. There’s a great New Yorker story that was published in April 2007 on Dan here that’s certainly worth a look. Here’s a teaser from this article:

The Pirahã, Everett wrote, have no numbers, no fixed color terms, no perfect tense, no deep memory, no tradition of art or drawing, and no words for “all,” “each,” “every,” “most,” or “few”—terms of quantification believed by some linguists to be among the common building blocks of human cognition.

It’s a very long article, but it paints a beautiful picture of linguistics, cognition, faith, and personal relationships. It’s packed full of great questions. There’s a LOT that’s in these writings I’ve linked to that I haven’t even brought up (including the idea of recursion in linguistics), so I urge you all to read more! There are also some great links for further reading in the Wikipedia article linked to above, including several scholarly papers.

I love this story because I can imagine Pedro first thinking about this problem while unpeeling a roll of tape. I don’t know if the inspiration actually came this way, but its a great mental image that conveys the idea that some of the most interesting problems to solve are right under our noses.

Here’s the abstract of the paper from Dr.  Reis’s website:

Thin adhesive films have become increasingly important in applications involving packaging, coating or for advertising. Once a film is adhered to a substrate, flaps can be detached by tearing and peeling, but they narrow and collapse in pointy shapes. Similar geometries  are observed when peeling ultrathin films grown or deposited on a solid substrate, or skinning the natural protective cover of a ripe fruit. In this work, we have shown that the detached flaps have perfect triangular shapes with a well-defined vertex angle; this is a signature of the conversion of bending energy into surface energy of fracture and adhesion. In particular, this triangular shape of the tear encodes the mechanical parameters related to these three forms of energy and could form the basis of a quantitative assay for the mechanical  characterization of thin adhesive films, nanofilms deposited on substrates or fruit skin.

The previous digression leads me to mention the Things of Interest blog, and their absolutely fantastic post on various forms of the “0.9999999… = 1″ proof. You can find that post here. In case one proof doesn’t do it for you, this site offers several, each of which occurs at a various level of mathematical rigor. There will definitely be a proof for you here that you’ll understand.

UPDATE: Daniel from the Logos Blog has contacted me about a great post over on that site. As he said, “Thought you might be interested in today’s Logos Blog post looking at The Top 50 People in the Bible and using the IBM Many Eyes visualization. Cool thing is that anyone can play around with the charts and data…” Check it out here.

Wolfram, the makers of the software Mathematica, are offering a $25,000 prize to the first person who can prove whether the above 2, 3 Turing machine is universal. From the website (here):

“A universal Turing machine is powerful enough to emulate any standard computer.

The question is: how simple can the rules for a universal Turing machine be?

Since the 1960s it has been known that there is a universal 7,4 machine. In A New Kind of Science, Stephen Wolfram found a universal 2,5 machine, and suggested that the particular 2,3 machine that is the subject of this prize might be universal.

The prize is for determining whether or not the 2,3 machine is in fact universal.”

What a great idea! I’m really curious to see how long it take for someone to claim the prize. If you’re interested in understanding more about what a Turing machine is, please check out the above links.

Most of the proofs I’ll be talking about from week to week won’t be overly intense. I’m sure that many of them will require some general knowledge background, but nothing too academic. My hope is that by explaining some interesting results that you too might see a little bit more of the grandeur contained in this subject. I remember when I took my first proof-based math class during my sophomore year of college. I knew that a lot of rigorous math had to do with proofs, but it wasn’t until my 20th year of life on this planet that I learned what they were really all about. And here’s one of the many revelations I came to rather quickly:

Math is nowhere near as objective as I thought it was growing up. In other words, I always thought that there was a unique answer to every problem. Because of this, I think that many people regard math as some sort of rigid 60 year old person wearing starched clothing who eats the exact same three meals a day and whose house is painted a single shade of grey. To use another image, many people view math problems as some sort of assembly line. You insert a problem at the beginning of the line, perform a bunch of robotic methods, and the answer plops out at the end of the line. If this is your view of math, no wonder you think it’s boring! There’s no art in these images. There’s no movement or color in these pictures.

Math is nowhere near as simple as an assembly line. At least not at its heart. But since most of us grow up learning rote methods to solve problems many of us find the subject to be too tedious or mundane. And I don’t blame you for thinking that. What I WOULD like for you to consider is that you’ve been misled. Like any other academic discipline, math is a growing organism. Hopefully in these “Proofs of the Week” I’ll be able to illuminate some of the beauty that is contained in math. The first of the series will be up in a day or two. Stay tuned!

Flatland: The Movie is an animated film inspired by Edwin A. Abbott’s classic novel, Flatland. Set in a world of only two dimensions inhabited by sentient geometrical shapes, the story follows Arthur Square and his ever-curious granddaughter Hex. When a mysterious visitor arrives from Spaceland, Arthur and Hex must come to terms with the truth of the third dimension, risking dire consequences from the evil Circles that have ruled Flatland for a thousand years.

Well, it sounds like there has definitely been some license given to modify the orginal plot of the novel. But I have to say that the plot modifications were immediately forgiven once I found out that Martin Sheen was going to do the voice of A. Square. Who can argue with that? Also, Tony Hale, of the late TV show Arrested Development, will be playing the King of Pointland. At any rate, as the website explains, “The movie will be part of an educational DVD, which will include the original text from Abbott’s book.” Also, it looks like it will be coming out in spring 2007, which isn’t too far away! If you’re dying to get a copy you can sign up on the website for priority access to the DVD. While you’re anxiously awaiting its release, I suggest reading Abbott’s original work. It’s a really quick read and is imaginative and original.

Perhaps the most interesting thing to note is that the article focuses not only on the result of the problem (the proof itself), but also the methods used to solve the problem. This is an hugely understated part of the mathematical process. I’m of the opinion that when the general populace thinks about math that they are fixated on two things: the problem and the answer. What people tend to overlook is the process of problem solving. In math, there are not always clear-cut methods that explain how to get from point A to point B. A lot of thought is sometimes necessary to figure out how to traverse the path. The Poincaré Conjecture is a monumental achievement not only because of the end result, but also because of the original steps the solvers of the problem (especially Grigori Perelman) took to get there. These steps will be used in other problems; they are not exclusively tied to this one specific problem. Once again, congratulations to Perelman and the other mathematicians who had a hand in making this historic achievement!