Pringles

This article about the saddle-shape of Pringles is a joy to read [ht: Prisca Chase]. I’ll give you an excerpt, but I encourage you to read the whole thing. It’s both mathematically stimulating and extremely funny:

Saddle up for maximum snack satisfaction (mathematically speaking)

Stephanie V.W. Lucianovic

My husband is a calculus professor and one who brings food items into the classroom with surprising regularity. No, he doesn’t bring pies on Pi day - though he can recite the string up to a couple dozen digits – but he does bring Pringles. As a teaching aid.

This afternoon when I walked into his study, I nearly tripped over a plastic Safeway bag filled with six red cans of Pringles. “Is it Pringles Day already?” I asked, nudging the bag. Pringles Day is the day Dr. Mathra lectures on the classification of critical points in multivariable calculus, and he uses the saddle-shaped Pringles to illustrate his points.

After class, the students get to eat his illustrations. It’s their favorite day.

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Later in the article, the fact that a Pringle can’t be made from a sheet of paper is mentioned. For a normal sheet of paper, this is true. But you can fold paper in such a way as to approximate a hyperbolic parabaloid. I’ve mentioned this before here and here. So go try it!

Math paper contains no math

I feel like I need to post about this too–just to get the word out.

At first I thought it was funny, but now it just makes me angry. I first heard about this paper thanks to my brother, Tim Chase, who shared this news via Retraction Watch. Then today I learned a bit more information by way of Alexander Bogomolny and his blog.

Okay, what’s going on? Authors M. Sivasubramanian and S. Kalimuthu have published this completely nonsensical math paper, and here’s what Retraction Watch had to say:

Have a seat, this one’s a howler.

According to a retraction notice for “Computer application in mathematics,” published in Computers & Mathematics with Applications:

This article has been retracted at the request of the Publisher, as the article contains no scientific content and was accepted because of an administrative error. Apologies are offered to readers of the journal that this was not detected during the submission process.

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Go read the whole paper in full text available here. At the very least, this paper has been retracted. That’s good.

But sadly, S. Kalimuthu and his coauthors are responsible for many other terrible papers too (seriously, go check them out!). How does this happen? Can anyone explain it? And why hasn’t he been stopped?

In one paper in particular, he has completely plagiarized Alexander Bogomolny’s site–as one commenter noticed. Check out Alexander’s blog CTK Insights for his coverage.

Like I said, this man needs to be stopped.

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When cars collide

[Another guest column from Dr. Gene Chase.]

Suppose two equally weighted cars collide in a head-on collision, each traveling at 50 miles per hour.  Do you think that the impact for one car will be more severe on the car and driver than the impact of that car’s hitting a brick wall?

To be fair, we have to assume that neither the cars nor the wall compress at all.  If the wall is as soft as a pillow, I’ll take the wall every time.

Marilyn vos Savant’s recent column in Parade Magazine says that hitting an oncoming car in that way is no more severe than hitting a solid wall.   They both stop dead, whether the wall or the other car causes it.

Each experiences a momentum change that is the same as if they hit a wall, not twice as much. That’s clear when I think of it now, using the law that momentum = impulse (that is, mass * velocity = force * time) but I’ve been mistaken when I’ve only thought about it casually, thinking it must be a 100 mph impact..

If a bike hits a car head-on, the situation is different, because the “bike-car” combination will continue to move in the direction of the car, so my intuition is correct in that case:  The bike driver fares worse than the car driver.  Comments at Marilyn vos Savant’s blog say as much.

I used to think that car bumpers that collapse at the slightest impact were poorly made.  In fact, if momentum is constant, extending the time of impact will decrease the force, to keep force * time constant.

Give me “cheap” bumpers and a wall made of pillows every time.

Minnesota Senator loves math!

I really enjoy reading J. Michael Shaughnessy’s column. He’s the president of the NCTM and always has interesting, timely things to say about math and math education. Here’s an excerpt from this week’s column, where he recounts his recent conversation with Senator Al Franken (D-Minn) as he eagerly shared a proof with President Shaughnessy. Go check it out!

 

Seen Any Good Proofs Lately? Raising the Social Currency of Mathematics

We all probably have had a friend or acquaintance, or even a perfect stranger, raving about a book she has just read, or a movie he has recently seen, and then saying, “Oh, you must read this book!” or, “You must see that film!” But how many of us have had this kind of experience in a social occasion where the person exclaimed, “Oh, you must see this proof!” So it was indeed refreshing to meet someone who really likes mathematics, as I did several weeks ago, in what might seem like a very unlikely setting—the Hart Senate Office Building in Washington, D.C.

On Wednesday mornings when Congress is in session, Senator Al Franken (D-Minn.) holds a breakfast gathering in his office for his constituents. Visitors to the breakfast consist primarily of people from Minnesota, but I received an invitation from a mathematics teacher who is spending the year working on the senator’s staff. A famous hearty porridge is served up at these breakfasts, and once guests have begun to circulate, Senator Franken drops in and greets everyone. I had been misinformed and thought that the Senator had been a mathematics major in college. When I asked him about this, he said that the rumor was false, but he agreed that his good grades in math had probably helped him get admitted to college.

After breakfast, the visitors were escorted to a terrace area in the hallway outside the office, where the senator spoke for a few minutes about events being debated in Congress and answered questions. Guests then lined up to have their pictures taken with the senator. I was at the end of the line, and as I shook his hand and introduced myself as the president of NCTM, he said, “Let me show you my geometric proof of the Pythagorean theorem!” Senator Franken then proceeded to grab scratch paper and a pen from one of his staffers and plopped down cross-legged on the hallway carpet. As I sat next to him, he began to sketch out his proof. He explained what he was doing, and why it worked, and I paraphrased each move he made so that it was clear to both of us how he was thinking and what he was doing.

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Cooking math

 

This article by Samuel Arbesman came through today on Wired.com

Are There Fundamental Laws of Cooking?

Cooking is a field that has in recent years seen a shift from the artistic to the scientific. While there are certainly still subjective and somewhat impenetrable qualities to one’s cuisine — de gustibus non est disputandum — there is an increasing rigor in the kitchen. From molecular gastronomy to Modernist Cuisine, there is a rapid growth in the science of cooking.

And mathematics is also becoming part of this. For example, Michael Ruhlman has explored how certain ingredient ratios can allow one to be more creative while cooking. Therefore, it should come as no surprise that we can go further, and even use a bit of network science, when it comes to thinking about food.

Yong-Yeol Ahn and his colleagues, in a recent paper titled Flavor network and the principles of food pairing, explored the components of cooking ingredients in different regional cuisines. In doing so, they were able to rigorously examine a recent claim: the food pairing hypothesis. The food pairing hypothesis is the idea that foods that go best together contain similar molecular components. While this sounds elegant, Ahn and his collaborators set out to determine whether or not this is true.

Using recipes from such websites as Epicurious, the researchers examined more than 50,000 recipes. They combined these recipe data with information about the chemical components in each of the ingredients, in order to create a network map of related ingredients. For example, shrimp and parmesan are connected in the network, because they contain the same flavor compounds, such as 1-penten-3-ol. A large flavor network of different ingredients is [above].

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He later gives a reference to George Hart’s “Incompatible Food Triad” problem and the associated website:

An example solution would be three pizza toppings — A, B, and C — such that a pizza with A and B is good, and a pizza with A and C is good, and a pizza with B and C is good, but a pizza with A, B, and C is bad. Or you might find three different spices or other ingredients which do not go together in some recipe yet any pair of them is fine.

Has any of this ever crossed your mind? Me neither.

 

 

 

LEGO math

This article was just posted to wired.com today and is an interesting summary of some research from 2002–but it is new to me. Here’s an excerpt from Samuel Arbesman’s article:

Most objects are made up of smaller parts, combined in complicated and diverse ways… In the wonderfully titled paper Scaling of Differentiation in Networks: Nervous Systems, Organisms, Ant Colonies, Ecosystems, Businesses, Universities, Cities, Electronic Circuits, and Legos,Mark Changizi and his colleagues set out to understand this concept. They found that in every single one of the systems in the wildly interdisciplinary list of the subtitle there was an increase in the number of types of components as the total number of pieces grew. The larger something is, the more types of building blocks it uses.

And this includes, of course, Lego bricks. Using a dataset of 389 Lego sets (this was done back in 2002, so if anyone can download the data easily, I would love to see if the results hold up with a richer dataset), they examined the number of distinct types of pieces in a set versus the total number of pieces in that set (examples of sets include “Air Patrol”, “Spy Boat”, and “Cargo Crane”, and a master list of Lego piece types is here).

They found that the number of piece types to total number of pieces could be fit nicely to a power law. Here it is on a log-log scale:

This curve demonstrates that as the number of pieces in a set grows, so do the number of piece types. However, the number of piece types grows sublinearly: while a larger set uses more piece types, as sets becomes larger, they use progressively fewer additional piece types (so larger sets actually use fewer types per piece). This is similar to other sublinear curves, where larger animals use less energy per cell for metabolism or larger cities actually need fewer gas stations per capita. Essentially, larger sets become more efficient, using the same pieces that smaller sets do, but in a more complex and diverse way.

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Now, just for fun, here’s a video of a  great LEGO contraption (HT: Tim Chase).

Also, just for fun, here’s a photo of a 4-foot Lego 737 that my friend Matthew and I built. We’ve actually finished most of it, I just don’t have a recent picture. (Notice in this photo the roof, tail, and wings are missing.) But this gives you a taste:

 

 

 

 

 

 

Teaching math with applications

Wow, if the title of this post didn’t grab you, I don’t know what will. Pretty riveting, right? Has anyone ever thought of teaching math with applications? </end sarcasm>

This is the basic thesis of a recent article from the NY Times, “How to Fix Our Math Education” by Sol Garfunkel and David Mumford:

THERE is widespread alarm in the United States about the state of our math education. The anxiety can be traced to the poor performance of American students on various international tests, and it is now embodied in George W. Bush’s No Child Left Behind law, which requires public school students to pass standardized math tests by the year 2014 and punishes their schools or their teachers if they do not.

All this worry, however, is based on the assumption that there is a single established body of mathematical skills that everyone needs to know to be prepared for 21st-century careers. This assumption is wrong. The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.

Today, American high schools offer a sequence of algebra, geometry, more algebra, pre-calculus and calculus (or a “reform” version in which these topics are interwoven). This has been codified by the Common Core State Standards, recently adopted by more than 40 states. This highly abstract curriculum is simply not the best way to prepare a vast majority of high school students for life.

For instance, how often do most adults encounter a situation in which they need to solve a quadratic equation? Do they need to know what constitutes a “group of transformations” or a “complex number”? Of course professional mathematicians, physicists and engineers need to know all this, but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.

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from toothpastefordinner.com

To be fair, the real thesis–if you read further in the article–is that we should primarily teach applications and math can swoop in and rescue us if and when it’s needed:

Imagine replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.

If you haven’t gathered already, I don’t agree at all with this thesis. It’s my opinion that math should be taught as math, respected as its own field of study, and a valuable part of a high school liberal arts curriculum. Students should value math for its inherent, abstract beauty. Applications are of course a must in any course. But I find that in the text resources I’ve used, the applications are often contrived. Extremely contrived. Doing math should feel like playing a game, like working on a puzzle, or like arguing.

In fact, when high school students ask why are we learning this?, I FIRST respond with the things I just said: It’s part of a liberal education; it will make you a well-rounded, intelligent person who can hold conversations with other smart people in other fields; and it’s fun. I mention SECOND what applications exist for the math we’re learning. For high school students, if we’re honest, most of them will never need any of the math we’re teaching. Seriously. If you’re not working in a math or science field, when was the last time you had to factor a polynomial?

The authors go on to say “Science and math were originally discovered together, and they are best learned together now.” But this is not universally true. In many cases, the ‘useless’ math was developed first (think of number theory for instance) and then only later were applications discovered (think of the RSA or El Gamal public key encryption schemes).

So why learn math? There was a nice post about this yesterday on one of my new favorite blogs, dy/dan, titled “Cornered By The Real World.” He highlights this great article by Samuel Otten in this August’s Math Teacher magazine. I highly recommend reading the whole article. As a taste, I’ll include the same snippet that Dan shared:

I believe that thinking and acting as if the justification for teaching and learning mathematics is found solely in everyday applications can be dangerous. Mathematics does not exist only to serve other professions, nor is it merely a collection of algorithms and procedures for dealing with real-world situations. Such a mind-set essentially paints our discipline into a weak and lonely corner and leaves undefended many of its greatest aspects.

I could say more about this, but I’ll spare you. I’m passionate about making math a subject worth learning all on its own. If I say more, I’ll start to sound like Paul Lockhart.

Longest mathematical proof

Here’s a recent article from NewScientist.com, Prize awarded for largest mathematical proof by Stephen Ornes:

The largest proof in mathematics is colossal in every dimension – from the 100-plus people needed to crack it to its 15,000 pages of calculations. Now the man who helped complete a key missing piece of the proof has won a prize.

In early November, Michael Aschbacher, an innovator in the abstract field of group theory at the California Institute of Technology in Pasadena will receive the $75,000 Rolf Schock prize in mathematics from the Royal Swedish Academy of Sciences for his pivotal role in proving the Classification Theorem of Finite Groups, aka the Enormous Theorem.

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John Milnor wins Abel Prize

Just two days ago, the Norwegian Academy of Science and Letters awarded mathematician John Milnor the 2011 Abel Prize. Though it was just awarded, the prize is in recognition of decades of work. Here’s a link to the short piece on NPR show All Things Considered. And here’s a direct link to the Abel prize website.  And here’s a snippet from the Scientific American article:

Dimension-Cruncher: Exotic Spheres Earn Mathematician John Milnor an Abel Prize

His discovery that some seven-dimensional spheres look different under the lens of calculus spurred decades of research in topology.

John Milnor, an American mathematician best known for the discovery of exotic hyperspheres, was awarded the 2011 Abel Prize, the Norwegian Academy of Science and Letters announced March 23.

Milnor, a professor at Stony Brook University in New York State, got a call at his Long Island home at 6 A.M. informing him he was receiving the $1-million prize—an honor first awarded in 2003 as mathematics’ answer to the Nobel Prizes. “I knew I was a possible candidate, but I certainly didn’t expect it,” says Milnor, 80, who had already earned numerous awards during his career, including a Fields Medal in 1962 and a Wolf Prize in 1989. Milnor is the second consecutive American-born Abel laureate; the 2010 prize went to John Tate of the University of Texas at Austin for his contributions to number theory.

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The reason I’m linking to lots of other sources is because I don’t understand Milnor’s results very well :-). But it sounds impressive.

[Hat tip: Raynell Cooper]