Category: geometry

The 10th Annual Year in Ideas

NY Times 2010 Year In Ideas

Another year has passed, which means it’s time again for the NY Times Magazine’s annual The Year in Ideas issue, “a high-to-low, silly-to-serious selection of ingenuity and innovation from 2010.” As with the 2009 list, a number of these ideas are based around some bit of mathematics and/or statistical analysis. The ones I’ve listed below are the ones that most prominently feature mathematics ideas, or feature mathematics and/or mathematicians centrally.

  • Perfect Parallel Parking by Jascha Hoffman mentions Simon Blackburn’s geometric analysis of parallel parking, which we covered on the blog previously. Updating that earlier story, Hoffman’s entry notes that Jerome White and some fellow teachers at Lusher Charter School in New Orleans subsequently improved the model. (White and company built in allowances for the driver to do a bit more maneuvering.)
  • Aftercrimes visits a topic seen already here in this blog: just as earthquakes typically beget aftershocks, some types of crime beget copycat crimes. Mathematician George Mohler has been able to show that “the timing and location of the crimes can be statistically predicted with a high degree of accuracy.” For more info, check out the entry and the earlier blog post.
  • The entry Social Media as Social Index describes some of the ways that researchers—academic, government, and corporate—are mining social networks like Twitter and Facebook for valuable information. For instance, algorithms analyzing millions of Twitter posts were able to predict how certain movies would perform at the box office and how the Dow Jones Industrial Average would perform in the near future. More social media data mining is undoubtedly in store, as the story ends with one Facebook officer quoted as saying that this is the future of opinion research.
  • Finally, two entries which illustrate the public appetite for data analysis. Do-It-Yourself Macroeconomics describes the growing legion of “ordinary citizens” who are making it their business to “pull apart the [economic] data and come to their own conclusions.” All this is possible, of course, due to the explosion in publicly available economic data, one example of which is described in The Real-Time Inflation Calculator. As the story concludes, thanks to this (freely available) software, “Data on prices, once monopolized by government gatekeepers, are now up for grabs.”

The Formula for Perfect Parallel Parking

Here’s a problem that may be harder than P vs. NP for some people: parallel parking. Looking for a chance to “show how we can apply mathematics to understanding something that we all share,” mathematician Simon Blackburn of University of London’s Royal Holloway College developed a formula that tells you exactly how much extra length (beyond the length of your car) you need to have to even consider parallel parking in a given space. Blackburn’s formula is fairly comprehensive, taking into consideration the following variables:

r is the radius of my car’s kerb-to-kerb turning circle, l is my car’s wheel-base (the distance between the centres of the front wheel and the corresponding back wheel), k is the distance from the centre of the front wheel to the front of the car, and w is the width of one of the parked cars: the one near the front of my car once I’ve parked.

(As most American readers have probably already guessed, ‘kerb’ is British for ‘curb’.) The story indeed has wide appeal, as the formula has seen mention in places as various as NPR, Fox News, the NY Daily News, and of course outlets in Britain like the Telegraph. Dr. Blackburn’s original paper can be found here.

Packing Tetrahedrons, and Closing In on a Perfect Fit

Another NY Times story, this time taking its starting point with Aristotle: “More than 2,300 years ago, Aristotle was wrong.” What he was wrong about was regular tetrahedrons, four-sided figures where each side is an equilateral triangle. (Dungeons & Dragons players will recognize the tetrahedron as the game’s 4-sided die.) Aristotle thought that tetrahedrons could be packed together perfectly, so that, for instance, you could fill a box with tetrahedrons with no unfilled space appearing anywhere in the box. 1800 years passed before somebody realized he was wrong, but even then nobody had a clue as to just how ‘well’ tetrahedrons could pack.

Fast-forward to the modern era, when tetrahedron packing underwent something of an arms race. John Conway and Salvatore Torquato from Princeton found a packing that they could mathematically prove would fill at least 72% of the space. Paul Chaikin at NYU then had high school students fill aquariums and other things with hundreds of the D&D dice, discovering that 72% was easy! You could always do better. But how much better? And was there a systematic way to do it (rather than dumping dice into an aquarium)?

This past year saw the arms race accelerate, with competing groups coming out with successively better packings: first Elizabeth Chen and Jeff Lagarias, from the mathematics department at Michigan, published a packing pattern that guaranteed 78% coverage. Then two distinct patterns emerged that got to 85%: an extremely complicated one from Sharon Glotzer, also from Michigan, but in the chemistry department (and coming at things from the point of view of developing new materials for the Air Force), and an extremely simple one by a group from Cornell University. Just weeks ago, Dr. Torquato and Yang Jiao, his graduate student, tweaked the Cornell pattern to get coverage of 85.55%. And finally, just this past Monday, Elizabeth Chen posted a pattern that reached 85.63%. Can you do better?

Between The Folds

The documentary Between the Folds looks at the art and the science of origami. Our blog has had an entry on the mathematics of origami previously; here is the chance for an extended look. The film debuts on PBS in December 2009. There apparently are also free screenings at various locations around the US. A description from PBS’s website:

Think origami is just paper planes and cranes? Meet a determined group of theoretical scientists and fine artists who have abandoned careers and scoffed at graduate degrees to forge new lives as modern-day paper folders. Together they reinterpret the world in paper, creating a wild mix of sensibilities towards art, science, creativity and meaning.

The film has won numerous accolades at various film festivals; check out the webpage for the production company.

Fold Everything

Appearing in National Geographic‘s “Big Idea” section, this article details some cool and surprising modern uses of the ancient art of origami, or paper folding. Mathematicians have recently been studying origami using geometric ideas and, as physicist Robert G. Lang is quoted, “It’s now mathematically proven that you can pretty much fold anything.” Some of the designs include folding mirrors and lenses for satellite use, arterial stents for medical use, and at least one common everyday use:

When engineers working on the design of car air bags asked Lang to figure out the best way to fold one into a dashboard, he saw that his algorithm for paper insects would do the trick. “It was an unexpected solution,” he says.

Medieval Mosques Illuminated by Math

This segment from National Public Radio examines the intricate geometric designs that often cover historic buildings in the Islamic world. When Peter Lu, a graduate student at Harvard University, first saw one of them he was reminded of ‘quasicrystal’ patterns he’d encountered in his classes. While it’s doubtful that the creators of the ancient designs knew anything about quasicrystals, it’s equally surprising that their designs–created over 500 years ago–echo these structures that were only identified fairly recently.

For a nice picture of a quasicrystal in another context, also see What Is This? A Psychedelic Place Mat? from Discover magazine here.

Of the Algorithms, by the Algorithms, for the Algorithms

This article in Slate describes the efforts of mathematicians to detect and combat gerrymandering. (Gerrymandering is a political trick whereby a political party in power restructures the voting districts in such a way that the party is extremely likely to remain in power.) The problem turns out to be pretty difficult, even from the mathematical standpoint of just figuring out what an “unnatural looking” district looks like.

Calculating the Geography of Crime

This article from Science News describes the work of Mike O’Leary, a mathematician at Towson University, in Maryland. O’Leary is developing a mathematical model that attempts to take information about a particular crime, information about crimes committed in that locale, and geographical information, to come up with likely locations where the crime’s perpetrator might live.

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