"He Perfects Animation Using Math and Science," by Patricia Cohen. The New York Times, 29 December 2010, Arts page C1.
The animation process combines art and physics: (from left to right) (a) an animator sculpts the character in an expressive walking pose, (b) and paints "guide" color patches that instruct the computer in (c) automatically introducing wrinkles and folds too tedious to draw by hand, (d) as well as additional physical effects, such as an electrical shock or a gust of wind. Image courtesy of the Columbia University Computer Graphics Group.
Complicated mathematical equations aren’t typically the first things that come to mind when watching animated movies, but they are, in fact, the key to some of the computer graphics that make the characters’ appearance so realistic. Eitan Grinspun, the head of Columbia University’s Computer Graphics Group, is the industry leader in one such element of appearance: hair. Using a relatively new branch of mathematics called discrete differential geometry, Grinspun has built innovative models for creating life-like simulations. Firms such as Disney and Pixar use his work, and his models will be the foundation for hair in the Rise of the Apes movie due out this summer. The utility of such realistic models also spans beyond Hollywood; Johns Hopkins Medical Center has begun using them to develop a program for simulating surgery using a virtual needle.
--- Lisa DeKeukelaere
"The US embassy cables enigma," by Jonathan Farley. Guardian, 23 December 2010.
The recent Wikileaks release of thousands of secret diplomatic cables provides the author with a springboard for a short discussion of the mathematics of cryptography. "[W]hat's surprising about the US state department/WikiLeaks scandal is that there is a scandal," Farley writes. "As a friend reminded me, these days, it's really quite easy to make sure no one can read your messages, unless you want them to." Farley then describes public key cryptography, which uses the difficulty of factoring numbers to encode messages. He speculates that officials at the State Department did not know about cryptography, or perhaps "were too lazy to care." Another possible explanation is proposed by someone who left the following comment about the article: "The bloke who stole the cables had the right to access them. I don't know how/if they were encrypted, but he would have been eligible to have the key in any case. The human element will always remain as a weak point in any secure system."
--- Allyn Jackson
"'Principia Mathematica' Celebrates 100 Years," an interview with Robert Siegel and Julie Rehmeyer on All Things Considered, National Public Radio, 22 December 2010.
On the 100^{th} anniversary of the publication of volume one of Principia Mathematica, science writer Julie Rehmeyer explains the significance of this work by Alfred North Whitehead and Bertrand Russell. She says that the authors "set out to show that math really boiled down to logic and to define at the very most basic level what mathematics was, and to show then that all math was logical consequences from some very, very simple principles." Rehmeyer also touches on the book's influence on Kurt Gödel--who later showed that not all of mathematics could be derived in this way--and how it laid the groundwork for computation.
--- Annette Emerson
"A scientist's finest hour," by Jan Mainka. The Budapest Times, 20 December, 2010.
The article recounts the genesis of the "now world-famous gömböc," defined as "the first known homogenous object with one stable and one unstable equilibrium point, thus two equilibria altogether on a horizontal surface." Domokos recalls a conversation in 1995 with world-renowned mathematician Vladimir Arnold about a topic of common interest: bodies and their stable points of equilibrium. "It can't have been for more than five minutes that we were standing and talking. We spoke about a few things related to Arnold's lecture... Almost in passing Arnold made the remark that in his opinion there must be bodies with just one stable and one unstable point of equilibrium." Thus was born the idea to create an object to demonstrate the concept. Learn more about the gömböc, its mathematical properties, its inventors, and see animations.
Photo (left to right): V.I. Arnold, Gábor Domokos, Péter Várkonyi with a gömböc. Photograph by Istvan Oravecz, courtesy gomboc.com.
--- Annette Emerson
"10th Annual Year in Ideas," New York Times Magazine, 19 December 2010.
Two ideas highlighted in this Year in Ideas issue for 2010 involve the creative use of mathematics. Perfect Parallel Parking," by Jascha Hoffman, recaps the work of mathematician Simon Blackburn, who came up with a "perfect parking" formula. Using geometry, the car's dimensions and turning radius, and the width of the vehicle in front, Blackburn calculated "the shortest parking space required for a back-in maneuver that puts the car flush with the curb without any shuttling back and forth." Then math teacher Jerome White used trigonometry to develop another formula that predicts "how much closer to the curb you could get using a back-and-forth maneuver." "Aftercrimes," by Clay Risen, recalls the work of mathematician George Mohler, who "showed that what holds for earthquakes also holds true for crime: not only does an initial crime beget future offenses, but these 'aftercrimes' also tend to occur according to a predictable distribution in time and space. He showed that the timing and location of the crimes can be statistically predicted with a high degree of accuracy."
--- Annette Emerson
"Antimatter exemplifies mathematics' prescience," by Tom Siegfried. "Science News, 18 December 2010 (From the Editor, page 2).
Two articles in this issue ("Antimatter, here to stay" and "Black holes in the bathtub") lead editor-in-chief Tom Siegfried to write about mathematics' predictive powers in his "From the Editor" piece. He writes how physicist Paul Dirac's equation describing the energy of electrons predicted the existence of an electron antiparticle before anyone suspected that such a particle existed and how another equation led Stephen Hawking to predict that radiation could escape a black hole and cause it to shrink. Both predictions were confirmed: The first about a year after Dirac's prediction in the 1920s and the latter recently in a laboratory experiment. Siegfried writes that the predictions illustrate "a recurring theme in science--the power of math to reveal realities not previously imagined."
--- Mike Breen
"Google Book Tool Tracks Cultural Change With Words," by Dan Charles. National Public Radio, 16 December 2010.
Mathematician and bioengineer Erez Lieberman Aiden and mathematician and biologist Jean-Baptiste Michel have created a searchable database of 500 billion words that they say "is a new and powerful tool to study cultural change." The source of the words is Google's book-scanning project, which to-date includes over 5 million books published over four centuries, now digitized. But this database just consists of "words and phrases, stripped of all context except the date in which they appeared." The researchers claim that this data will enhance literary and cultural studies. They tracked themes and phrases through time and found correlations, clusters and peaks related to treaties, fame, censorship, changes in grammar, and epidemics, to name a few. The article shows graphs, and comments following the article reflect some skepticism about attributing too much value to trends in commonly-used words and phrases.
--- Annette Emerson
"Fairfield U. professor devises math tool to help streamline cancer research," by Mike Lauterborn. Connecticut Post, 15 December 2010.
Why would the results from cancer experiments done with microarrays performed simultaneously differ from results of the same experiments performed individually? Vera Cherepinsky (Fairfield University) is part of a team that developed a mathematical model to explain what her team calls competitive hybridization. Their model also helps in the design of better experiments. The research is published in "Competitive hybridization models," Physical Review E, Vol. 82, issue 5 (read the abstract).
--- Mike Breen
"Math Puzzles' Oldest Ancestors Took Form on Egyptian Papyrus," by Pam Belluck. New York Times, 6 December 2010.
What do grade school geometry books and Egyptian papyrus scrolls have in common? Formulas for calculating the area of a hemisphere. This article provides a summary of the math problems contained in ancient Egyptian texts dating to 1650 B.C., currently held by museums in Britain, Egypt, and Russia. The scrolls demonstrate how to solve the practical problems of their time, such as measuring a ship’s rudder and evaluating equivalence between bread and beer for trading purposes. Although scholars have found a few errors, they note that the mathematics is largely accurate and even contains a fair estimation of pi. The author notes that the difficulty in the scrolls lies not in the complexity of the mathematical problems, but instead in the deciphering of their meaning.
--- Lisa DeKeukelaere
"Non-Transitivity in Baseball, Medicine, Gambling and Politics," by John Allen Paulos. ABCNews.com, 5 December 2010.
In this month's "Who's Counting?" column on ABCNews.com John Allen Paulos writes about how statistical correlation and transitivity can sometimes trick the human mind. Paulos uses real life examples from baseball statistics and medical science to explain how simple mathematics can help us better understand why for example taking certain expensive vitamin supplements, might not correlate with good health.
--- Baldur Hedinsson
"Small College Goes Big in Research for Recruiting," by Eric Hoover. The Chronicle of Higher Education, 5 December 2010.
In this article, writer Eric Hoover tells the story of Lincoln Memorial University, a small private college in eastern Tennessee with big ambitions. In order to "become more selective, and to enroll more high-achieving students," the college hired a private consultant to "help design and run a large admissions experiment." After soliciting over 100 ideas from administrators, staff, and students on how to attract more applicants, the college and consultant chose 22 of the most feasible ideas to test among a few dozen groups of high schools. The consultant used multivariable testing, a method in which different combinations of factors are tested at the same time. As a result, the university was able to identify the effectiveness of each strategy, with some surprising results. For example, some seemingly good ideas-—having faculty getting in touch with potential students and having a college Facebook page-—were a disincentive to students. On the other hand, "calling students within 24 hours of their first inquiry and including a letter about financial aid in a follow-up mailing" proved effective. Hoover notes that, "in the end, even the most advanced statistical analysis is better at revealing what than why." He speculates that socioeconomic differences among students, and high schools, probably had an impact on the results.
--- Claudia Clark
"Mathematical immortality? Give a theorem your name," by Jacob Aron. New Scientist, 3 December 2010.
What do you get for the person who has everything? Well, for £15, you can purchase a brand-new theorem, complete with naming rights, from TheoryMine. Current managing director, Flaminia Cavallo, came up with the idea for TheoryMine while pursuing an undergraduate degree in mathematics and computer science at the University of Edinburgh. Writer Jacob Aaron describes the completely automated process by which the TheoryMine software generates theorems:
From the library of mathematical knowledge, the program generates a set of mathematical axioms, then combines them in different ways to produce a series of conjectures. It then uses the library to discard a portion of these on the basis that there are already counter-examples... Overly complex conjectures are also ignored. Then it applies a technique known as "rippling," in which it tries out various sequences of logical statements until one of these sequences turns out to be a proof of the theorem. |
--- Claudia Clark
"U.S. 'Connects The Dots' To Catch Roadside Bombers," by Tom Gjelten. Morning Edition, National Public Radio, 3 December 2010.
These days, "social network" is synonymous with "Facebook," but the term really means just that: a network of people who are connected as acquaintances or friends. In Iraq and Afghanistan, social network analysis is a key tool in uncovering the individuals responsible for roadside bombs, also known as improvised explosive devices or IEDs. Kathleen Carley, a computer science professor at Carnegie Mellon, and Army Maj. Ian McCulloh, deputy director of the Counter-IED Operation Integration Center in Baghdad and one of Carley’s former students, are at the forefront of this research. The thinking is that these attacks are not planned by just one individual, and so the motto of the anti-IED effort is "Attack the Network." The main idea behind their work is to identify the individuals with the most connections so that, if stopped, this would disrupt the entire network. McCulloh teaches soldiers to "mathematically quantify influential network nodes." He shows his students to visualize a network, where people are represented by nodes and are connected to other nodes by edges. Nodes are scored on their "betweenness" and average shorter distance to the other nodes. While an experienced analyst can gather information in a few days, McCulloh claims that with his software and mathematical analysis he can take 15 or 20 minutes to reach the same conclusions. Another advantage to having numbers is that it gives the military a bit more confidence when trying to capture or kill a target, rather than relying on intuition and hunches.
--- Adriana Salerno
"The Incredible, Edible Foam--and the Mysterious Mathematics Behind It," by W. Wayt Gibbs and Nathan Myhrvold. Scientific American, December 2010, page 22.
Scientific American feasts on the mathematical mysteries of edible foam in its December issue. The fizzy stuff is made up of interlocked bubbles, such as the foam in a frothy cappuccino. The mathematical rules underlying how the bubbles self-organize has been studied for more than a 100 years and is still not completely understood. It is only in recent years that world-leading chefs have been able to put the mathematics to use and turned a variety of foods such as cod, mushrooms and potatoes into edible foam.
--- Baldur Hedinsson
"The Science of 'Disestimation'," by Charles Seife. Scientific American, December 2010, page 31.
Seife's definition of disestimation is "taking fuzzy numbers way too seriously." His example is reaction to a poll by the Pew Forum on Religion and Public Life. The average number of correct answers by atheists and agnostics was 20.9 (of 32) while the overall average was 16.0, which prompted several media outlets to conclude that "nonbelievers" knew more about religions than believers. The flaws in this conclusion cited by Seife are
Seife concludes, "The press leaped on the atheists versus believers headlines without critically examining the numbers. The Pew study revealed less about our faith in God than it did about our faith in polls--which, far too often, is blind."
--- Mike Breen
"A Geometric Theory of Everything," by A. Garrett Lisi and James Owen Weatherall. Scientific American, December 2010, page 55.
A Grand Unified Theory (or GUT) is pretty much the current Holy Grail of Physics. Since Newton's time, physicists and mathematicians have been working together to come up with a model that explains everything. Newton discovered a universal law of gravitation. Maxwell developed a theory of electromagnetism. One hundred years later this theory was combined with weak nuclear forces into what physicists call electroweak theory, to which strong nuclear forces were added to form what is now known as the Standard Model of particle physics. The latter is the current best theory of nongravitational forces. Since the 1980s, string theory has been the dominant research program in theoretical physics and the favorite candidate for a GUT. But it is not the only effort in this direction. In this article, Lisi and Wetherall explain a theory with a framework similar to that of the Standard Model, E8 theory, where all forces (including gravity) and matter are described as the twisting of a single geometric object. "Many physicists share an intuition that at the deepest level, all physical phenomena match the patterns of some beautiful mathematical structure," say the authors.
The main geometric idea behind the Standard Model is that of a fiber bundle, where a different fiber corresponds to a different particle. In electromagnetism the fiber bundle to study consists of circles attached to every point of spacetime (a circle, or U(1), is the simplest example of a Lie group). The weak force is associated with a three-dimensional Lie Group fiber called SU(2). The strong nuclear force that binds quarks into nuclei corresponds to the eight-dimensional Lie Group SU(3). When trying to find a good GUT, physicists are interested in one larger Lie group for all forces instead of different Lie groups for each force. Georgi and Glashow found, in 1973, that the combined Lie Group of the Standard Model fit nicely into the Lie group SU(5) as a subgroup, but this model was quickly discarded after some heavy experimentation. A related GUT developed around the same time around the Lie group Spin(10), which in turn fits nicely into the exceptional Lie group E6. This embedding suggests that bosons and fermions, long thought to be completely different, are actually parts of a single fiber. The force missing from this picture is gravity, which is described by the Lie group Spin(1,3), which combined with Spin(10) using a single Lie group, Spin(11,3), should yield a Gravitational Grand Unified Theory, as described last year by Roberto Percacci and Fabrizio Nesti. This last group fits nicely into the exceptional Lie group E8 (much like Spin(10) fit into E6), and seems to be an all-encompassing model with a wonderfully intricate structure.
The authors acknowledge that a lot of work still needs to be done to test this theory, and they believe the Large Hadron Collider should be able to glean some information about whether their suspicions are correct, or disprove their conjectures. But if they are correct, they say "we will have achieved a complete unification and have the satisfaction of knowing we live in an exceptionally beautiful universe."
--- Adriana Salerno
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