This is an absolutely mind-blowing idea: robotic “paper” that can fold itself into an arbitrary three dimensional object. Be sure to watch the short video accompanying the article.
http://web.mit.edu/newsoffice/2010/programmable-matter-0805.html
Tying (or folding?) all of the physics and engineering together here is the mathematics of origami. How can you fold a square sheet into a boat? A plane? A tetrahedron? A super-intelligent robotic giraffe?
In 2007 Jon McLoone used Mathematica to create a Hangman game pitting the computer guesser against the human word-selector. As his daughter became old enough to play against the demonstration, and old enough to get frustrated with the computer guesser always winning, she asked her dad the obvious question: to beat the computer, what are the best words to choose?
Surprised that he had not considered such a good question himself, McLoone set about playing 15 million games of Hangman (automated, I imagine) using every word in the dictionary and arming the computer with a number of different letter-guessing-strategies. The word that the computer failed to guess the most often was somewhat surprising.
So what kinds of strategies make the best guesser? And to counter that, what kinds of strategies should the word-selector employ?
This is a nice article in the NYT about a recently proposed solution to the famously unsolved mathematical question “Does P = NP?”
http://www.nytimes.com/2010/08/17/science/17proof.html
Essentially this question is about how long it takes to solve certain kinds of problems: if a proposed solution to a problem can be checked in some reasonable amount of time, does that mean we can always solve the problem in a reasonable amount of time? [Warning: the definition of reasonable here may seem unreasonable.]
For example, it doesn’t require many operations to determine whether or not 7411 divides 748511; even by hand, you can work it out in a few steps. It requires significantly more operations, however, to find the prime factors of, say, 837751. Essentially, P v NP asks “are problems that can be checked by computers (maybe lots and lots of computers working in parallel) necessarily solvable by computers?” It is still an open question.
Another fascinating aspect of this particular open question is the role that the internet has played in bringing great mathematical minds together. Proposed solutions can be instantly accessed and vetted by those capable of evaluating the arguments. Such a community can work quickly and efficiently, not just to ascertain a proof’s validity, but to improve and refine it together.
This is a very cool (and somewhat technical) description of a simple way to use Mathematica to hide a secret message in an image file.
http://blog.wolfram.com/2010/07/08/doing-spy-stuff-with-mathematica/
The basic idea is that you erase the last digit (in binary) of each pixel’s “color channel”, and then use that spot to store part of the secret message. Given an image, you can then recover the secret message by looking at that last digit in each channel.
This process does change the image somewhat, but not in a way that the normal human eye (with normal viewing equipment) would ever notice. In the post, the author subtracts the adjusted image from the original:
To most viewers, the difference looks like this:
Only under extreme contrast can you actually see the real difference in the two. Here, we see the invisible ink reappear!
All that’s left is to decode the colors and read your message. But be careful–sending and receiving secret messages might get you some unwanted attention.
This is an interesting computer simulation of the potential spread of the gulf oil disaster throughout the Caribbean and Atlantic:
http://www.youtube.com/watch?v=nAiG-TPYIFM
This reminds me of weather-prediction models: input some initial conditions, set up the propagation rules, and then iterate, iterate, iterate. Of course, modeling the mathematics of the initial conditions and the propagation rules is a huge challenge.