I’ve seen better tricks, and setting up and executing this trick is no simple task, but there is definitely some cool mathematics behind this.
http://www.telegraph.co.uk/science/7950022/The-magic-of-mathematics-amaze-your-friends.html
Application Technology
P v NP and Collaborative Mathematics
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.
Challenge Probability
Events with Probability Zero
I just donated this shirt to the Salvation Army. I estimate the probability that I will ever see this shirt again to be zero. In fact, forget estimating: I think the probability that I will see this shirt again is exactly zero.
In fact, I’ll go even further than that: I claim that the probability that anyone who reads this ever sees this shirt is exactly zero. Not .0000000000001. Zero.
Even if there are infinitely many possible scenarios in which this shirt is seen by someone, this number is dwarfed by the infinitude of possible scenarios in total. Thus
moderate infinity ÷ really big infinity = zero
I dropped it off at the Salvation Army in Brooklyn. Let me know if you see it: evidence will be required to claim your prize.
Application Challenge
Comprehensive Calorie Counting
There’s an interesting op-ed in the NYT about how mathematics is used to put food production and consumption into context:
http://www.nytimes.com/2010/08/20/opinion/20budiansky.html
For example, we can measure food transportation costs in calories (as a calorie is, indeed, a unit of energy), and so we can look at production and transportation costs of certain foods and compare them to the calorie content of the food itself. For example, it takes about 5000 calories to produce a 100-calorie head of lettuce.
The author’s intent is to poke holes in some of the common arguments used by proponents of “eating local”: Transportation costs for most foods are neglible compared to household storage and preparation costs, says the author, thus “eating local” is not an especially eco-friendly strategy. However, the author hilmself makes a number of weak and erroneous arguments, comparing “apples to rocks” in some cases.
The piece offers some interesting mathematical ideas and a good critical reading exercise.
Appreciation Geometry
Garden-Variety Fractals
My Mom gave me some clippings of a Christmas Cactus (Chris, pictured at the right), and after taking a while to get accustomed to her new home, Chris is finally starting to grow.
The new leaves sprouting out, smaller but similar to the original, put me in mind of the Mandelbrot set. The bulbs that “grow” out of the Mandelbrot set are perfectly similar to the original, and no matter how much you zoom in, you’ll always see the exact same sort of object.
I can’t say for sure if the leaves of the Christmas Cactus are infinitely self-similar, but it’s close enough for my eyes.