My summer of modest carpentry continued with the staining of this unfinished bench:
That one-pint can of Black Cherry stain claims that it will cover 75 square feet of surface. If so applied, how thick would that layer of stain be? Let’s go with inches first, and convert to microns later. Or perhaps a more reasonable question is how does the thickness of the stain compare to the thickness of a sheet of paper?
Hopefully someone will figure that out and tell me.
Whenever I spend a lot of time driving, navigating, and map-reading, I find myself making a lot of rough estimates of distances. The process reminds me of how one estimates and ultimately evaluates, using Calculus, the arclength of a curve.
To find the length of a curve, we approximate the curve with a series of line segments. It’s easy to find the length of a line segment, and so by sacrificing exactness, you turn a hard problem into an easy one. This is a fundamental technique in Calculus.
I made my line segments equal in length to 40 miles on the map. Now just add up the lengths of the line segments to approximate the length of the curve.
Each of the seven line segments is (roughly) equal to 40 miles, so the approximate length of the path from Brooklyn to Burlington is 280 miles (not a terrible estimate).
There are plenty of ways to improve the approximation, and the straightforward, but complicated, calculus approach eventually produces the arclength integral.
On the actual drive my approximations weren’t as good, as I was using an inferior distance estimator.
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?
