I like the natural parabola formed by these hills. Or maybe it’s an inverted bell-curve. Whatever this North side of Eagle Lake, Acadia National Park is, it’s lovely.
Application Sports
A Beautiful Presentation of Tennis and Mathematics
This video is a superb lesson on the inherent mathematics of tennis from the New York Times:
In breaking down a single return selection–down the line or cross-court–you get the flavor of how geometry, physics, probability, and game-theory all factor in to this instantaneous decision.
These folks at the New York Times really do an excellent job producing these videos.
Appreciation Challenge
Passing Time and Tolls
I enjoy traveling and I enjoy driving, but I don’t enjoy paying highway tolls. One way I try to diffuse that annoyance is by hoarding coins when I travel, with the intent of paying every toll on the way back entirely in loose change.
I highly recommend the activity–it’s a fun little counting game, it keeps the coins in circulation, and there is a sense of satisfaction that accompanies using 18 assorted coins in a transaction.
But be warned–if you are travelling along the Hudson River in New York state, on I-87, and you decide to get off at Saugerties, one of the toll-booth operators there does not like pennies. I mean, really, does not like pennies. Especially more than 40 of them. Trust me.
Appreciation History
Socks and the Axiom of Choice
Every time I buy socks I think of the Axiom of Choice.
About a century ago, mathematicians were arguing about exactly which basic axioms, or assumptions, were needed in order to justify all of our mathematics. Because of the personalities involved and the nature of mathematical discourse at the time, Set Theory was the starting point, and one of the axioms under consideration was the Axiom of Choice.
Deciding on axioms is tough business: an axiom has to be powerful enough to do something but obvious enough for people to accept it as true without evidence. But deciding on axioms has to be done: before we can prove anything, we need to assume something is true.
The Axiom of Choice essentially says that if you have an infinite number of sets, you can form a new set by choosing an arbitrary element from each of those sets. It seems sensible enough, but fierce mathematical debate raged for years about whether this was obvious enough to be true. Some mathematicians still don’t accept this principle.
So why would someone object to this sensible-enough idea? That’s where shoes and socks come in.
Suppose you had infinitely many pairs of shoes. There’s a straightforward way to define a new set that contains one shoe from each pair: choose every left shoe. This explicit rule make its clear how to construct this new set, and so forming this new infinite set seem reasonable.
But imagine you had infinitely many pairs of socks. Since the socks are identical, you can’t give a specific rule that says “for each of the pair of socks, give me that one”. You need to believe in the Axiom of Choice in order to believe that such a set, one containing one sock from each pair of socks, can really be formed without giving an explicit rule.
As it turns out, deciding to believe in this set of socks has substantial consequences for what you can prove in mathematics. So there’s something to think about the next time you are sock shopping!
Related Posts
Application Geometry
Limits and Oar Making
A boat-builder once described to me one procedure for making oars.
Start with a long piece of lumber with a square cross-section. Take off a certain amount from each of the four corners. Now their are eight corners. From each of the eight corners, take off a smaller amount. The progression of cross-sections looks something like this:
Through practice, the builder knows how much wood to remove at each stage. You can continue to repeat this process, but some sanding at this point will probably get you pretty close to what you are looking for: a circular cross-section.
This naturally brings to mind the hallmarks of Calculus: approximations and limits. At each stage of the process, the cross-section of the oar becomes a better and better approximation of a circle. Indeed, the limit of such a process is indistinguishable from a circle.
I’m not sure if anyone making oars is thinking about Calculus, but sometimes it’s hard for me not to think about it!