[math-fun] nuclear pennies / archimedes codex / indian clerk
1) Nuclear pennies, a game on a semi-infinite strip http://sigfpe.blogspot.com/2007/09/arboreal-isomorphisms-from-nuclear.html later on, the blog entry devolves into a discussion of functional programming, Haskell, and monads, but still, interesting 2) I'm enjoying this recent book by Reviel Netz and William Noel, The Archimedes Codex: http://www.amazon.com/Archimedes-Codex-Reviel-Netz/dp/030681580X 3) The recent book by David Leavitt, "The Indian Clerk," a novel about Hardy and Ramanujan, is fun too http://www.amazon.com/Indian-Clerk-Novel-David-Leavitt/dp/1596910402/ -- Thane Plambeck tplambeck@gmail.com http://www.plambeck.org/ehome.htm
The really fun part of nuclear pennies is the link at the very end, Blass' paper "seven trees in one." There, he shows how since a full binary tree is either a leaf or two trees, we have T = T^2 + 1 and, solving for T, we get a 6th root of unity for the cardinality of the groupoid T. T^6 isn't isomorphic to 1, but T^7 is isomorphic to T. Now, that's nothing special, since the set of full binary trees is countable. But this isomorphism doesn't need to go deeper into the trees than a fixed amount illustrated on the page. Baez and Dolan showed how groupoids are like "fractional sets," that it makes sense to talk about algebraic structures with rational nonnegative cardinality: |X| = \sum_{equivalence classes [x]} 1/Aut(x) Many others have extended their work. Fiore and Leinster follow up the seven-in-one paper with Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers, math/0212377 and show that their definition has Baez & Dolan's cardinality as a special case. Many more references here: http://www.math.ucr.edu/home/baez/counting/ -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
Bob, Thanks! The game reminds me of the Game of the Life. Marlin -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Thane Plambeck Sent: Monday, October 01, 2007 7:41 PM To: math-fun Subject: [math-fun] nuclear pennies / archimedes codex / indian clerk 1) Nuclear pennies, a game on a semi-infinite strip http://sigfpe.blogspot.com/2007/09/arboreal-isomorphisms-from-nuclear.ht ml later on, the blog entry devolves into a discussion of functional programming, Haskell, and monads, but still, interesting 2) I'm enjoying this recent book by Reviel Netz and William Noel, The Archimedes Codex: http://www.amazon.com/Archimedes-Codex-Reviel-Netz/dp/030681580X 3) The recent book by David Leavitt, "The Indian Clerk," a novel about Hardy and Ramanujan, is fun too http://www.amazon.com/Indian-Clerk-Novel-David-Leavitt/dp/1596910402/ -- Thane Plambeck tplambeck@gmail.com http://www.plambeck.org/ehome.htm _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Thanks for the "Indian Clerk" reference -- I already have the Archimedes Codex book (and the SLAC physicist who worked on the project spoke at a lunch meeting at my school last spring!). As for nuclear pennies, I did the activity with my 9-year-old and then with a group of middle school math students at the San Jose Math Circle (http://geometer.org/sjcircle or http://sanjosemathcircle.org ). One thing that my 9-year-old pointed out is that "if you split the first penny, it should really be half a penny on each space, not a whole penny". Then we discussed what happened with further splitting ... and yeah, if you put a quarter penny on each space, that's OK, but it gets hard to keep track when you have 5/8 of a penny somewhere -- how do you represent that? So then we decided that each place should have its value (that's me trying to steer him toward the interesting invariant, instead of having the invariant be that all the penny pieces add up to one whole penny). Starting with the 1 splitting in half, you pretty quickly get a pattern of period 6: 1, 1/2, -1/2, -1, -1/2, 1/2, 1. It's the real part of the complex numbers! So you can prove that translations of a single penny must be multiples of 6 without needing to solve x^2 + 1 = x or dealing with complex numbers. And of course you can start with any two numbers, not just 1 and 1/2. I teased the kids by asking them to assign values to each space that will avoid the fractions ... which they could do ... and then by asking them to avoid the negative numbers ;) --Joshua Zucker
participants (4)
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Joshua Zucker -
metaweta＠gmail.com -
Thane Plambeck -
Thomas, Marlin