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Wednesday, January 18, 2006

Translations and Duality

Imagine playing a game in which cards numbered from 1 to 9 are placed face up on the table, and players take it in turns to pick up a card. Your aim is to be the first to collect a hand in which there are three cards which sum to 15. Now, if you know an optimal strategy for noughts and crosses, or tic-tac-toe, you'll be able to play this game optimally too. The solution is at the end of the post.

This translation between problems I found in a paper by Offer Shai to introduce more useful kinds of transformations between engineering problems. For example, to check whether a complicated configuration of trusses is stable, you form the dual of its graph, thereby yielding a new graph which represents a certain mechanism. Your question has now become whether this mechanism is mobile, something you already know how to solve. (See p. 16 for a more thorough table of the analogy.) Shai goes on to use this translation to find unthought of analogues to known concepts.

In recent years string theorists have pointed to extraordinarily rich mathematical dualities thrown up by their understanding that certain physical models should be dual. Cumrun Vafa's Geometric Physics is a gentle introduction (at least to begin with) to some of these. Again, analogues can end up looking very different: "...the question of quantum corrections for one manifold gets transformed to the question involving the variation of complex structure on the other, which is classical."

I've been chatting with John Baez about another opportunity for analogizing, this time between using different rigs. Rigs are essentially rings but without necessarily having inverses for addition. like the natural numbers. If you follow the link, you'll be able to see how sums of amplitudes of paths from quantum mechanics, the path of least action from the calculus of variations, and path connectedness from homotopy theory are all related.

Answer to puzzle: Each time you or your opponent takes a card imagine a O or an X being placed in the corresponding square in this grid:

2 7 6
9 5 1
4 3 8


Georg said...

The tic-tac-toe duality isn't exact, though, is it? E.g. 9+6=15 gives a winning hand in the card game, but the corresponding two noughts or crosses don't give a win in tic-tac-toe.

February 10, 2006 5:53 PM  
david said...


Notice that I specified that 3 cards need to sum to 15, so it is an isomorphic problem. It's not a bad party trick. People sense something familiar about the situation but usually don't quite know why.

February 10, 2006 10:28 PM  

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