By Igor Kriz and Paul Siegel, at Scientific American: Rubik's Cube Inspired Puzzles Demonstrate Math's "Simple Groups".
The Rubik's cube can be said to be a physical embodiment of the Rubik group, which is a certain subgroup of S48. (Why 48? There are 54 "facets" of the Rubik's cube, nine on each side; six of these don't move, leaving 48.) This subgroup has an easy presentation with six generators, namely rotations of the six faces. As the authors point out, other "Rubik's-type" puzzles embody groups of the same general sort.
The authors have invented puzzles based on certain sporadic groups, namely theMathieu groups M12 and M24 and the Conway group Co0. These right now only exist as computer programs, although the authors claim that a physical version of the M24 puzzle could be built.
A possibly interesting, although not-at-all well-defined question -- which groups are "buildable", in the sense that one can build a physical object that represents them?
The programs (which run on Windows machines) can be downloaded from Igor Kriz's home page.
Thanks to John Armstrong for pointing me to the article.
A random walk through mathematics -- mostly through the random part.
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4 comments:
Must the realization of a group be in three dimensions or less?
I'd wager that if you can use any number of dimensions, you can build some kind of puzzle to represent any finite group.
Perhaps the smallest number of dimensions to "construct" a given group might be related to representation-theoretic quantities. But my knowledge of representation theory is pretty much nill.
So the presentation is going to be defined by a set of permutations? Then I suspect that a model exists.
Take the presentation permutations and break them into cycles. Use gearing to gang these cycles so they all happen together. In the Rubik's cube, this is done by putting them on the same axis.
Mechanical problems arise when a single element has too many presentation cycles that it is required to be a member of. The usual Rubik's cube maximum for this number is 3, that is, the corners are on three presentation cycles.
To increase this number, first let the elements being permuted be represented by spheres (so the cycles do not require any particular symmetry). I don't see any limit in the number of marble cycles that go through a single marble, if the mechanics arranges for the marbles to be other than adjacent. In fact, I think that a horizontal model exists, where the various marbles on a cycle get moved together by gears.
For example, you can push the marbles by arranging for them to be shifted. They do not have to be moved along by a solid circular device.
Isn't the question of buildability just representation theory? Which groups admit faithful linear representations, etc.
And to ben allen, I think the smallest dimension of a linear group into which a group has a faithful representation *is* a representation-theoretic quantity. I don't recall it being terribly interesting thing, but it's probably somewhere in the literature.
Hey! I learned vector calculus from Igor Kriz when I was a wee freshman! It was my first college math class, and I enjoyed it a lot. ^_^ I only found out years later that Prof. Kriz has an Erdos number of two! o_O
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