tag:blogger.com,1999:blog-264226589944705290.post2417071908577737519..comments2022-05-19T07:55:48.367-07:00Comments on God Plays Dice: Rubik's deck of cards?Michael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-264226589944705290.post-8233937060745395922008-06-10T17:54:00.000-07:002008-06-10T17:54:00.000-07:00Hey! I learned vector calculus from Igor Kriz when...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_OAaronhttps://www.blogger.com/profile/18281785407407667986noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-60742437573413947362008-06-09T20:26:00.000-07:002008-06-09T20:26:00.000-07:00Isn't the question of buildability just representa...Isn't the question of buildability just representation theory? Which groups admit faithful linear representations, etc.<BR/><BR/>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.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-81336167371262850912008-06-09T16:05:00.000-07:002008-06-09T16:05:00.000-07:00So the presentation is going to be defined by a se...So the presentation is going to be defined by a set of permutations? Then I suspect that a model exists.<BR/><BR/>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.<BR/><BR/>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.<BR/><BR/>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.<BR/><BR/>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.CarlBrannenhttps://www.blogger.com/profile/17180079098492232258noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-47506902630959673552008-06-09T15:37:00.000-07:002008-06-09T15:37:00.000-07:00Must the realization of a group be in three dimens...Must the realization of a group be in three dimensions or less? <BR/><BR/>I'd wager that if you can use any number of dimensions, you can build some kind of puzzle to represent any finite group.<BR/><BR/>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.Ben Allenhttps://www.blogger.com/profile/15594823641514744644noreply@blogger.com