Wikipedia:Reference desk/Archives/Mathematics/2008 November 1

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November 1

Rubik's cube and Square one

A friend of mine was showing me the Square one - it has a similar concept to the Rubik's cube but changes shape when turning faces, so doesn't always stay a cube (in fact, it's not exactly a cube even in solved state).

I was wondering how it is possible to give a sensible group structure on it - as trying to do it naively as with the Rubik's cube fails and we often end up with non-composable operations.

(You can refer yourself to this applet http://f2.org/cgi-bin/sq1.cgi for the description)

For example, if you start by twisting the top and then twist the right hand side, you get to a position where there are only 4 valid positions for the top where you can twist the right hand side (instead of 8 originally), but still 8 distinct configurations. (and that's just one example of the many pathological properties). Whereas in the star configuration (see applet), there are only 6 distinct positions for the top, from all of which it is possible to twist the right hand side.

I'm not sure how to phrase this - but is there a way to put a meaningful group structure on this ? I'll always be able to consider it as a group just by taking the symmetric group on however many possible configurations there are and making up a silly group operation, but that won't correspond to physically turning faces of the cube.

I'm thinking of maybe having something with three generators, one being "turn the top until next possible configuration where you can twist down", one being "twist down" and the last being "turn the bottom until next possible configuration where you can twist down". But I have a feeling that that would go wrong too (though I'm not sure), so I'm at a loss to think what could work.

Thanks. --Xedi^Talk 09:21, 1 November 2008 (UTC)

The best you can do to get a group is simply to allow as your group elements only those sequences of moves which return the puzzle to its initial shape (though possibly permute colours). If you want to model the puzzle more closely you can consider a groupoid in which each distinct "shape" of the puzzle is a different object. —Blotwell 23:28, 1 November 2008 (UTC)

This is covered on about page 216 of Joyner's Adventures in Group Theory. I believe he explains both the group and groupoid ideas. 74.140.212.193 (talk) 00:29, 3 November 2008 (UTC)

Adding two waves

Is there a formula for adding two waves of different frequency and amplitude together? IE: asin(ft+x) + Asin(Ft+X) = ???

Thanks207.203.88.15 (talk) 16:35, 1 November 2008 (UTC)

what you've written is it. You can't simplify adding different frequencies any more (unless one is an integer multiple of the other). -- SGBailey (talk) 22:14, 1 November 2008 (UTC)

You can, at least in some cases. See Beat (acoustics). --Tango (talk) 23:39, 1 November 2008 (UTC)