Some Reflections
Oct. 26th, 2008 12:37 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowA couple of people have asked for an explanation of the gesticulations I described in this post. I'll try.
First, picture a row of five glasses, labelled 1 through 5. Part of the question deals with all the ways of rearranging those glasses; the set of all of those rearrangements is called "the symmetric group on five letters", or S5 for short. Of particular interest are the "transpositions", which simply swap the positions of two glasses; each transposition is named simply by giving the numbers of the glasses which are to be interchanged. So, for instance, (12) is the name for the act of swapping the glasses labelled 1 and 2.
Now, each transposition has "order two"; that is, if you perform the operation twice, you're back where you started. Where things get interesting is the combination of two different transpositions. If, for example, you apply (12) to the initial order 12345, it becomes 21345. If you then apply (13), the order becomes 23145. This combined rearrangement isn't a transposition; it has order three - that is, you'd have to do it three times to restore the original position. We say, "the composition of (13) and (12) has order three". (Yes, that's the order we put them in; the first operation to be applied goes on the right, and the second on the left. There are reasons for this.) On the other hand, the composition of (12) and (45) has order two, although it isn't a transposition. (The composition turns 12345 into 21354.)
That's one side of the story. The other has to do with operations, not on that row of glasses, but on the whole of three-dimensional space. One kind of operation is a "reflection"; every point on one side of a specified plane gets moved to a point the same distance on the other side, just like a mirror. Reflections also have order two; if you reflect, and then reflect back in the same plane, you're back where you began. Once again, things get interesting when you combine two different reflections.
To follow this next bit, it might be best to think in two dimensions instead of three; think of a sheet of paper, and of flipping it over a specified line. If you flip the sheet over one line, and then flip it over another, the sheet winds up right-side-up again, but it's been rotated. If the angle between the two lines is, say, 60 degrees, the sheet gets rotated through twice that, or 120 degrees. If the two lines are perpendicular, the sheet rotates through 180 degrees. Now, that 120-degree rotation has order three; three 120-degree rotations put you back where you started. The 180-degree rotation has order two.
What we were wondering about was this: is it possible to, so to speak, mimic the behavior of S5, using operations on three-space? My first gesture - left arm forward, palm vertical - represented a vertical plane, which I was associating with (12). The second gesture - right arm at a 60-degree angle to the first, palm vertical - represented another vertical plane, sixty degrees away from the first one, and I associated that with (13). Just as the composition of (13) and (12) has order three, so the composition of the reflections in those two planes also has order three. The third gesture - left arm sweeping horizontally - indicated a horizontal plane, which would be perpendicular to the first two; I associated that with (45). Again, just as the composition of (45) with either (12) or (13) has order two, so the composition of reflection in one of the vertical planes with reflection in the horizontal plane has order two. The question was whether we could associate some operation in three-space to every element of S5, and have the compositions behave in the same way. (It turns out to be impossible, but I didn't come up with a proof of that for another couple of days.)
Anyway, that's what was going on.
First, picture a row of five glasses, labelled 1 through 5. Part of the question deals with all the ways of rearranging those glasses; the set of all of those rearrangements is called "the symmetric group on five letters", or S5 for short. Of particular interest are the "transpositions", which simply swap the positions of two glasses; each transposition is named simply by giving the numbers of the glasses which are to be interchanged. So, for instance, (12) is the name for the act of swapping the glasses labelled 1 and 2.
Now, each transposition has "order two"; that is, if you perform the operation twice, you're back where you started. Where things get interesting is the combination of two different transpositions. If, for example, you apply (12) to the initial order 12345, it becomes 21345. If you then apply (13), the order becomes 23145. This combined rearrangement isn't a transposition; it has order three - that is, you'd have to do it three times to restore the original position. We say, "the composition of (13) and (12) has order three". (Yes, that's the order we put them in; the first operation to be applied goes on the right, and the second on the left. There are reasons for this.) On the other hand, the composition of (12) and (45) has order two, although it isn't a transposition. (The composition turns 12345 into 21354.)
That's one side of the story. The other has to do with operations, not on that row of glasses, but on the whole of three-dimensional space. One kind of operation is a "reflection"; every point on one side of a specified plane gets moved to a point the same distance on the other side, just like a mirror. Reflections also have order two; if you reflect, and then reflect back in the same plane, you're back where you began. Once again, things get interesting when you combine two different reflections.
To follow this next bit, it might be best to think in two dimensions instead of three; think of a sheet of paper, and of flipping it over a specified line. If you flip the sheet over one line, and then flip it over another, the sheet winds up right-side-up again, but it's been rotated. If the angle between the two lines is, say, 60 degrees, the sheet gets rotated through twice that, or 120 degrees. If the two lines are perpendicular, the sheet rotates through 180 degrees. Now, that 120-degree rotation has order three; three 120-degree rotations put you back where you started. The 180-degree rotation has order two.
What we were wondering about was this: is it possible to, so to speak, mimic the behavior of S5, using operations on three-space? My first gesture - left arm forward, palm vertical - represented a vertical plane, which I was associating with (12). The second gesture - right arm at a 60-degree angle to the first, palm vertical - represented another vertical plane, sixty degrees away from the first one, and I associated that with (13). Just as the composition of (13) and (12) has order three, so the composition of the reflections in those two planes also has order three. The third gesture - left arm sweeping horizontally - indicated a horizontal plane, which would be perpendicular to the first two; I associated that with (45). Again, just as the composition of (45) with either (12) or (13) has order two, so the composition of reflection in one of the vertical planes with reflection in the horizontal plane has order two. The question was whether we could associate some operation in three-space to every element of S5, and have the compositions behave in the same way. (It turns out to be impossible, but I didn't come up with a proof of that for another couple of days.)
Anyway, that's what was going on.
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2008-10-26 11:09 pm (UTC)*
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(Yes, yes you are going to be sorry.)
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P-L-A-N-E.
Heh.
no subject
Date: 2008-10-26 11:24 pm (UTC)(There is more than one pun in the line above.)
no subject
Date: 2008-10-27 01:43 am (UTC)But it's always fun to read you, talking maths. I can grasp that the explanations would be perfectly clear if only I had the mathematical background--if that makes sense. Perhaps more like: it's that I can see the structure of what you are saying, even if the depth of meaning doesn't click. It's like looking at beautiful scenery through fog. I might as well not be frustrated because it's foggy, but just appreciate that I can glimpse a line here, a shadow there. Best I can do.
no subject
Date: 2008-10-28 02:32 am (UTC)For the rest... well, I tried.
no subject
Date: 2008-10-28 03:18 am (UTC)And Subcreator, thanks for letting me know that there are others who read SF's maths posts with enjoyment, even if we lack complete comprehension. I like your abstract art analogy--that works, too.
no subject
Date: 2008-10-28 02:49 am (UTC)(I have to admit, I too see only the obvious pun.) :-(
-Peter
* I follow your description, but I lack the ability to make an application. Alas, I have only basic skills in algebra and geometry - 28 years removed.
no subject
Date: 2008-10-28 02:50 am (UTC)-P