Addition of commutator GIF
I change PI to PN for consistency reasons. I got rid of the table because it is now extraneous. However it is commented out in the raw text of this talk page in case you want to re-include it.
Circular17, I don't want to mess with your work and make large changes, but I think the part that is directly above the gif and reproduced below is not longer needed. I think in this case "a picture/animation says a thousand words"
We notice that N A and N B are used as temporary storage :
- The content of the intersection, PN, is first hidden by A in its storage N A, and comes back with A', so that the only transformation applied to these pieces is B'.
- The content of N A' is brought by A, and is then hidden by B in N B, so that the only transformation applied to these pieces is A.
- The content of N B' is brought by B, and then is placed immediately in N A' by A'. So the transformation applied to these pieces is B A'.
In other words, all the commutator does is :
- PA goes A
- PB goes B A'
- PN goes B'
That being said, I think what might be really helpful is if below the animation, we talk about a "real life example". I'll see if I can put something together.Dave3457 (talk) 15:18, 28 November 2012 (EST)
Changes by Circular17
- I would be happy to look over your work. Keep in mind that I not sure that I always receive an email notification when a page I'm watching on Speedsolving Wiki (SSW) is changed . I did receive one for your March 30 changes however. My watchlist only showed one of your March 30 edits. (Maybe that's intended)
- I am looking things over now, it might take a couple of days.
- Aren't commutators wonderful ! :) I just discovered (using a program) a wonderful 2-gen one that twists 2 corners. Its the only one of its kind that I know of, it may be my favorite comm being a 2-gen and all. Check it out below. (Note: I personally cut and paste new algorithms I'm learning into CubeTwister)
- Its not that hard to understand how this comm works, you don't have to memorize it, follow the two corner-edge pairs on each side of the twisted corner and how the twisted corner moves between them.
- R2DRD'RDR2 D2 R2D'R'DR'D'R2 D2 (16,22)
- [ R2DR D' RDR2, D2] OR [ R2DR D' RDR2, D] - Dave3457 (talk) 04:05, 31 March 2013 (UTC)
- I don't know how these notifications works.
- That's an interesting sequence indeed.
- I would reflect it like this :
- R2U'R'UR'U'R2 U' R2URU'RUR2 U
- so that it can be used for orienting corners of the last layer on top.
- or for left hand :
- L2ULU'LUL2 U L2U'L'UL'U'L2 U'
- --Circular17 (talk) 02:46, 31 March 2013 (EDT)
- Ya, any commutator I post will certainly be designed for those who solve the top layer last. Myself, I don't solve for speed and solve the bottom layer last. In fact I often do not rotate the cube at all but keep each face pointing in the same direction all of the time. That way I don't use muscle memory to perform an algorithm and force myself to understand the algorithms more deeply.
- - Dave3457 (talk) 23:07, 8 April 2013 (EDT)
- I created two images to help with the defining of the variables and took what you wrote and expanded it. I've noticed that your writing is very "dense". Judging from your WP User page, I'm assuming that English is not your first language although you wouldn't guess judging from your grammar. (you do occasionally "miss" an odd word however). (For the record, I fully appreciate that my own grammar and spelling is very poor).
- You are right, English is not my first spoken language. But I started it very early with programming languages.
- With regards to your writing, may I suggest that you make it less "dense". I suspect that you are a mathematician. [I can't be sure because I can't read your user page :)] Whether or not you are a mathematician you tend to take the symbolic shortcuts that mathematicians tend to.
- You are right, I've studied mathematics and IT.
- For example I changed "When J and K have an empty intersection, A and B commute,..." to "When J and K have no intersection, A and B commute,..."
- Oh that's right. I'll try to think about it. It's not really a shortcut, but that there is a mathematical object describing the intersection which is called the empty set. But I completely understand that most people would be confused.
- That being said I appreciate that as things get more technical as you go into more and more detail, not using the concise language of mathematics becomes more and more difficult.
- Yeah, that's why I've done that, because I was lost myself.
- Because of my desire to share the wonders of the commutator to the general non-mathematical population I, personally, will be focusing on the basics here and will be trying to make those basics as clear as possible using images and "extended" language. (Also I'm not sure I'm going to understand, or be able to take the time to understand, the later stuff)
- Your explanations are very good. Thanks.
- I hope you don't mind but I'm planning on modifying and "extending" the first and second sections quite heavily.
- Have fun :)
- I've only glanced at the new stuff that you have written, but from what I've read it seems pretty "heady".
- Yes, it was hard for me to do this mathematical work, so I can understand that it's not very readable.
- My general thoughts at this point are that we can expect each reader to stop reading the page when things get to difficult for them. Or to put it another way, they will stop when the amount of effort required is not worth the reward. Toward this end, I personally, would strongly suggest beginning with the easiest stuff and working toward the more difficult, even if its not the "correct" order from a mathematician's point of view. For example, you presently begin with the example case [M', U2] = M' U2 M U2 which rotates edges, however the case you mention later, [R' D' R D R' D' R, U] which twists two corners would be much easier to understand. I could see me creating a series of images or an animation for that one.
- Absolutely. I'm ok with that.
- Also, I'm thinking that, at least when we are dealing with the basic stuff, that things would be easier to understand if we dealt with an example right from the start rather than talk in the abstract. For example you wrote...
- If NB' = N, the sequence B only moves affected pieces that are inside the intersection. It may also move pieces that are outside the intersection, but those moves will be cancelled at the end. Affected pieces will only be N and NA'. So those pieces will be in J, i.e. among pieces that are directly affected by A.
- While I know I'm speaking from the point of view of a guy with only an average IQ, I think a lot of people besides me would not follow that easily either. If a diagram was associated with it I'm sure it would be a lot easier to follow.
- In fact, I'm thinking about specific moves while writing this. My goal was to make it general so that it can be applied to any situation. The quirk part and so on are the result of that exhaustive description.
- Personally, I thinking that since the B move is usually the rotation of the Up side, we should start with A inplace situations rather than the B inplace situations and that way we can talk about the easier to understand examples ...
- A = "rotating a corner" like [R' D' R D R' D' R, U]
- A = "flipping an edge" like [R' E' R2 E2 R', U]
- A = "exchanging two corners" like [R' L' D2 R L, U]
- A = "exchanging two edges" like [M2 D2 M2, U]
- Odds and ends:
- ...I noticed that you referred to a single corner being "rotated". Personally I prefer to reserve the word "rotate" to when I'm dealing with 3-cycles and such. I instead try to refer to them as being "twisted"
- - Dave3457 (talk) 00:45, 9 April 2013 (EDT)
- I want to make a strong suggestion. Do you know how later on you use subscripts to identify pieces such as PA,PB,PN . May I suggest that we also use subscripts to identify the locations of J and K. May I suggest the capital letter L which would stand for Location. ie J=>LA, K=>LB. This would greatly relieve the "mental strain" one experiences trying to remember what they stand for. Since J and K are bolded one could simply do a mass "find and replace" operation. (although not in the images).
- While technically we should do it for N, NA' and NB' as well, given how often those locations of referenced, I think that might be to much clutter and not really necessary. I would be happy to do the switch over.
- - Dave3457 (talk) 15:23, 9 April 2013 (EDT)
- I was tempted to use the letter L at some point, but I noticed that it can be confused with the move L (left). With subscript however, this may be ok. About N, are you thinking about LN ? Not sure it would be better, so I suppose we can keep it that way. And it sounds good, I think. N like intersection, like N-tersection. --Circular17 (talk) 07:49, 12 April 2013 (EDT)
Relationship of NA’=N and NB’=N
- Circular17, I suspect that there is something misleading about the way things are being presented right now. There is a bit of a suggestion that NA’=N and NB’=N are not perfectly symmetrical. (or maybe they aren’t)
- Specifically…the two NA’=N examples that I have discussed…
- [R' D' R D R' D' R, U] and [M2 D2 M2, U]
- have their NB’=N symmetric counterparts which accomplish the exact same thing…
- [U, R’ D R D’ R’ D R] and [U, M2 D2 M2]
- It seems that in these two cases [A,B] = [B,A’].
- However it isn’t true for…
- [U2,M] which is a NA’=N version
- [M', U2] which is a NB’=N version
- ie [A,B] ≠ [B,A’] in this case.
- I guess it is because of the “overlap”. That Quirk thing that you talk about but that I don’t understand yet. Am I right on that?
- Whatever the case, we don’t want to give the impression that NA’=N and NB’=N are any different if they aren’t. I guess it comes down to whether or not you can do the same thing with both by just exchanging and reversing things. To what degree are they different?
- -- Dave3457 (talk) 04:29, 12 April 2013 (EDT)