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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 29, 2009 9:26 pm 
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elijah wrote:
and this is what virtual (or imaginary) pieces are?
Or... how exactly does this relate to imaginary pieces?


These new face centers and edges can almost be thought of as Matt's inverted faces (piece type 2) and Matt's inverted edges (piece type 4), two of the imaginary pieces. They aren't quite a perfect match as their movment isn't tied to the faces but as Bram says above they are so close they may be hard to differentiate them from the imaginary pieces. Even if they are different its still a nice puzzle.

The best way to answer what the virtual pieces are is to think of a 3x3x3 where every face turn also turns the slice layer next to it. This way each face center would be moved with 5 of the 6 face turns and each edge would be moved with 4 of the 6 face turns. Note, one face turn moves 5 faces and 8 edges. These faces and edges are Matt's imaginary pieces (types 2 and 4). If you could somehow get Bram's idea, these floating face centers and edges, to move in this way they would certainly be Matt's imaginary pieces.

The virtual pieces that Andreas talks about in the first post of this thread appear to be a subset of the imaginary pieces. The 3x3x3 does have imaginary pieces but doesn't have any virtual pieces according to the way they are defined above.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Wed Dec 30, 2009 7:32 am 
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Seems liek the last 20 posts were about implementing the imaginary pieces of the 3x3x3. That is okay to me but I can't contribute there.

But I wasn't unproductive the last days:
I analyzed the Circle-faceturning dodecahedrons of Gelatinbrain:1.1.[22-25]
I was successful for the circle Pyraminxes (5.1.18 and 5.1.19) too.
In every case no imaginary pieces were needed. The virtual ones were needed in some cases.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Wed Jan 06, 2010 5:31 pm 
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Andreas Nortmann wrote:
Seems liek the last 20 posts were about implementing the imaginary pieces of the 3x3x3.

Here is an animation for Bram's idea that I left rendering while I was off to my interview...
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Andreas Nortmann wrote:
But I wasn't unproductive the last days:
I analyzed the Circle-faceturning dodecahedrons of Gelatinbrain:1.1.[22-25]
I was successful for the circle Pyraminxes (5.1.18 and 5.1.19) too.
In every case no imaginary pieces were needed. The virtual ones were needed in some cases.

I think that has to do with the way the virtual and imaginary pieces are defined. Virtual pieces can remain stationary under a particular (full yet independant) set of twists as can all real pieces. The imaginary pieces can not remain stationary and I believe to put these pieces to use in an actual puzzle or in applet form would require very odd turns like the complex 3x3x3 which looks like a 5x5x5 but still only has U, D, L, R, F, and B as allowed twists. All these circle puzzles expose pieces with can be held on to (i.e. remain stationary) under a full/independant set of twists.

Carl

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