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 Post subject: Piece orbits on the Master Skewb
PostPosted: Mon Feb 25, 2013 4:52 pm 
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I've been solving the Master Skewb for awhile now and I noticed something. Whenever I scramble and solve it the square-point pieces seem to behave as if they are in an "orbit" similar to the triangles on a helicopter cube. After studying the geometry of the pieces and grouping them here's my observation: am I right in saying that the pieces highlighted in light blue (below) can never switch or change places with the pieces highlighted in gray? If true, why is this so?
Attachment:
File comment: The light blue pieces lie in one orbit, and the gray pieces lie in another orbit, so they can never switch places.
master skewb geometry.jpg
master skewb geometry.jpg [ 230.21 KiB | Viewed 1046 times ]

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 Post subject: Re: Piece orbits on the Master Skewb
PostPosted: Mon Feb 25, 2013 5:04 pm 
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benpuzzles wrote:
...am I right in saying that the pieces highlighted in light blue (below) can never switch or change places with the pieces highlighted in gray? If true, why is this so?
...
Your assumption is correct!
Just try a sample where one blue piece wants to travel to a grey place. This is not possible by legal actions.
Like the Helicopter centres can never exchange orbits without jumbling.

The corners live in two orbits, too.

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 Post subject: Re: Piece orbits on the Master Skewb
PostPosted: Mon Feb 25, 2013 6:03 pm 
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Most of the pieces on the FTO / VTC are in two orbits. It isn't restricted to just to the X centers on the Master Skewb. Ignoring orientation, there are
25617399782746378052141555666743402716019309867392
16028988632254755015360675679513378669945809209917
440000000000000000000000000000000000
distinct permutations for the Complex VTC. If you factor this number it suggests that there are no orbits with more than 12 pieces.

If you're looking for a physical reason for this, take a look at TomZ's Tetrahedral Twins. You can think of a FTO / VTC as two tetrahedral puzzles fused together and only partially interacting with each other. This is also why Skewb Corners are in two different orbits. A move on one of the tetrahedra may affect pieces in the other but the two can never exchange pieces.

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Last edited by Brandon Enright on Mon Feb 25, 2013 7:27 pm, edited 1 time in total.

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 Post subject: Re: Piece orbits on the Master Skewb
PostPosted: Mon Feb 25, 2013 6:53 pm 
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bmenrigh wrote:
You can think of a FTO / VTC as two tetrahedral puzzles fused together and only partially interacting with each other. This is also why Skewb Corners are in two different orbits. A move on one of the tetrahedra may affect pieces in the other but the two can never exchange pieces.
It's not just the pieces with cant be exchanged. On the FTO/VTC puzzles there are cuts that clearly belong to one tetrahedral puzzle that never align with the cuts of the other "fused" tetrahedral puzzle. You can really highlight this difference in orbits by varying cut depth for example and create puzzles like the Offset Skewb and F-Skewb. And without varying cut depth you can vary cut shape and get into the idea I proposed and grigr rendered here. There the corners which are in different orbits are now actually different shapes and its clear the edges can never be flipped in place. The edges can never be flipped on the Breese Cube either but its possible to make it obvious. Note if you did this to a master skewb it would be possible to make your light blue and gray pieces actually different shapes.

Carl

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 Post subject: Re: Piece orbits on the Master Skewb
PostPosted: Tue Feb 26, 2013 10:53 pm 
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Interestingly enough, Drewseph's Master Skewb used a tetrahedral core, so the "square point pieces" are physically different as far as the internal mechanism is concerned.

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 Post subject: Re: Piece orbits on the Master Skewb
PostPosted: Tue Feb 26, 2013 11:06 pm 
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Puzzlemaster42 wrote:
Interestingly enough, Drewseph's Master Skewb used a tetrahedral core, so the "square point pieces" are physically different as far as the internal mechanism is concerned.

Oh, thanks for letting me know! I really asked this question to be sure such a mechanism could be used (I'm designing a new puzzle) :wink:

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 Post subject: Re: Piece orbits on the Master Skewb
PostPosted: Tue Feb 26, 2013 11:41 pm 
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Puzzlemaster42 wrote:
Interestingly enough, Drewseph's Master Skewb used a tetrahedral core, so the "square point pieces" are physically different as far as the internal mechanism is concerned.

I've played with a few VTC puzzles with a tetrahedral mechanism rather than octahedral. It simplifies the design but it's much harder to get the tension uniform on the puzzle and it's much easier for big groups of pieces to pop out. I recommend a octahedral core if it's feasible.

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 Post subject: Re: Piece orbits on the Master Skewb
PostPosted: Wed Feb 27, 2013 5:11 pm 
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bmenrigh wrote:
I've played with a few VTC puzzles with a tetrahedral mechanism rather than octahedral. It simplifies the design but it's much harder to get the tension uniform on the puzzle and it's much easier for big groups of pieces to pop out. I recommend a octahedral core if it's feasible.

You do bring up a good point but there is actually a simple way around the popping problem. By using both symmetries of the tetrahedral mechanism in two separate layers, it becomes more stable.
I'll use the Skewb as an example. A regular Skewb has just one groove underneath the center pieces and half of the corner pieces. I've realized it's possible to add a second layer of grooves underneath the first one to make the pieces more stable, as you can see below:
Attachment:
File comment: Center
skewb center.jpg
skewb center.jpg [ 177.51 KiB | Viewed 792 times ]

Attachment:
File comment: Floating corner
floating corner.jpg
floating corner.jpg [ 163.16 KiB | Viewed 792 times ]

Attachment:
File comment: Anchored corner and core
anchored corner and core.jpg
anchored corner and core.jpg [ 160.56 KiB | Viewed 792 times ]

Here's an image of the type of interaction between pieces:
Attachment:
File comment: Interaction 1
interaction 1.jpg
interaction 1.jpg [ 177.73 KiB | Viewed 792 times ]

Attachment:
File comment: Interaction 2
interaction 2.jpg
interaction 2.jpg [ 174.78 KiB | Viewed 792 times ]

Keep in mind that it still uses a 4 armed core, but because all of the pieces are gripped evenly on all sides, the puzzle becomes (nearly) as stable as an octahedral core.

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