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 Post subject: What IS Shim's F-Skewb?
PostPosted: Thu Sep 30, 2010 2:15 am 
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While trying to answer this question myself I think I've come up with the answer I'm most happy with.

With TomZ's Tetrahedral Twins puzzle in mind I asked myself what the Order=2 version would look like. Keep in mind Tomz's Tetrahedral Twins puzzle I consider to be Order=3 with 3 cut planes per axis of rotation. And here is what I came up with...

Image

Notice what this is? Its a shape mod of this puzzle... (the exact same cut planes as in the image above)

Image

So you have a Master Skewb. But in the first form above you can notice that no matter how you scramble the puzzle Blue parts NEVER mix with Red parts and the Blue cuts never mix with the Red cuts. Its really two Tetrahedral puzzles in one and as I don't have a Master Skewb I didn't realize till now that you couldn't flip the edge pieces. This picture shows you why. So tie the Red cuts to the Red puzzle and the Blue cuts to the Blue puzzle and let's pull them apart. What are they?

Image

So we have 2 Pryaminxes coming together to make 1 puzzle. Just like TomZ put 2 of Okamoto's Master Pyraminxes into the same puzzle. If you'd like you can also call these Gelatinbrain 5.1.2 and they are Pryaminxes minus the trivial tips. But what ties these two puzzles together? Their cuts don't mix, their pieces don't mix, but they do share a common center. That fixes their location... but not their size. So now ask yourself how do we make this a "different" puzzle by changing the relative size of the two tetrahedra? To answer that let's take another look at the tetrahedra but now with the cut planes of both puzzles in place.

Image

Notice the cut planes belonging to one Pyraminx cross into the body of the other and vice versa. But we also know the Pyraminx has a tetrahedral core. Its edges are formed from the lines conecting the face centers of the Pyraminx. So if we shrunk down one of the two Pyraminxes such that it fit in the core of the other we should have something different.... correct? Let's try...

Image

Put your cursor/pointer over the face centers of the Red tetrahedron. Notice how the Blue tetrahedron is inscribed in the Red one? And you can now also notice that while the Red tetrahedron looks very similiar to the one above the Blue tetrahedron is now no longer cut up by the cut planes of the Red puzzle. So... what do we have here? Let's shape mod it into a cube.

Image

Look familiar? So the F-Skewb is an Order=2 puzzle formed by merging two Order=1 Pyraminxes together. Just as the Master Skewb is. The only difference is the relative size of the two tetrahedra. And its the fact that the two Pyraminxes don't share each other's pieces or cut planes that allows us to vary their relative sizes.

So the Order=1 version of the Master Skewb is the Skewb. Knowing what we know now what does the Order=1 version of the F-Skewb look like. Well let's combine an Order=0 tetrahedral puzzle and an Order=1 tetrahedral puzzle. Namely these two...

Image

And here is what we have.

Image

Or maybe you would prefer its cubic shape mod.

Image

In all the above images the camera and cut planes are fixed and never move to help give you an idea of the relative sizes of these puzzles. Nothing is bandaged in any of these puzzles and all are doctrine.

Its now 2am my time and its late and all these puzzles you have now seen before so I suspect most of you will say there is nothing "new" here. But I believe now knowing what I've shown here there are two families of corner turn cube puzzles.

Image

I think ??? is something new and I'll see if I can figure out what it looks like. I also think there may be several puzzles that fit in this new gap too.

Enjoy,
Carl

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Last edited by wwwmwww on Thu Sep 30, 2010 9:24 am, edited 1 time in total.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Thu Sep 30, 2010 8:49 am 
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Thank you for the interesting post, I didn't expect that the puzzle would deserve such a profound investigation!

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Fri Oct 01, 2010 4:37 pm 
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Nice analysis. For future readers of the forums, here's a link to Timur's thread for the F-Skewb.

wwwmwww wrote:
Its now 2am my time and its late and all these puzzles you have now seen before so I suspect most of you will say there is nothing "new" here. But I believe now knowing what I've shown here there are two families of corner turn cube puzzles.

Image

I think ??? is something new and I'll see if I can figure out what it looks like. I also think there may be several puzzles that fit in this new gap too.
(EDIT: I've since come up with a better way to describe this. The method of drawing the cuts in this post is very crude and should be taken with little weight. The relationship between cuts is way more complicated that just a 2 bit relative depth. There's still a huge number of families though, more than I show here.)

I'm not so sure families are the best way to describe these. Families will be more numerous than cuts on higher orders. I'll try to explain, but first I'll start with gingervergo's image showing his interpretation of the cuts slightly modified with annotation.
Attachment:
vergo.png
vergo.png [ 2.08 KiB | Viewed 6777 times ]
Now lets flip one onto the other so we can see them relative to each other.
Attachment:
vergo2.png
vergo2.png [ 1.58 KiB | Viewed 6777 times ]
Family I as you describe it are puzzles where the cuts are mirrored across the red line. Also, I think the cuts must be equidistant but that doesn't matter until order 4+.
Attachment:
Family1.png
Family1.png [ 3.25 KiB | Viewed 6777 times ]
I hope this notation makes sense. Now lets look at all of the Order=3 puzzles, but this time staggering the cuts like what is done on the F-Skewb. I don't any pretty POV-Ray pics of the cross-section to prove it, but I think these are all different. You can easily pick out Family 1, but which ones are Family 2-9?
Attachment:
Order3.png
Order3.png [ 3.43 KiB | Viewed 6777 times ]
Here's my proposed method to label these. Start at the black line and move outwards naming the cuts as you see them.

Key:
D = Deep Cut, can only appear once at the beginning. The cut is on Pyraminx A and B.
L = Cut is on the left, or Pyraminx A.
R = Cut is on the right, or Pyraminx B.
E = There are two cuts equidistant from the center. The cut is on Pyraminx A and B.

Example labels for the Order=3 pic above:
RRR
DRR
LRR
ER
RLR
RE
RRL
DRL
DE

Under this scheme, the F-Skewb has the signature of RL.

I have no idea how to work the spacing of the cuts into it. It may seem like spacing doesn't matter much, but I'm nearly positive that the relative position of the cuts isn't always enough to define the puzzle. For example, the Skewb and the Offset Skewb appear to have the same pieces on the outside, but the Offset skewb has a core that the Skewb doesn't. My only thought is to calculate which pieces are real and virtual and use that as part of the signature. Actually, would a list of real/virtually pieces be enough to distinguish between families too?

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Last edited by GuiltyBystander on Mon Oct 04, 2010 1:58 pm, edited 1 time in total.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sat Oct 02, 2010 11:40 am 
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GuiltyBystander wrote:
I'm not so sure families are the best way to describe these. Families will be more numerous than cuts on higher orders
I'm not sure. The more I play with it the more I think there are just 2 families. Let me first start with where I left off above... I.e. trying to push this to a new puzzle.

Here we start with the F-Skewb:
Image

But are all the pieces defined by these cuts present in this puzzle? No... there are some outside this cube. As GulityBystander points out we already know one of the virtual cores of the Skewb is made real in the Offset Skewb.
GuiltyBystander wrote:
For example, the Skewb and the Offset Skewb appear to have the same pieces on the outside, but the Offset skewb has a core that the Skewb doesn't.
So I had hopes of finding more virtual pieces of the Master Skewb in this volume outside the F-Skewb. Let's check...
Image

No such luck... these are just the normal Master Skewb pieces and I'd be tempted to call this puzzle the Offset Master Skewb.

So if I can't find much new here (though I guess the Offset Master Skewb does qualify as a new puzzle... at least I haven't seen one before) let's move on to order 3.

If we make an Order=2 corner turn cube by adding two Order=1 tetrahedral puzzles together then to get to Order=3 we can add an Order=2 tetrahedral puzzle to an Order=1 tetrahedral puzzle. I was initially thinking you could also get there by adding three Order=1 tetrahedral puzzle together but then I realize that at least 2 of those tetrahedral puzzles HAD to be in the same orientation so this thought doesn't add anything new. The two Order=1 tetrahedral puzzles with the same orientation are equivalent to one Order=2 tetrahedral puzzle.

And I should point out I'm calling these tetrahedral puzzles and not Pyraminxes for a reason.
GuiltyBystander wrote:
D = Deep Cut, can only appear once at the beginning. The cut is on Pyraminx A and B.
The tetrahedral puzzle which is deep cut is NOT a Pryaminx. Its the Halpern-Meier Tetrahedron.

So a starting point to make an Order=3 version of the F-Skewb is to start with a Pryaminx WITH the trivial tips this time. This is our Order=2 puzzle.
Image

Add that to the Order=1 Pyraminx WITHOUT the trivial tips seen here in Red with the Red cuts.
Image

Let's start to mod this into a cube.
Image

And now to complete the mod.
Image

This I would call the Order=3 version of the F-Skewb and the next puzzle in Family II. It too has pieces outside this volume and if you include those you end up with the Offset Elite Skewb but again there are no new virtual pieces that have been made real. :cry:
Image

Now to address the notion of non equidistant cuts etc.
Image

1 is an Order=3 tetrahedral puzzle. You only have turns about 4 corners of the cube, not all 8 so this is in another class all together.

2, 8, 9 as drawn are Order=4 puzzles. No, I'm wrong. I was looking at it as you put the deep cut on both of the base tetrahedral puzzles so option 8 would be two Order=2 tetrahedral puzzles added together but in this case it DOES produce an Order=3 puzzle as the deep cuts on both tetrahedral puzzles ARE the same cuts.

And I'm sure some of these options are equivalent. If we consider the yellow cut on the Blue Order=2 tetrahedral puzzle above as a corner turn notice we could have also added this same cut to the Red Order=1 tetrahedral puzzle except there it would have been a face turn. If you want to treat all turns as face turns then you need to account for negative cut depths as some cuts are on the opposite side of the origin from the face they are turning about.

So I think if you pick odd spacings you'll create other puzzles but it will be something akin to this:
Image

Here we see 4 different Order=2 puzzles but they are all the same pieces found in the Order=2 MultiSkewb. And I think what we end up with here is something that will look like this.

The Multi-Offset Skewb is equal to the MultiSkewb except one virtual core has been made real.
The Multi-Offset Master Skewb is equal to the MultiMasterSkewb except one virtual core has been made real.
The Multi-Offset Elite Skewb is equal to the MultiEliteSkewb except one virtual core has been made real.

Changing cut depths I *think* will change the shape of the pieces but I'm not sure it can be used to make any more pieces "real". I haven't proven that and I'd LOVE to be proven wrong but I'm not seeing how it can be done at the moment. And there may very well be a good chance I'm wrong.

Carl

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Last edited by wwwmwww on Sun Oct 03, 2010 8:21 am, edited 1 time in total.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sun Oct 03, 2010 2:04 am 
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wwwmwww wrote:
And I should point out I'm calling these tetrahedral puzzles and not Pyraminxes for a reason.
GuiltyBystander wrote:
D = Deep Cut, can only appear once at the beginning. The cut is on Pyraminx A and B.
The tetrahedral puzzle which is deep cut is NOT a Pryaminx. Its the Halpern-Meier Tetrahedron.
I'm sorry, I was using a Pyraminx as a generic term for a tetrahedral puzzle, I wasn't aware you made a distinction.

wwwmwww wrote:
1 is an Order=3 tetrahedral puzzle. You only have turns about 4 corners of the cube, not all 8 so this is in another class all together.
I think 1 (pure tetrahedral puzzles) are very closely related to the Skewb/F-Skewb class of puzzles and should be included in the class. If you think about the master skewb as a tetrahedra and measure the distance from the center to the cut, you'll have one positive cut and one negative cut. Pure tetrahedral puzzles have all their cuts in the positive section. On some level, families are based on the ratio of positive and negative cuts.

wwwmwww wrote:
Changing cut depths I *think* will change the shape of the pieces but I'm not sure it can be used to make any more pieces "real". I haven't proven that and I'd LOVE to be proven wrong but I'm not seeing how it can be done at the moment. And there may very well be a good chance I'm wrong.
I've realized now that my previous drawings are a very crude representation and don't show much detail. They don't really show the critical points where changes happen. Here's a better attempt to show how changing the depth makes virtual pieces real. We'll start with a simple Order=2 Tetrahedron. Here's I'm rendering just the cutting planes. For now I'd like to focus on the pieces between the red and green planes. I'll be calling these the red-green pieces.
Attachment:
Order2.png
Order2.png [ 12.85 KiB | Viewed 6678 times ]
Now lets add a very large Order=1 Tetrahedron which is inverted (btw, what are we calling the relationship between these two opposite tetrahedrons?). I'm just rendering 1 cutting plane (blue) for now so you can see the rest of the puzzle better.
Attachment:
Order3-1.png
Order3-1.png [ 14.46 KiB | Viewed 6678 times ]
All of the red-green pieces remain intact and how they move/twist remains the same. Let's take that blue tetrahedron and shrink it a bit so that the cut is a little deeper.
Attachment:
Order3-2.png
Order3-2.png [ 13.83 KiB | Viewed 6678 times ]
You can now see the red-green corner poking through the blue cutting plane. This piece is now cut in two so that each piece has different twistability properties. This new piece used to be a virtual piece but like Pinoccio, he's now a real piece. I think we can say that this is a new Family right? Let's keep shrinking that blue tetrahedron.
Attachment:
Order3-3.png
Order3-3.png [ 13.79 KiB | Viewed 6678 times ]
Again we've created a new piece by cutting up the red-green edge. This is a new Family again right? We aren't just changing the shape of pieces. Let's keep going.
Attachment:
Order3-4.png
Order3-4.png [ 12.93 KiB | Viewed 6678 times ]
Now the blue tetrahedron is nearly a Halpern-Meier Tetrahedron/skewb. Yet again we've create a new family but for a different reason. There is only 1 red-green corner again, but it's not the one we started with. It's the new piece we created when we first shrank the blue tetrahedron. The old red-green corner piece has disappeared. What was a real piece is now virtual.



I think this same thing can be done with 2 opposite Order=1 but it would be far less interesting and harder to show.

If you think about the complex F-skewb type puzzles for higher orders, families are just puzzles with different pieces being real or virtual.

Now the only thing I'm worried about here is that pieces are possibly popping out of existence in one place and reappearing somewhere else. I can show you what I mean by repeating the process but this time with 2 opposite Order=1 Triangle Prism puzzles. These would look like Order=2 Hexagonal Prisms just like 2 Order=1 Tetrahedra looks like 1 Order=2 Octahedron.
Attachment:
Triangle.png
Triangle.png [ 21.37 KiB | Viewed 6678 times ]
As you shrink the larger triangle, piece 2 disappears but reappears somewhere else due to the symmetries of the triangles. I want to say that making one of the triangular prisms so it's Order=2 would make the reappearing piece different, but I'm not positive.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sun Oct 03, 2010 8:56 am 
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I just realized something. The Order=1 F-Skewb isn't the Offset Skewb.

These all belong together. I call this the Offset Family.
Image

The F-Family is this.
Image

Not sure what to call the trivial Order=1 member of this family or if its been made before but there it is.

GuiltyBystander wrote:
I'm sorry, I was using a Pyraminx as a generic term for a tetrahedral puzzle, I wasn't aware you made a distinction.
I'm ok with it if you spell it out like that but to me the Pyraminx is a specific tetrahedral puzzle, as is the Halpern-Meier Tetrahedron. Just wanting to avoid confusion.
GuiltyBystander wrote:
I think 1 (pure tetrahedral puzzles) are very closely related to the Skewb/F-Skewb class of puzzles and should be included in the class. If you think about the master skewb as a tetrahedra and measure the distance from the center to the cut, you'll have one positive cut and one negative cut. Pure tetrahedral puzzles have all their cuts in the positive section. On some level, families are based on the ratio of positive and negative cuts.
I agree. All of these are tetrahedral puzzle if you allow positive and negative cuts which I obviously have with the Pyraminx with trivial tips above. You are correct.

Let me read the rest of your post a few more times before I comment...
Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sun Oct 03, 2010 9:28 am 
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wwwmwww wrote:
These all belong together. I call this the Offset Family.
{image1}
The F-Family is this.
{image2}
Quick question: Are we looking at these puzzles as multicubes? If so, aren't those the same?

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sun Oct 03, 2010 9:50 am 
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GuiltyBystander wrote:
Here's I'm rendering just the cutting planes. For now I'd like to focus on the pieces between the red and green planes. I'll be calling these the red-green pieces. Now lets add a very large Order=1 Tetrahedron which is inverted (btw, what are we calling the relationship between these two opposite tetrahedrons?).
It's a mirror relationship. When they are the same size their verticies form a cube. But I already see a problem here. Your red-green pieces are Order=2 pieces. Your red-green-blue pieces are Order=3 pieces, not virtual Order=2 pieces.

Now having said that can the red-green pieces also be Order=3 pieces? YES!!!! The blue cut is still there its just outside this piece. Ok... let's go back to this:
Image

Here I show all the "real" pieces inside an Order=2 MultiMasterSkewb... but have I shown all the real pieces inside an Order=2 Multi Corner-Turning Cube? Probably (almost certainly) NOT! Here I fixed the cut planes and changed the size of the cube. But we now know the two Tetrahedral base puzzles don't need the cuts at the same depth. So to see ALL the pieces I need to vary two parameters, the size of the puzzle and the cut depth of one of the tetrahedral puzzles. And that depth that I vary should include both positive and negative values. And I agree, if I do that I should be able to make some of the virtual pieces of the MasterSkewb real. So let me think about how to do that...

What you are talking about is Order=3 and the complexity of showing/counting all those pieces gets exponentially more complicated. I think the point you are trying to make can be made at Order=2. And even there an easy way to show it isn't jumping out at me.

Actually I guess it can be made at Order=1 too. And there its easy. Displace the deep cut planes from the origin and one of the Skewb's virtual cores becomes real.

GuiltyBystander wrote:
Now the only thing I'm worried about here is that pieces are possibly popping out of existence in one place and reappearing somewhere else. I can show you what I mean by repeating the process but this time with 2 opposite Order=1 Triangle Prism puzzles. These would look like Order=2 Hexagonal Prisms just like 2 Order=1 Tetrahedra looks like 1 Order=2 Octahedron. As you shrink the larger triangle, piece 2 disappears but reappears somewhere else due to the symmetries of the triangles. I want to say that making one of the triangular prisms so it's Order=2 would make the reappearing piece different, but I'm not positive.

Image
Look at your second and fourth figures. You are also saying piece 1 is the same as piece 0. That's not correct and for the same reason I think your two piece 2's are already different pieces. They are pieces which are the mirror image of each other. I think its akin to the two virtual cores of the Skewb. Both can be made real... just not at the same time and they are also mirrors of each other.

Carl

P.S. Error on my part... "You are also saying piece 1 is the same as piece 0." That is NOT implied in the above figure. I see my reading error now. However I still think your two piece 2's are different parts already. Though I'm now less sure. Notice only in your third figure is rotation by 60 degrees allowed. All the other figures MUST be rotated by 120 degrees and red cuts and blue cuts never mix. So if you include the third figure in this set is it somehow implied the rotations must be multiples of 120 degrees? And if not the distinction between blue cuts and red cuts can get lost... that never happens on these tetrahedral puzzles.

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Last edited by wwwmwww on Sun Oct 03, 2010 11:04 pm, edited 4 times in total.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sun Oct 03, 2010 9:54 am 
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GuiltyBystander wrote:
Quick question: Are we looking at these puzzles as multicubes? If so, aren't those the same?
Yes, but you don't have to look at them as Multicubes.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 1:54 am 
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wwwmwww wrote:
What you are talking about is Order=3 and the complexity of showing/counting all those pieces gets exponentially more complicated. I think the point you are trying to make can be made at Order=2. And even there an easy way to show it isn't jumping out at me.
Yep. I have a plan to show all the Order=2 Tetrahedron puzzles and I'm really psyched about sharing it. But it's late, so pics will come in the morning. I'll also have a table of real/virtual pieces too.

wwwmwww wrote:
..Both can be made real... just not at the same time and they are also mirrors of each other....
P.S. Error on my part... "You are also saying piece 1 is the same as piece 0." That is NOT implied in the above figure. I see my reading error now. However I still think your two piece 2's are different parts already. Though I'm now less sure. Notice only in your third figure is rotation by 60 degrees allowed. All the other figures MUST be rotated by 120 degrees and red cuts and blue cuts never mix. So if you include the third figure in this set is it somehow implied the rotations must be multiples of 120 degrees? And if not the distinction between blue cuts and red cuts can get lost... that never happens on these tetrahedral puzzles.
I want to say that the two "2"s are the same kind of piece. I'm alright with calling them different pieces because it gets confusing on the Order=3 version, but it would mean we'd have to break the "1"s up into two groups as well. You'd also have to relabel the pieces in your famous MultiMasterSkewb animation.
Attachment:
MasterSkewb.png
MasterSkewb.png [ 23.22 KiB | Viewed 6576 times ]
There are a lot of piece on the Complex MultiMasterSkewb that are very similar to these "2" pieces. Do you know which ones I'm talking about? If not, tomorrow's pics should help.

I know Andreas Nortmann counted all the pieces and permutations of the Complex3x3x3. Any idea if you or he has done the same for the Complex MultiMasterSkewb?

And you're right, the two sets of "2" pieces cannot both be real at the same time.

As for the 60/120 turns, I was thinking of these as just floppies so that they are the 2D analog of the 3D Tetrahedron. I shouldn't have used the word prism I guess.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 8:02 am 
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I still think it is just a skewb with the added dino cuts. The shape of the pieces do not affect how it is solved, ie it functions exactly the same as a normal skewb plus partial dino cuts.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 10:08 am 
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PuzzleMaster6262 wrote:
I still think it is just a skewb with the added dino cuts. The shape of the pieces do not affect how it is solved, ie it functions exactly the same as a normal skewb plus partial dino cuts.



I think its pretty clear that that's not what the puzzle is. For more clarification than what these two have provided (which is more than ample guys, reading all of this is a real treat) check out Tony Fisher's cubominx

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 Post subject: Re: What Is Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 1:02 pm 
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Argh!
Why do these interesting theory-heavy threads always come to live when I am a few days absent from my computer? With the following post I try to catch up with the expalantions already given. Please be patient with me.
wwwmwww wrote:
Here I show all the "real" pieces inside an Order=2 MultiMasterSkewb... but have I shown all the real pieces inside an Order=2 Multi Corner-Turning Cube?
I thought I had an agreement with Carl about definitions of "real", "virtual" and "imaginary" pieces.
"Real": All pieces with can be visible in puzzles made with pairs of equidistant cuts.
"Virtual": All pieces present in a twistability analysis which are not real.
"imaginary": All pieces present in Matt Gallas method of analyzing which are not real and not virtual.
To understand "twistability analysis" see this thread if you are a masochist: viewtopic.php?f=1&t=15667
Carls animation therefore contains all real pieces of HC2. Not more not less.
GuiltyBystander wrote:
I know Andreas Nortmann counted all the pieces and permutations of the Complex3x3x3. Any idea if you or he has done the same for the Complex MultiMasterSkewb?
No for the permutations. But I already mentioned (hidden in a similar long thread) that there are 22 complex pieces for HC2. 6 of them are real. 3 are virtual.

All 3 virtual pieces have now be shown in physical volume:
  • The already mentioned virtualCore. This is visible in Gelatibrain 5.1.1 as well, since it is a pieces which is present in order=1-puzzles as well. Please note that only one (of 2 theoretical ones) have be shown yet.
  • The 6 (out of 12 theoretical) edges of the MasterPyraminx.
  • The 4 (out of 8 theoretical) "Pinoccio"-pieces of GuiltyBystander.

Edit: Refined the definitions of Real/Virtual/Imaginary; corrected spelling


Last edited by Andreas Nortmann on Mon Oct 04, 2010 11:50 pm, edited 1 time in total.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 1:08 pm 
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I'm sorry, but I have to agree with PuzzleMaster6262. If you remove the Dino cuts, then you're left with an Offset Skewb, which is solved like a normal Skewb. That's all I can see it as.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 1:51 pm 
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(edit: I was making this post while Andreas posted so he answers a few things I was rambling about. I'll read it more depth later and reply if needed in a separate post. For now I'll just say I'm glad you made it to this thread and I'm curious of your interpretations.)

Alright, as promised, here's all of the Order=2 Tetrahedral puzzles where the two cuts are not on the same side of the origin. I'm excluding the Order=2 Pyraminx because the piece analysis is rather boring and the it gets confusing naming the pieces because we're basically making duals of tetrahedrons which turns faces into corners and corners into faces.

One thread that might really help with discussions is The Order=3 Corner-Turn MultiCube (Plus One). I know it's for Order=3 Tetrahedra, but there's still some good discussion on the Order=2 variety. This thread talks a lot about real/virtual/imaginary pieces. I'm not completely sure what the difference is between virtual/imaginary so for my own sanity I will use just real/virtual to signify that the piece has actual volume or not.

For people confused about the idea of the multicube, please take a brief look at wwwmwww's Multidodecahedron thread. It helps make it clear why we care about more than the pieces on the surface of the F-Skewb.

First lets look at the Order=1 Tetrahedrons. There are 5 pieces. There's the Core=0, Face=1, Edge=2, Corner=3, and Inverted Core=4. The "=" notion is showing how many axes can be used to move/rotate that piece type. A normal Pyraminx only has pieces 0-3. A Halpern-Meier Tetrahedron or Skewb is deep cut so the core has 0 volume so it only has pieces 1-3. Piece 4 is nearly always virtual unless you do special bandaging tricks.

The Order=2 pieces will just be the intersection of a piece on one tetrahedron with a piece on the other tetrahedron. For example a the intersection of a Blue Core and a Red Corner is a Order=2 piece. Andreas Nortmann and label which Master Skewb pieces are real/imaginary/virtual here and it helps see things here a bit. At that time we weren't thinking of the Master Skewb as 2 tetrahedra so he labels pieces based on the corners of a cube instead of what the two tetrahedral intersected pieces were. I haven't tried to link my system with his yet but it would be nice to do at some point.

Here's the picture showing all 9 possible Order=2 Tetrahedrons. I am rendering just the cores of two tetrahedra. The relationship between two tetrahedra is all that is need to define the puzzle. You can see that 6-9 are mirrored versions of 4-1. Whether theses are different or not goes back to the discussion about piece 2 in the triangular floppy.
Attachment:
Order=2.png
Order=2.png [ 46.31 KiB | Viewed 6495 times ]
1. Red is Deep-Cut
1->2: Core-Core becomes real
2. Red is completely inside Blue
2->3: Corner-Core becomes virtual
3. Red is inscribed by Blue. This is the F-Skewb.
3->4: Core-Face becomes real
4. Red is {can't think of the relationship name} Blue
4->5: Edge-Core becomes virtual
5. Red is {can't think of the relationship name} Blue. This is the Master Skewb.
5->6: Core-Edge becomes real
6. Red is {can't think of the relationship name} Blue
6->7: Face-Core becomes virtual
7. Red circumscribes Blue. This is the F-Skewb.
7->8: Core-Corner becomes real
8. Red completely contains Blue
8->9: Core-Core becomes virtual
9. Blue is deep-cut.

Here's a simple table I made to see some of the properties of the pieces.
Attachment:
pieces.png
pieces.png [ 11.26 KiB | Viewed 6495 times ]
The first table shows how many axes will move/rotate the piece.
The second table shows when the piece first becomes real in my 1-9 line up of puzzles.
The third table shows the last puzzle in my 1-9 line up where the piece is real.
The fourth table shows the ranges of the puzzles it's in. The coloring indicates if it's in the F-Skewb, Master Skewb or both. You can see that the Edge-Core is real on the F-Skewb but virtual on the Master Skewb. Also, the Core-Face is virtual on the F-Skewb but real on the Master Skewb. I think this shows definitively that the Multi F-Skewb is not the same as Multi Master Skewb.



I haven't actually counted the number of Order=3 puzzles yet, but I am currently estimating that there are 30-40 Order=3 tetrahedral puzzles.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 3:20 pm 
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GuiltyBystander wrote:
The Order=2 pieces will just be the intersection of a piece on one tetrahedron with a piece on the other tetrahedron. For example a the intersection of a Blue Core and a Red Corner is a Order=2 piece. Andreas Nortmann and label which Master Skewb pieces are real/imaginary/virtual here and it helps see things here a bit. At that time we weren't thinking of the Master Skewb as 2 tetrahedra so he labels pieces based on the corners of a cube instead of what the two tetrahedral intersected pieces were. I haven't tried to link my system with his yet but it would be nice to do at some point.
I'm looking at Andreas post now to try to link his notation to mine and I've realized I've made a critical error. The problem is that saying Edge-Edge isn't enough to define a piece. There are several ways two edge pieces can intersect. There are in fact 3 Edge-Edge type pieces. I need to walk though all notation and figure out which one of his I'm using. Sorry for any mixup. I'm kind of mad at myself for not thinking of this.

Using this image as a key
Attachment:
labels.png
labels.png [ 6.69 KiB | Viewed 6474 times ]
Here's Andreas's notation converted to my lame notation.
Attachment:
andreas-guilty.png
andreas-guilty.png [ 11.08 KiB | Viewed 6474 times ]
And here's the counts to show the duplicates.
Attachment:
andreas-guilty-counts.png
andreas-guilty-counts.png [ 2.37 KiB | Viewed 6474 times ]
The 0's are nothing to worry about. I'm pretty sure it means he would say that the two "2" in the triangular floppy are the same kind of piece.

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 Post subject: Re: What Is Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 3:35 pm 
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Andreas Nortmann wrote:
wwwmwww wrote:
Here I show all the "real" pieces inside an Order=2 MultiMasterSkewb... but have I shown all the real pieces inside an Order=2 Multi Corner-Turning Cube?
I thought I had an agreement with Carl about definitions of "real", "virtual" and "imaginary" pieces.
"Real": All pieces with can be visible in puzzles made with pairs of equidistant cuts.
"Virtual": All pieces present in a twistability analysis which are not real.
"imaginary": All pieces present in Matt Gallas method of anylzing which are not real and not virtual.
Carls animation therefore contains all real pieces of HC2. Not more not less.

Sloppy sloppy language on my part. I guess it stems from two possible uses of the word "real". There are "real" pieces that fit the definition here. And I've also used the expression when talking about "virtual" piece that can be made "real" and by that I mean to have a positive/physical volume.

I need to slow down a bit and digest this info before I get ahead of myself again.

Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Mon Oct 04, 2010 3:48 pm 
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gingervergo wrote:
PuzzleMaster6262 wrote:
I still think it is just a skewb with the added dino cuts. The shape of the pieces do not affect how it is solved, ie it functions exactly the same as a normal skewb plus partial dino cuts.
I think its pretty clear that that's not what the puzzle is. For more clarification than what these two have provided (which is more than ample guys, reading all of this is a real treat) check out Tony Fisher's cubominx
You can't go that far. PuzzleMaster6262's view is perfectly correct. The only reason I didn't like it was because it alone doesn't answer why you can get away with just adding 4 of the 8 cuts present on the dino cube. And I also don't like calling them dino cuts. The Offset Master Skewb has the "same" cuts now present in a larger cube and you wouldn't call them dino cuts there? That term ties the cuts to the shape the puzzle is in. And I like to look at a puzzle as being defined by its cuts... not the other way around. Its more of a personal preference thing.

Now those that call this a bandaged this or that I'd argue with... it's NOT a bandaged puzzle.

Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Tue Oct 05, 2010 10:19 am 
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To came back to the original topic I want to give my final answer:
The F-Skewb is:
HC2 [E, X/2, C4/2][C4/2]
Please compare it with the MasterSkewb
HC2 [E, F, X, C4][F, C4]
see this posting for the notation:
viewtopic.php?p=190396#p190396

There has been some discussion about a "Multi F-Skewb" and its differences to a "Multi Master Skewb". As I understood this thread (viewtopic.php?f=1&t=14875) a MultiXYZ is a puzzle where all REAL pieces (back then virtual pieces weren't well defined) of the class XYZ are present and have to be included into the solution. So there is a MultiCornerturningHexahedronWith2Cuts (or MultiHC2) and it is a superset of the MasterSkewb which is a superset of the F-Skewb.


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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Tue Oct 05, 2010 10:50 am 
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GuiltyBystander wrote:
I want to say that the two "2"s are the same kind of piece. I'm alright with calling them different pieces because it gets confusing on the Order=3 version, but it would mean we'd have to break the "1"s up into two groups as well. You'd also have to relabel the pieces in your famous MultiMasterSkewb animation.
I'm ok with calling them the same kind of piece but they are different pieces.
GuiltyBystander wrote:
Do you know which ones I'm talking about? If not, tomorrow's pics should help.
Still need to study the pics below (your post... above this one). This thread is growing faster then I have time to keep up with at the moment.
GuiltyBystander wrote:
I know Andreas Nortmann counted all the pieces and permutations of the Complex3x3x3. Any idea if you or he has done the same for the Complex MultiMasterSkewb?
No... and I hadn't known Andreas had done that for the Complex3x3x3 till I saw your post. So THANKS for the link!!!!
GuiltyBystander wrote:
And you're right, the two sets of "2" pieces cannot both be real at the same time.
Well not in one physical puzzle with one set up cut planes that divide up all space. With finite cut planes, i.e. two puzzles and some geary like connections between those two puzzles it CAN be done in principle. I'll try to post a Skewb puzzle animation I made that shows BOTH virtual cores being played with in the same puzzle later night if I can find it. It's unfinished and its something that fell through the cracks when I started working again but I think it's enough to make this point.
GuiltyBystander wrote:
As for the 60/120 turns, I was thinking of these as just floppies so that they are the 2D analog of the 3D Tetrahedron. I shouldn't have used the word prism I guess.
Ok... I was extending them into prisms and adding another cut parallel to the plane of your floppies. Not use to seeing 2D twisty puzzles with the axis of rotation in the same plane as the pieces. I guess the puck puzzles can be looked at in that light.

Carl

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 Post subject: Re: What Is Shim's F-Skewb?
PostPosted: Tue Oct 05, 2010 11:54 am 
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Andreas Nortmann wrote:
GuiltyBystander wrote:
I know Andreas Nortmann counted all the pieces and permutations of the Complex3x3x3. Any idea if you or he has done the same for the Complex MultiMasterSkewb?
No for the permutations.
I took a peek at GAP. It doesn't look like it would be that hard to do. Maybe if I manage to get some free time down the road I'll take a shot. It will be several months as I have a few other projects going first so don't let me stop anyone from jumping in.
Andreas Nortmann wrote:
But I already mentioned (hidden in a similar long thread) that there are 22 complex pieces for HC2. 6 of them are real. 3 are virtual.
Link... please. I need a refresher myself as my brain is having to do over time to keep up with this thread. If you can't find it I'll look for it tonight as I have a few ideas where it might be.
Andreas Nortmann wrote:
All 3 virtual pieces have now be shown in physical volume:
They have? WOW!!! I really need to get up to speed.
Andreas Nortmann wrote:
  • The already mentioned virtualCore. This is visible in Gelatibrain 5.1.1 as well, since it is a pieces which is present in order=1-puzzles as well. Please note that only one (of 2 theoretical ones) have be shown yet.
  • The 6 (out of 12 theoretical) edges of the MasterPyraminx.
  • The 4 (out of 8 theoretical) "Pinoccio"-pieces of GuiltyBystander.
I can show BOTH virtual cores of the skewb... I will later tonight I hope. I suspect showing them both on a Master Skewb would be similiar... if not the identical solution.
I'm not up to speed yet on the other two and has the term "Pinoccio"-pieces come up before this thread? I'm thinking of that as any virtual piece that can be given real volume. Need to read more of GuiltyBystander's posts again.

If there are only 3 virtual pieces I should have able to show them all with a reasonable set of animations.

Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Tue Oct 05, 2010 12:18 pm 
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GuiltyBystander wrote:
I'm not completely sure what the difference is between virtual/imaginary so for my own sanity I will use just real/virtual to signify that the piece has actual volume or not.
Real and Virtual pieces can serve as holding points. They will never move under one "complete" set of linearly independent moves. This isn't true for imaginary pieces.
GuiltyBystander wrote:
First lets look at the Order=1 Tetrahedrons. There are 5 pieces. There's the Core=0, Face=1, Edge=2, Corner=3, and Inverted Core=4. The "=" notion is showing how many axes can be used to move/rotate that piece type. A normal Pyraminx only has pieces 0-3. A Halpern-Meier Tetrahedron or Skewb is deep cut so the core has 0 volume so it only has pieces 1-3. Piece 4 is nearly always virtual unless you do special bandaging tricks.
Actually no special bandaging tricks are needed here as your Core=4 is the second virtual core of the Skewb or Order=1 deep-cut Tetrahedron. Your Core=1 is the first virtual core. As I recall the Skewb doesn't have any imaginary pieces... and now that I think about it I'm not sure its possible for an Order=1 puzzle to have imaginary pieces. Andreas, is that correct?

Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Tue Oct 05, 2010 12:55 pm 
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Andreas Nortmann wrote:
There has been some discussion about a "Multi F-Skewb" and its differences to a "Multi Master Skewb". As I understood this thread (viewtopic.php?f=1&t=14875) a MultiXYZ is a puzzle where all REAL pieces (back then virtual pieces weren't well defined) of the class XYZ are present and have to be included into the solution. So there is a MultiCornerturningHexahedronWith2Cuts (or MultiHC2) and it is a superset of the MasterSkewb which is a superset of the F-Skewb.
"A is superset of B" means that A has all of the pieces of B.
I would say that the MultiHC2 is the generic term for the class of puzzle that contains the "Multi Master Skewb" and the "Multi F-Skewb." The "Mulit Master Skewb" is a superset of the Master Skewb. The "Mulit F-Skewb" is a superset of the F-Skewb. In my line up, puzzle 4 is a superset of 3 (Multi F-Skewb) and 5 (Multi Master Skewb)
Going one step higher, the ComplexHC2 has all real/imaginary/virtual pieces and is the superset that contains all MultiHC2 puzzles.


wwwmwww wrote:
Andreas Nortmann wrote:
But I already mentioned (hidden in a similar long thread) that there are 22 complex pieces for HC2. 6 of them are real. 3 are virtual.
Link... please. I need a refresher myself as my brain is having to do over time to keep up with this thread. If you can't find it I'll look for it tonight as I have a few ideas where it might be.
Link. I'm working on labeling which ones I was looking at now.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Tue Oct 05, 2010 2:03 pm 
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GuiltyBystander wrote:
Link. I'm working on labeling which ones I was looking at now.
And done. I think I found a error/typo in your list Andreas.
Andreas Nortmann wrote:
06[ULF ULB URF] real - X9
07[ULF ULB DLF] virtual
I'm pretty sure these are the same piece. I think the second one should be [ULF ULB DRB]. Please confirm. (Edit: confirmation received and new scheme appears in the image below)

Here's an updated table with Andreas's labels. "Never Real" means that the piece is never real because it contains 2 corners that are on opposite sides of the cube.
Now a lot of these pieces come in 2 sets. The "2" piece on the Multi-Triangle Floppy is a good example of this. Red means that one of the sets is real. Green means that the other set is real. Yellow means that both sets are real.
Attachment:
andreas-guilty-2.png
andreas-guilty-2.png [ 15.18 KiB | Viewed 6269 times ]
Here's the differences between the Master and F-Skewb.
  • One set of piece 4 is real on F-Skewb but not on the Master Skewb. Red from one row may or may not correspond to red from another row. This is kind of weird because it's labeled as virtual.
  • The Master Skewb has both sets of piece 2 while the F-Skewb only has one set.

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Last edited by GuiltyBystander on Wed Oct 06, 2010 1:33 pm, edited 2 times in total.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Tue Oct 05, 2010 7:51 pm 
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I need to know the maths behind all this stuff, cuz these topics always seem to really boil down to that.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Tue Oct 05, 2010 11:10 pm 
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wwwmwww wrote:
I'll try to post a Skewb puzzle animation I made that shows BOTH virtual cores being played with in the same puzzle later night if I can find it. It's unfinished and its something that fell through the cracks when I started working again but I think it's enough to make this point.
Ok here it is:
Image
If this were a physical puzzle then the central Skewb Diamond would be a controller of sorts. It would be wirelessly connected to motors in the two tetrahedral puzzles on either side such that they would turn in sync in the fashion shown with the Skewb Diamond being manipulated by the solver. So these 3 puzzles become one puzzle that must be solved.

The top 3 puzzles in this view are the front view of the puzzles. The bottom 3 puzzles are the same puzzles above but just now viewed from the rear. In principle it should be possible to make this type of puzzle as an app too and places like Gelatinbrain could let us solve them.

And as far as I know this is the first and only implementation of the Skewb that gives BOTH virtual pieces physical volume in the same puzzle. The piece GuiltyBystander calls Core=0 is in the right tetrahedron. And the piece Core=4 is in the left tetrahedron.

And if I ever find the time to finish this animation (5 more moves are needed) it would show you that you can solve the Skewb Diamond without solving the neighboring tetrahedrons at the same time. Despite that there really isn't a new solving experience here... but I believe that isn't true in general. I believe a similiar puzzle could be made that allowed you to solve all the real and virtual pieces of the Master Skewb. I'd call it the Augmented Master Skewb. Real+Virtual=Augmented. If you could then throw in the imaginary pieces it would be the Complex Master Skewb. Real+Imaginary=Complex. And it should be noted the Virtual pieces are a subset of the Imaginary pieces so there is no need to talk about Real+Virtual+Imaginary. It would still just be the Complex version of the puzzle.

Now I just need to wrap my mind around these other 2 virtal pieces in the Master Skewb and see if I can make an animation that would show what the Augmented Master Skewb would look like to solve and I'm fairly certain it would offer a new solving experience.

Note the Augmented Skewb = the Complex Skewb as I don't believe it has imaginary pieces. I don't think any Order=1 puzzle does.

Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Wed Oct 06, 2010 12:28 am 
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I've thought about trying to do that with the 3x3x3 but I think I lack the self control to maintain the state on 2 cubes and the solving know how to complete it. Perhaps I should start with a highly bandage version that only allows U & R moves.

It's funny you mention the wireless controller idea. I've been thinking about what it would take to cram some electronics into one of those jumbo 10cm 3x3x3 and use it as a USB controller to play with virtual cubes.

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 Post subject: Re: What Is Shim's F-Skewb?
PostPosted: Wed Oct 06, 2010 12:15 pm 
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wwwmwww wrote:
I'm not up to speed yet on the other two and has the term "Pinoccio"-pieces come up before this thread?
Use CTRL+F to find it in his thread.
wwwmwww wrote:
I'm not sure its possible for an Order=1 puzzle to have imaginary pieces. Andreas, is that correct?
I can't answer that. I still do not know how to use Matts system on order=1 puzzles. We discussed it but the results don't satisfy me.
GuiltyBystander wrote:
I would say that the MultiHC2 is the generic term for the class of puzzle that contains the "Multi Master Skewb" and the "Multi F-Skewb." The "Mulit Master Skewb" is a superset of the Master Skewb. The "Mulit F-Skewb" is a superset of the F-Skewb. In my line up, puzzle 4 is a superset of 3 (Multi F-Skewb) and 5 (Multi Master Skewb)
Going one step higher, the ComplexHC2 has all real/imaginary/virtual pieces and is the superset that contains all MultiHC2 puzzles.
So far we agree. There are many supersets to the MasterSkewb. Which one is the MultiMasterSkewb? The smallest which makes sense is the one with ALL real pieces. If we start with the MasterSkewb or with the F-Skewb doesn't change this. Hence my earlier post.
GuiltyBystander wrote:
I think I found a error/typo in your list Andreas.
Yup. There is a mistake but your suggestion is wrong too. I corrected the table like this:
07[ULF ULB DLF] virtual becomes 07[URF ULB DLF] virtual
08[ULF URB DLB] imaginary becomes 08[ULF URB DLF] imaginary
Two derived mistakes:
15[ULB URF DLF DRF DRB] imaginary - inversion of piece 08
becomes
15[ULB URF DLB DRF DRB] imaginary - inversion of piece 08
16[URF URB DLB DRF DRB] imaginary - inversion of piece 07
becomes
16[ULF URB DLB DRF DRB] imaginary - inversion of piece 07

These errors explain why there is a virtual piece in your table that you have classified as "never real" and an imaginary piece which seems to be "sometimes real".

@Carl: Cool animation. If I can make a suggestion. First proove your "Diamond solved, but tetrahedrons"-point before goning further to the MasterSkewb.

@elijah: I admire you for still reading all this.


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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Wed Oct 06, 2010 5:15 pm 
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that's fine, but I really wish the difference between virtual and imaginary pieces could be magnified.
It seems both types can be given physical volume, except don't exist on the normal versions of the puzzles...
Can anyone sum this difference up in simple terms?

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 Post subject: Re: What Is Shim's F-Skewb?
PostPosted: Wed Oct 06, 2010 6:27 pm 
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Andreas Nortmann wrote:
GuiltyBystander wrote:
I would say that the MultiHC2 is the generic term for the class of puzzle that contains the "Multi Master Skewb" and the "Multi F-Skewb." The "Mulit Master Skewb" is a superset of the Master Skewb. The "Mulit F-Skewb" is a superset of the F-Skewb. In my line up, puzzle 4 is a superset of 3 (Multi F-Skewb) and 5 (Multi Master Skewb)
Going one step higher, the ComplexHC2 has all real/imaginary/virtual pieces and is the superset that contains all MultiHC2 puzzles.
So far we agree. There are many supersets to the MasterSkewb. Which one is the MultiMasterSkewb? The smallest which makes sense is the one with ALL real pieces. If we start with the MasterSkewb or with the F-Skewb doesn't change this. Hence my earlier post.
Yes, there are many supersets to the MasterSkewb. I would say that the MultiMasterSkewb is the puzzle that clearly can represent all of the pieces in wwwmwww's image below. These should be all the ones you labeled as real. The MultiMasterSkewb should have no more or no less pieces than these. When I say pieces, I mean the ones that identify if the puzzle is solved or not, not the ones that are there for structural support or the mech. I'm not completely sure what you mean by "start with." I hope I'm clarifying something and not just repeating myself.Image

Andreas Nortmann wrote:
GuiltyBystander wrote:
I think I found a error/typo in your list Andreas.
Yup. There is a mistake but your suggestion is wrong too. I corrected the table like this:...
It looks like you took my suggestion, but moved it to a different set of corners with the same relation and then swapped the rows. I guess your ordering has meaning that I'm missing. In any case, I've updated my image.

Andreas Nortmann wrote:
@Carl: Cool animation. If I can make a suggestion. First proove your "Diamond solved, but tetrahedrons"-point before goning further to the MasterSkewb.
Didn't you prove that the imaginary/virtual pieces add something to permutations on the 3x3x3 with this thread? Is there any particular reason that you think they wouldn't do the same on tetrahedral based puzzles? I guess with the augmented order=1 tetrahedron it's just an additional orientation not permutation, but still.

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 Post subject: Re: What Is Shim's F-Skewb?
PostPosted: Wed Oct 06, 2010 9:11 pm 
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GuiltyBystander wrote:
Andreas Nortmann wrote:
So far we agree. There are many supersets to the MasterSkewb. Which one is the MultiMasterSkewb? The smallest which makes sense is the one with ALL real pieces. If we start with the MasterSkewb or with the F-Skewb doesn't change this. Hence my earlier post.
Yes, there are many supersets to the MasterSkewb. I would say that the MultiMasterSkewb is the puzzle that clearly can represent all of the pieces in wwwmwww's image below. These should be all the ones you labeled as real. The MultiMasterSkewb should have no more or no less pieces than these. When I say pieces, I mean the ones that identify if the puzzle is solved or not, not the ones that are there for structural support or the mech. I'm not completely sure what you mean by "start with." I hope I'm clarifying something and not just repeating myself.
Well since I started this use of Multi-X I guess I better offer some definitions.

Multi - X = is the puzzle that contains all the pieces created by the volumes of space that the cut surfaces (usually planes but not always) of puzzle - X creates.

Super Multi - X = Has all the pieces of Multi - X plus each piece has a fixed orientation and position in the solved state.

Augmented - X = Has all the pieces of Multi - X plus all the virtual pieces of puzzle - X.

Super Augemented - X = Has all the piece of Augmented - X plus each piece has a fixed orientation and position in the solved state.

Complex - X = Has all the pieces of Multi - X plus all the imaginary pieces of puzzle - X. Virtual pieces are a subset of the imaginary pieces so they are included here.

Super Complex - X = Has all the piece of Complex - X plus each piece has a fixed orientation and position in the solved state.

The Multi Skewb is NOT equal to the Multi Offset Skewb as the Offset Skewb has a "real" core that the Skewb doesn't have. But this piece is one of the virtual pieces of the Skewb so the Augmented and Complex versions of the Skewb and Offset Skewb would be the same.

It's similiar to the Multi Skewb Diamond is NOT the Multi Skewb but the Super Multi Skewb Diamond and the Super Multi Skewb are the same puzzle.
GuiltyBystander wrote:
Andreas Nortmann wrote:
GuiltyBystander wrote:
I think I found a error/typo in your list Andreas.
Yup. There is a mistake but your suggestion is wrong too. I corrected the table like this:...
It looks like you took my suggestion, but moved it to a different set of corners with the same relation and then swapped the rows. I guess your ordering has meaning that I'm missing. In any case, I've updated my image.
It takes me time for this stuff to sink in and you both are ahead of me here so far.
Andreas Nortmann wrote:
@Carl: Cool animation. If I can make a suggestion. First proove your "Diamond solved, but tetrahedrons"-point before goning further to the MasterSkewb.
You think I would have said what I did if I hadn't? The way I have this coded in POV-Ray I'm sure is far from the best and for reasons that aren't obvious to me the frames on this puzzle animation get slower and slower with each move. Let's say the 10 frames that make up the first move render in 10min. The next 10 frames of move 2 render in over 20mins. And by the time I'm to move 5 the frames are taking a day to render. The final frame at the end of the 10th move would take a month to render at the settings I'm using for the others. I've just started it again at greatly reduced settings and it still has yet to color a single pixel.
Andreas Nortmann wrote:
@Didn't you prove that the imaginary/virtual pieces add something to permutations on the 3x3x3 with this thread? Is there any particular reason that you think they wouldn't do the same on tetrahedral based puzzles? I guess with the augmented order=1 tetrahedron it's just an additional orientation not permutation, but still.
You are correct... the tetrahedral puzzles turn the Skewb Diamond into a Super Skewb Diamond so solving the Augemented Skewb Diamond or Complex Skewb Diamond is no different then solving the Super Skewb Diamond but I suspect that isn't the case when it comes to the Master Skewb.

Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Wed Oct 06, 2010 10:50 pm 
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elijah wrote:
that's fine, but I really wish the difference between virtual and imaginary pieces could be magnified.
It seems both types can be given physical volume, except don't exist on the normal versions of the puzzles...
Can anyone sum this difference up in simple terms?
Thanks for sticking around. I love this math stuff but at the end of a long work day when the brain is mush I find it hard to keep all this strait myself. I've got next Monday off so I'll try to catch up then...

But back to virtual vs imaginary. It goes back to Andreas's vs Matt's twistability analysis methods. Matt really only defined his method for what I call even orders but I think I've generalized it to include odd orders here:

http://twistypuzzles.com/forum/viewtopic.php?f=1&t=18651

But instead of pointing you to more math, the key lies in what can serve as a holding point and what can't. So a holding point must be defined. Its a piece that never moves under a complete set of a puzzles independent moves.

Let's take a Complex 3x3x3 for example. There are the 6 face turns and 3 slice turns that can be made on a 3x3x3... correct? But only 6 of these 9 turns are independent of each other. We can pick only 2 turns per axis of rotation and to have a complete set we MUST pick 2 turns per axis so lets look at the options we have:

Choice 1:
axis A: two face layers
axis B: two face layers
axis C: two face layers
This make our core a holding point as it never moves under this complete set of moves

Choice 2:
axis A: a face layer and a slice layer
axis B: two face layers
axis C: two face layers
This make one of the face centers (the one on the A axis in the face layer that doesn't turn) a holding point as it never moves under this complete set of moves.

Choice 3:
axis A: a face layer and a slice layer
axis B: a face layer and a slice layer
axis C: two face layers
This make an edge piece a holding point.

Choice 4:
axis A: a face layer and a slice layer
axis B: a face layer and a slice layer
axis C: a face layer and a slice layer
This makes a corner piece a holding point.

Do we have any other choices? No. Yet the Complex3x3x3 has 10 piece types. They must be "purely" imaginary as pieces that are real or virtual can serve as holding points.

I'm using "purely" imaginary to mean imaginary but not virtual as the imaginary pieces include the virtual ones too.

So how can a piece NOT serve as a holding point... if it turns with more then one layer along any given axis of rotation then it MUST be imaginary. Only one layer along each axis is never turned so an imaginary piece WILL move under a complete set of independent turns. Think of a piece that is in both the L and R layers.

If we hold the R layer fixed and just move the L and Slice layer it moves with the L layer.
If we hold the L layer fixed and just move the R and Slice layer it moves with the R layer.
If we hold the Slice layer fixed and just move the L and R layer it moves with both of them.

So any piece that moves with both the L and R layers is "purely" imaginary.

From a puzzle making standpoint virtual pieces can sometimes be given a real volume and have in several puzzles.

To bring a "purely" imaginary piece into play always requires tying layers together in very odd ways and to date I don't believe any have appeared on puzzles that have actually been constructed aside from a few simulations.

How is this for a short version:

Imaginary = a piece with zero volume on a puzzle with equidistant cut planes (maybe surfaces) centered on the origin.
Virtual = an imaginary piece that can serve as a holding point.
Real = a piece with finite (or infinite) non-zero volume on a puzzle with equidistant cut planes (maybe surfaces) centered on the origin and can serve as a holding point.

I hope this helps,
Carl

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 Post subject: Re: What Is Shim's F-Skewb?
PostPosted: Thu Oct 07, 2010 12:00 pm 
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GuiltyBystander wrote:
It looks like you took my suggestion, but moved it to a different set of corners with the same relation and then swapped the rows. I guess your ordering has meaning that I'm missing. In any case, I've updated my image.
What you suggested is the definition of a imaginary piece, not a virtual piece and I wanted to leave index=7 for a virtual pieces.

@Carl:
I have a problem with your definitions. Up to now I assumed that in MultiXYZ XYZ has to be a class of puzzles, in this case MultiHC2 which is short for MultiHexahedronCornerturning2Cuts. For the audience: HC denotes an axis system, not a specific shape.
The point is: The virtualCore is still a virtual piece despite that you can make it visible. Your posting assumes that the virtualCore is a real piece in the offset Skewb. No. It is visible (although "under" other pieces) but still virtual.
Maybe you understand now why I suggested to redo the definitions of real/virtual/imaginary. But that is a different story.
You now might say that you have to consider the OffsetSkewb and the F-Skewb as subsets of the AugmentedHC2 (I adapted your definitions) and not as subsets of the MultiHC2 but please remember that this problem only appears in HCx.
wwwmwww wrote:
Real = a piece with finite (or infinite) non-zero volume on a puzzle with equidistant cut planes (maybe surfaces) centered on the origin and can serve as a holding point.
A minor correction: Every piece with non-zero volume can serve as holding point so the second condition is useles.


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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Thu Oct 07, 2010 10:09 pm 
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Andreas Nortmann wrote:
@Carl:
I have a problem with your definitions. Up to now I assumed that in MultiXYZ XYZ has to be a class of puzzles, in this case MultiHC2 which is short for MultiHexahedronCornerturning2Cuts. For the audience: HC denotes an axis system, not a specific shape.
If XYZ is a class of puzzles then yes I would agree it should be assumed the MultiXYZ is simply the set of all real pieces.

Andreas Nortmann wrote:
The point is: The virtualCore is still a virtual piece despite that you can make it visible. Your posting assumes that the virtualCore is a real piece in the offset Skewb. No. It is visible (although "under" other pieces) but still virtual.
I agree it IS a virtual piece but I came up with the notion on the MultiXYZ before the concepts of real/virtual/imaginary pieces was around. When I came up with it... it was simply to address the concept of solving the volume of a puzzle and not just its surface. And as such I think I would use the terms a bit differently then you. I'd be tempted to say the virtual core isn't a part of either the Skewb or the Offset Skewb. Neither puzzle lets you "solve" this piece as it's "unstickered". The virtual core isn't a part of the MultiSkewb but it IS a part of the MultiOffsetSkewb.

So I guess I need to offer 3 definition of MultiXYZ. And they depend on what XYZ is. Lets look at HC2... all the puzzles seen here:

Image

If XYZ is a class of puzzles: (ie HC2)
(1) MultiXYZ = the set of all real pieces in this class.

If XYZ is a specific puzzle and the deepest cut version of its class. (ie Master Skewb is the deepest cut HC2 puzzle. If the above animation were to contine you'd never see any more new puzzles till the cuts merged and you had a Skewb which is an HC1 puzzle, not HC2)
(2) MultiXYZ = the set of all pieces created by the volumes of space that the specific cut planes of puzzle XYZ create, be these pieces real or virtual.

If XYZ is a specific puzzle and its NOT the deepest cut version of its class. (ie MultiDinoCube)
(3) MultiXYZ = the set of all pieces created by the volumes of space that the specific cut planes of puzzle XYZ create INSIDE the volume of puzzle XYZ, be these pieces real or virtual.

Now having said that... if I have definition (3) then definition (2) becomes redundant because if puzzle XYZ is the deepest cut version of its class there are no new pieces OUTSIDE the volume of puzzle XYZ.

So lets change that to:

If XYZ is a class of puzzles: (ie HC2)
(1) MultiXYZ = the set of all real pieces in this class.

If XYZ is a specific puzzle (ie MultiDinoCube or MultiMasterSkewb)
(2) MultiXYZ = the set of all pieces created by the volumes of space that the specific cut planes of puzzle XYZ create INSIDE the volume of puzzle XYZ, be these pieces real or virtual.


Better?

See my post here:
http://twistypuzzles.com/forum/viewtopic.php?p=222270#p222270
This is what I mean when I called TomZ's Compy Skewb A Order = 3 Corner-Turning MultiCube. It is not THE Order = 3 Corner-Turning MultiCube.

Specifically its the MultiDinoSkewb its NOT the MultiHC3. That is what I tried to make here and actually ended up with a bit more. I got a puzzle which turned out to be a subset of AugmentedHC3.

Andreas Nortmann wrote:
Maybe you understand now why I suggested to redo the definitions of real/virtual/imaginary. But that is a different story.
I think the set we have now works well. We just need to get everyone on the same page.

Andreas Nortmann wrote:
You now might say that you have to consider the OffsetSkewb and the F-Skewb as subsets of the AugmentedHC2 (I adapted your definitions) and not as subsets of the MultiHC2 but please remember that this problem only appears in HCx.
Actually no. I don't consider the OffsetSkewb or the F-Skewb to have any virtual pieces. The OffsetSkewb IS the same puzzle as the Skewb. The F-Skewb IS a subset of the MasterSkewb. The MultiOffsetSkewb IS a subset of the AugmentedHC1 (not 2). The MultiF-Skewb IS a subset of the MultiOffsetMasterSkewb which is in turn a subset of the AugmentedHC2.

Andreas Nortmann wrote:
wwwmwww wrote:
Real = a piece with finite (or infinite) non-zero volume on a puzzle with equidistant cut planes (maybe surfaces) centered on the origin and can serve as a holding point.
A minor correction: Every piece with non-zero volume can serve as holding point so the second condition is useles.
Its a bit more then simply "non-zero volume" but I guess I am being redundant again as I had this bit "on a puzzle with equidistant cut planes". Think of the Complex3x3x3 that looks like a Multi5x5x5. In that puzzle each of the imaginary pieces has a non-zero volume too... but its not possible to get that Multi5x5x5 look-a-like using just 2 equidistant cut planes. So I'm ok with:

Real = A piece with finite (or infinite) non-zero volume on an order=N puzzle with N equidistant cut planes [per axis of rotation] centered on the origin.

I realized I needed to specify the order of the puzzle as the ComplexHF2 does have equidistant cut planes centered on the origin... it just has more then 2 of them. I also need to drop the bit about the cuts being surfaces as a pair if cut planes could be considered 1 disconnected surface.

HF2 = HexahedronFaceturning2Cuts aka 3x3x3

Sound better?

Carl

[edit: needed to add the per axis of rotation bit]

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Thu Oct 07, 2010 11:44 pm 
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I've been working on trying to count the number of Order=3 Tetrahedral. But before we look at this directly, I think it's kind of helpful to look at the same puzzle with lower order and dimension. Tetrahedra are 3D objects, triangles are 2D objects, and lines are 1D objects. FYI, simplex is the generic term for these n-dimensions objects. So with that in mind, here's what I propose is all the Order=1 Simplexes for dimensions 1-3. For all my drawings, I will just be drawing the cores/cuts of these puzzles. For 1D I will be re-using the very simple cut diagram I tried using earlier in this thread.
Attachment:
D123-O1.png
D123-O1.png [ 2.46 KiB | Viewed 6147 times ]
Now I'm sure you can understand what a puzzle in 2D and 3D from this picture means. What you wondering is, "What are the pieces in 1D supposed to mean and how does it turn?" Well, you have me at a lose there too, but what's important (in my mind at the moment) is the relationship between the cores. Hopefully this will make more sense as go to higher orders. So, let's go to higher orders. I've included the versions where the cuts are on the same side because they're kind of essential building block for higher orders.
Attachment:
D1-O2.png
D1-O2.png [ 1.44 KiB | Viewed 6147 times ]
Attachment:
D2-O2.png
D2-O2.png [ 3.15 KiB | Viewed 6147 times ]
Attachment:
D3-O2.png
D3-O2.png [ 7.51 KiB | Viewed 6147 times ]
So there are 7,9,11 different "puzzles" for Order=2 Simplexes Dimensions 1-3 respectively. There are also 3,4, and 5 what I call critical points on these "puzzles." A critical point is an alignment of cuts where two pieces have 0 volume and moving one of the cuts one way or the other will make one of the pieces have volume. The puzzles between the critical points are kind of like the hybrids of the two critical points. Sort of like how gelatinbrain 1.1.2 is a hybrid of the megaminx and the pyraminx crystal.
In 1D, the critical points are red deep-cut, red vertex = blue vertex, and blue deep-cut. Again, I can't imagine what a "piece" means here, but lets pretend it does mean something :P
In 2D, the critical points are red deep-cut, red vertex = blue edge, red edge = blue vertex, blue deep-cut. Here you should be able to see what pieces are appearing and disappearing.
In 3D, the critical points are red deep-cut, red vertex = blue face, red edge = blue edge, red face = blue vertex, blue deep-cut. It's a little harder to see the pieces, but if you need help, use the table I posted a few posts up.

Now on to Order=3. It's fairly easy to draw all of them for dimensions 1,2.
Attachment:
D1-O3.png
D1-O3.png [ 3.11 KiB | Viewed 6147 times ]
Attachment:
D2-O3.png
D2-O3.png [ 12.59 KiB | Viewed 6147 times ]
Now counting all of the dimension=3 order=3 ones may take some time, but by extrapolating from the lower dimensions, I think I can safely predict it would look something like this.
Attachment:
D3-O3.png
D3-O3.png [ 16.86 KiB | Viewed 6147 times ]
I didn't take the time to render all 61 squares. I just fudged the first row where all order=2 puzzles are completely inside of a tetrahedron that is the same "direction" as the ones in the top left.

For all of the diagrams above I assumed there was an absolute ordering to the cuts. Green>Blue>Red. This may seem a bit inconsistent because I include the inversion of ever puzzle. If you want to know how many puzzles there are where the cut order isn't defined (like if you wanted to build some kind of complex HC3 out of every puzzle), just multiply by o! where o is the order. If you want to know how many puzzles there are without the inverted versions, just divide by 2 and round up.

And just for fun I started to do some Order=4 work.
Attachment:
D1-O4.png
D1-O4.png [ 5.77 KiB | Viewed 6147 times ]
Anything with higher dimensions would get hard to manually draw out/count. I want to find a formula to calculate it. I'm sure there's a generic recurrence relations similar to the Fibonacci sequence for all dimensions/orders, but I can't quite put my finger on it. For the 1D version I think this works. n is the order of the puzzle.
Code:
T(0) = 1
T(1) = 3
T(n) = 2*T(n-1) + T(n-2)
or
T(n) = T(n-1) +2*(1 +sum(T(x),x,0,n-2) )
In the mean time, here's a summary of the counts so far. I'm a little hazy on whether I should say there's 1 order=0 puzzle or not.
Attachment:
table.png
table.png [ 1.96 KiB | Viewed 6147 times ]

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Fri Oct 08, 2010 12:22 am 
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wwwmwww wrote:
If XYZ is a specific puzzle (ie MultiDinoCube or MultiMasterSkewb)
(2) MultiXYZ = the set of all pieces created by the volumes of space that the specific cut planes of puzzle XYZ create INSIDE the volume of puzzle XYZ, be these pieces real or virtual.
I'm a bit unhappy with the "INSIDE" part but I do understand the value of defining it this way. Defining it this way leaves a void for what I want it to mean. What do you call the relationship between the Dino Cube and the MultiMasterSkewb if you have never ever seen a MasterSkewb? In my previous posts talking about the MultiF-Skewb, I was using this definition for MultiXYZ:
MultiXYZ = the set of all pieces created by the volumes of space that the specific cut planes of puzzle XYZ
If I can't use the name MultiF-Skewb, what do I call that? Without reverting to saying MultiOffsetSkewb of course.

wwwmwww wrote:
Real = A piece with finite (or infinite) non-zero volume on an order=N puzzle with N equidistant cut planes centered on the origin.
It kind of bugs me that a piece's realness is based on it's existence in a very specific set of cuts for that set of axes. It just seems weird that you can say that a piece is real but has no volume. Or that a piece is imaginary, but I can point to it and show you where it exists.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Fri Oct 08, 2010 10:55 am 
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GuiltyBystander wrote:
If I can't use the name MultiF-Skewb, what do I call that? Without reverting to saying MultiOffsetSkewb of course.
Well we could call the Offset Skewb the "deepest cut" F-Skewb going back to the definition I dropped above. So maybe call it the Deepest MultiF-Skewb.

Here are a few another ideas...

How about Boundless MultiF-Skewb to say you are looking at all space and not just the inside of the F-Skewb? Or simplify Boundless Multi to just Boundless, ie Boundless F-Skewb. Actually there must be a better term. You want the puzzle that contains "all possible" boundaries put on the space cut up by the cut planes of the F-Skewb not just the infinite one. One could take Boundless F-Skewb to mean the Offset Skewb, not the MultiOffsetSkewb.

Isn't there a term from linear algebra that means something like a linear combination of all possible solutions? Not being able to think of it... how about Combination F-Skewb?

Maybe I need to come at this from another angle. Let's define a new term:

InverseXYZ = the set of all pieces created by the volumes of space that the specific cut planes of puzzle XYZ create OUTSIDE the volume of puzzle XYZ, be these pieces real or virtual.

Then what you are after is MultiXYX + InverseXYZ. In that case maybe the best term would be UnitaryXYZ.

How does Unitary F-Skewb sound?

GuiltyBystander wrote:
It kind of bugs me that a piece's realness is based on it's existence in a very specific set of cuts for that set of axes. It just seems weird that you can say that a piece is real but has no volume. Or that a piece is imaginary, but I can point to it and show you where it exists.
It's only the virtual pieces that will be given volume if the cut planes are moved around in the fashion you mention above. And yes real pieces can go out of existance too. So the real and the virtual pieces have more in common with each other then the other imaginary pieces. So questions I'm not sure of the answers to...

(1) Can all virtual pieces be given physical volume this way? On all puzzles?
(2) Can all real pieces be pushed out of existance this way? On all puzzles?

Look at the 2x2x2. Three mutually orthogonal planes, regardless of where they are pushed to, always cut all space up into 8 corner pieces. Continue with the 3x3x3 and you'll always find the same 27 regions in space.

As long as the axes of your puzzle are orthogonal you never have virtual pieces. You just have real and imaginary and you will never be able to "point" to an imaginary piece making your figures the way you are. It's the non-orthogonal axes of rotation that muddy the waters and creates this middle ground between real and imaginary.

Think of a 4D Skewb where are 4 axes were orthogonal. Actually wouldn't that just be the 2x2x2x2? Ok, maybe that doesn't help.

Ok... maybe Andreas has a good point about redo the definitions of real/virtual/imaginary. If all real pieces can be pushed out of existance on non-ortorthogonal puzzles by moving the cut planes AND if all virtual pieces can be pulled into existance on non-ortorthogonal puzzles by moving the cut planes that maybe we should murge these two groups.

Say something like all ortorthogonal puzzles have Real and Imaginary pieces and all non-ortorthogonal puzzles have Pinocchio and Imaginary pieces. The Pinocchio pieces are the set of Real+Virtual and say what is real and what is virtual depends on the specific location of the cut planes.

The only other approach I see is to pick one specific set of cut planes and say they define the Real and Virtual pieces and that is what we've done to date even though my language has been a bit sloppy in that regard to date. It seems fair as there is really only one logical choice for which set of cut planes to choose.

If there are Real pieces that can never be pushed out of existance or Virtual pieces that can never be pulled into existance [on non-ortorthogonal puzzles] by simply moving the cutting planes around things can get much more complicated. That complication if it even exists is avoided by picking one specific set of cut planes.

Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sat Oct 09, 2010 6:26 am 
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@GuiltyBystander: Nice work with the tetrahedrons.

@Concerning the MultiXYZ:
The interesting thing is that under the definition of GuiltyBystander we have this:
MultiMasterSkewb=MultiDinoCube=MultiFaceTurningOctahedron (etc.) = MultiHC2 (in Carls definition)
So far so easy. Then I thought the MultiFSkewb (under the definition of GuiltyBystander) is equal to the AugmentedHC2 of Carls definition but it isn't.
I suggest MultiXYZ (AugmentedXYZ, ComplexXYZ) should be somehow defined independently from cutting depths which is true if XYZ is an axis system.
My 2 cents: If XYZ shall be a puzzle, the term Multi- should be avoided.

wwwmwww wrote:
(1) Can all virtual pieces be given physical volume this way? On all puzzles?
(2) Can all real pieces be pushed out of existance this way? On all puzzles?
The problem here is that "moving the cutting planes around" is at least complicated on axis system except HC. Please remember Carl that I wanted to ignore the virtual pieces completely before I realized I need them for special cases like MasterPyraminx, Gelatibrain 5.1.1, CircleMegaminxes and your MultiHC3PlusOne. Even on HF it is not as easy as it seems. You have always 27 reagions of space in HF2 regards of the distances from an origin but only if the inner space is a cube you get a 3x3x3-90-90-90, otherwise you get a 3x3x3-90-180-180 or a 3x3x3-180-180-180. And now imagine what happens when you move a single cutting plane of a megaminx somewhat deeper or shallower.

But lets focus on HC2: The core can never pushed out of existence although it can be hidden under the others.
Regarding the other pieces: Think about what the cutting planes of a DinoCube do with inifinite space. All real pieces are always present but they might be invisible depending on where you cut the infinite space down to the puzzles outer form.
Next example: In DF2 (DodecahedronFaceturning2Cuts) there are 2 kinds of (real) face pieces. Now lets add another sets of equidistant cuts. Depending on where it is place you can have two different kinds of additional face pieces but not both at the same time.
In conclusion: I consider it unavoidable to work with cutting depths.

GuiltyBystander wrote:
It kind of bugs me that a piece's realness is based on it's existence in a very specific set of cuts for that set of axes. It just seems weird that you can say that a piece is real but has no volume. Or that a piece is imaginary, but I can point to it and show you where it exists.
Therefore are here my new tries for redefining pieceTypes with direct reference to cutting depths:
P = (NonZero)volumePieces under at least one configuration of equidistant cuts (REAL)
ZPH = ZeroVolumePieces under equidistant cuts that can act as a holding point (VIRTUAL)
ZPN = ZeroVolumePieces under equidistant cuts that can NOT act as a holding point (pure IMAGINARY)
PH = P+ZPH = Pieces which can act as a holding point
ZP = ZPH+ZPN = ZeroVolumePieces under equidistant cuts (IMAGINARY)

I need some sleep!


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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sat Oct 09, 2010 6:33 pm 
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Andreas Nortmann wrote:
@GuiltyBystander: Nice work with the tetrahedrons.
Agreed! I'll come back to this when I have time to let it all sink in. I must be getting old and slow.
Andreas Nortmann wrote:
@Concerning the MultiXYZ:
The interesting thing is that under the definition of GuiltyBystander we have this:
MultiMasterSkewb=MultiDinoCube=MultiFaceTurningOctahedron (etc.) = MultiHC2 (in Carls definition)
Yes we do and I don't like that either. That is why I limited MultiXYZ to only the volume of puzzle XYZ. It also comes in very handy.

Look at TomZ's Dino Skewb. As Tom himself said:
TomZ wrote:
The puzzle is a build-up from the Twins' mechanism, a fully functional Tetrahedral Twins (minus the trivial tips) is present on the inside of the Dino Skewb. Forum member wwwmwww suggested this build-up when he saw the Tetrahedral Twins,
And inside that Tetrahedral Twins is a Skewb Diamond. In the spirt of the MultiXYZ concept I wanted to be able to play with ALL three puzzles at the same time. I wanted to solve the VOLUME of TomZ's Dino Skewb, NOT just the surface. And after a few emails (my part was small but I like to think I helped push things the the right direction) the The Compy Skewb was born.
TomZ wrote:
This is the Compy Skewb, a variation on my Dino Skewb, suggested by Carl Hoff (wwwmwww). When I mentioned that the Dino Skewb is built on a Tetrahedral Twins, which in turn is built on a Skewb he asked if I could design a modification for him that would have all the pieces visible. The final result I came up with is far from a modification, but it looks great.
Untrivial tips
The puzzle is essentially a combination of a Skewb and Compy Cube (shallow cut Dino), hence the name. It's not quite that actually. That design wouldn't have the tips. Originally I had the tips chopped off for aesthetics, but Carl mentioned that while I showed the permutation of the Skewb corners (you can tell from the piece below the tip), I did not show their orientation. He suggested to add tips to show the orientation.
I wanted the Skewb corners in there and in my mind this WAS a MultiDinoSkewb. What better way is there to describe the relationship between the Dino Skewb and the Compy Skewb?
Andreas Nortmann wrote:
So far so easy. Then I thought the MultiFSkewb (under the definition of GuiltyBystander) is equal to the AugmentedHC2 of Carls definition but it isn't.
No its not. If you limit it to the volume of the F-Skewb it is missing a few real pieces and only has one of the virtual pieces. And if you want to talk about this set I can't think of a better name then MultiFSkewb.
Andreas Nortmann wrote:
I suggest MultiXYZ (AugmentedXYZ, ComplexXYZ) should be somehow defined independently from cutting depths which is true if XYZ is an axis system.
My 2 cents: If XYZ shall be a puzzle, the term Multi- should be avoided.
Not sure about Augmented and Complex at the moment but I disagree with regard to Multi. Lets look at its history. I believe Jared was the first to use Multicube in this context here in Sept. 2009. The idea of solving the volume of a puzzle has been around much longer then that. Tony Fisher called then Hidden Cubes back in Nov. 2006. I struggled with what to call them here in Sept. 2009 and before that here in Oct. 2008. Till Jared threw out Multicube the best terminology I could come up with was [s]Super[/s] Super Cubes where the [s][/s] was intended to strike out the first Super and that was based on the CubixPlayer2 program made by Per Kristen Fredlund that allowed you to Super Super solve NxNxN puzzles and that dates back to at least April 2004 as seen here. I assume there was a CubixPlayer1 and I suspect this concept dates back even further but it didn't appear to have a good name till the Multi- term came up last year so I jumped and ran with it. I don't really want to back slide now. I agree the use of Augmented and Complex could be problematic as where are the virtual and imaginary pieces? If they have zero volume how do you tell if they are INSIDE or OUTSIDE puzzle XYZ? If they are used with puzzle XYZ I think they'd have to mean:

AugementedXYZ = All the real pieces INSIDE PuzzleXYZ plus all the virtual pieces of the axis system belonging to PuzzleXYZ.
ComplexXYZ = All the real pieces INSIDE PuzzleXYZ plus all the imaginary pieces of the axis system belonging to PuzzleXYZ.
Andreas Nortmann wrote:
Even on HF it is not as easy as it seems. You have always 27 reagions of space in HF2 regards of the distances from an origin but only if the inner space is a cube you get a 3x3x3-90-90-90, otherwise you get a 3x3x3-90-180-180 or a 3x3x3-180-180-180.
Agreed... and there is also the problem of negative cut depths. Imagine the cut for the right face on the left side of a 3x3x3 and the cut for the left face on the right side of the 3x3x3. You still have 27 pieces BUT the core, edges, and face centers are now imaginary pieces.
Andreas Nortmann wrote:
And now imagine what happens when you move a single cutting plane of a megaminx somewhat deeper or shallower.
My brain turns to mush at this point... it gets ugly... I agree.
Andreas Nortmann wrote:
But lets focus on HC2: The core can never pushed out of existence although it can be hidden under the others.
Are you sure about that? I'm not. What happens when one of the cut planes is moved through the origin and onto the same side as the other cut. If the core contains the origin isn't there now another piece in its place? Granted I'm still sorting through all the info above so maybe I'm out in left field.
Andreas Nortmann wrote:
Now lets add another sets of equidistant cuts. Depending on where it is place you can have two different kinds of additional face pieces but not both at the same time.
In conclusion: I consider it unavoidable to work with cutting depths.
Ahh... but you are now up to Order=4 puzzles. For Order=1, 2, and 3 I believe only one set of equidistant cuts is needed to define what is real, virtual, and imaginary. The other cut used in these 3 orders is the deep cut. On Order=4 there is now a potential problem. I'd propose for Order=4 if the two inner cuts are at 1 and -1 on the axis of rotation then the two other cuts be at 3 and -3 so all layers are the same width. For Order=5 the cuts would be at -2, -1, 0, 1, 2 so again all layers are the same width. Etc. Pieces present (non-zero volume) with these cuts are Real, zero volume pieces are Imaginary, and the Imaginary pieces that can serve as holding points are Virtual.
Andreas Nortmann wrote:
Therefore are here my new tries for redefining pieceTypes with direct reference to cutting depths:
P = (NonZero)volumePieces under at least one configuration of equidistant cuts (REAL)
ZPH = ZeroVolumePieces under equidistant cuts that can act as a holding point (VIRTUAL)
ZPN = ZeroVolumePieces under equidistant cuts that can NOT act as a holding point (pure IMAGINARY)
PH = P+ZPH = Pieces which can act as a holding point
ZP = ZPH+ZPN = ZeroVolumePieces under equidistant cuts (IMAGINARY)
Up to Order=3 is this any different then what I proposed above? At Order=4 and greater your P set can be greater then my Real set. I think your PH would still be equal to my Real plus Virtual sets even at Order=4 and greater. The same I think is true for your ZPN and my Imaginary set minus my virtual set. Considering I'm still working with Order=1 in my examples above and I'm struggling to keep up at Order=2 I hadn't seen this issue with the naming at Order=4 before.

I'm not sure which is best... but I do see one perk of picking my definition above. It means I could easily make a single animation to count all the "Real" pieces in an Order=4 puzzle. But I agree... what is real and what is virtual is highly dependent on your choice of cut planes. In that regard I think I like your definitions better in the more general case. And when I've discussed the difficulty of making animations to count the real Order=4 pieces in the past I've been using your definitions without even realizing it.

Carl

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Last edited by wwwmwww on Sun Oct 10, 2010 8:26 am, edited 4 times in total.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sun Oct 10, 2010 3:48 am 
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Sounds like we have agreements here. Lets see when I can use it again...
wwwmwww wrote:
Agreed... and there is also the problem of negative cut depths. Imagine the cut for the right face on the left side of a 3x3x3 and the cut for the left face on the right side of the 3x3x3. You still have 27 pieces BUT the core, edges, and face centers are now imaginary pieces.
This can be solved by redefining what a movement is: A redefined movement would be one movement in your proposed cutting depths and afterwards twisting the axissystem as a whole. The ZPNs (aka purely imaginary) pieces are only needed when this kind of inversion doesn't help.


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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Fri Oct 15, 2010 1:19 am 
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It's taken me a little while to respond. I've been working on a project that I'll share in a bit

wwwmwww wrote:
Andreas Nortmann wrote:
@Concerning the MultiXYZ:
The interesting thing is that under the definition of GuiltyBystander we have this:
MultiMasterSkewb=MultiDinoCube=MultiFaceTurningOctahedron (etc.) = MultiHC2 (in Carls definition)
Yes we do and I don't like that either. That is why I limited MultiXYZ to only the volume of puzzle XYZ. It also comes in very handy.
Right. Which is why I added "but I do understand the value of defining it this way."

Sometimes I think every post should include a short dictionary. So many terms being used differently... For example, I was using (although not claiming dibs on the word or anything) inverse to represent the relationship between 4 & 6 on my 1-9 tetrahedron line up.



And now my muse. Ever since Carl showed us the inner workings of the MultiDodecahedron, I've wanted to explore it a little better. So to that end, I've been working on a program to cut up an arbitrary solid with an arbitrary axis system. Here's a few pics of me trying to recreate some of Carl's great pics in that thread.
Image
Image
Image
Image

Controls:
Top 2 sliders select what "depth" of pieces to show. Here, depth means the number of axis that the piece is moved by. The All/Even/Odd to the right also filter depth.
The 3rd slider controls the depth of the cut.
The 4th slider controls the depth of the viewing plane. You can also use the mouse wheel. The viewing plane is useful for cutting the puzzle in half to look at the insides. Right-click & drag to rotate it. The Off/Select/Cut control the behavior of the viewing plane.
The two boxes below that control the shape of the puzzle and what axis system to use.

Download Link

Now I'm not done yet, but I really wanted to share my progress so far. I know there's lots of stuff I need to add so don't go building a bug/wish list quite yet. I want to have support for multiple axis systems with varying depth so we can watch these pieces pop in and out of existence or see the insides of awesome new puzzles.

Disclaimer: I kind of hacked the cutting up function so it's not completely perfect. It may freeze or crash if there's too many pieces. Piece counts aren't likely to be accurate for lots of deep cuts. Puzzles with lots of deepish cuts may take a while to compute & render.


I'm trying to come up with a way to count the different kinds of pieces, but I'm at a loss atm. So far I've been using # of axes the piece is on, but that's not enough. And trying to compare and identify the relationship of the axes of a piece is NP-C I think. Can I count the number of triangles, squares, etc. on the piece and use that? One problem is that you will get more pieces on shape mods and I'm also afraid it won't be enough. Any thoughts/tips here would be nice.

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sat Oct 16, 2010 3:15 am 
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That tool is a great addition to Carls animations. Well made.
GuiltyBystander wrote:
I'm trying to come up with a way to count the different kinds of pieces, but I'm at a loss atm.
Sounds like you want to identify all NonZeroVolume pieces (P) of a specific class of puzzles. You can easily (?) write programs to list all HoldingPointPieces PH of a given class. Comparably you can write programs to list all Pieces (P+ZP) for puzzles with even number of cuts per axis. So far I haven't found a way to filter the PositivVolumePiecesUnderEquidistantCuts (P) out of these lists.
GuiltyBystander wrote:
And trying to compare and identify the relationship of the axes of a piece is NP-C I think.
Is it? I don't know what you want to say here. So I can't answer this.
GuiltyBystander wrote:
Can I count the number of triangles, squares, etc. on the piece and use that? One problem is that you will get more pieces on shape mods and I'm also afraid it won't be enough.
You could use something like that.
Shape mods don't really change anything, they just change the shape of the pieces: Think of a solid of infinite size which is cut into pieces with a specific axis system. In puzzles with at most 3 cuts per axis, every piece with nonzerovolume will be present although some hidden behind others.
In more general cases (like puzzles with 4 cuts per axis) this doesn't work that easily.


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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sat Oct 16, 2010 1:58 pm 
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Andreas Nortmann wrote:
GuiltyBystander wrote:
I'm trying to come up with a way to count the different kinds of pieces, but I'm at a loss atm.
Sounds like you want to identify all NonZeroVolume pieces (P) of a specific class of puzzles.
Right. I was thinking of something along the lines of the list of pieces Carl put on this animation. I'm less concerned with the naming.

Andreas Nortmann wrote:
GuiltyBystander wrote:
And trying to compare and identify the relationship of the axes of a piece is NP-C I think.
Is it? I don't know what you want to say here. So I can't answer this.
You can classify a piece by the number of axes it is moved by and the relation ship between them. On the pentultimate, there are 2 pieces visible. Both pieces are moved by 6 axes but the relationship between the six axes is different so the two pieces are different types. I've graphed them below. Vertices represent axes and edges represent the relationship between them. The left graph is for the center piece, the right is for the corner piece.
Attachment:
pentultimate.png
pentultimate.png [ 2.41 KiB | Viewed 5942 times ]
To tell if two pieces are the same, we need to know if the two graphs are equivalent. This problem is know as graph isomorphism. Looks like I may have tacked on the -C a little too hastily. This problem is only known to be NP atm. Actually, our problem has a few more restrictions because the edges have lengths (angels between the two vertices).
I'm hoping there's a better/simpler way to look at this.

Here's another way of stating the problem if my above ramblings don't help. The pentultimate has 32 pieces. How do I know that there are 20 of one kind and 12 of another?

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Sat Oct 16, 2010 6:52 pm 
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GuiltyBystander wrote:
Now I'm not done yet, but I really wanted to share my progress so far.
WOW!!! Just WOW!!! Great new toy we have here. What takes me days to do in POV-Ray can now be done in seconds. I LOVE it. And this thread has turned into something much more then I expected.

Sorry I've been quite of late but I'm still getting a puzzle that is stuck in my head out into a form I can share. I hope it will be worth the wait.

Carl

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 Post subject: Re: What IS Shim's F-Skewb?
PostPosted: Mon Oct 18, 2010 7:14 am 
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GuiltyBystander wrote:
You can classify a piece by the number of axes it is moved by and the relation ship between them. On the pentultimate, there are 2 pieces visible. Both pieces are moved by 6 axes but the relationship between the six axes is different so the two pieces are different types. I've graphed them below. Vertices represent axes and edges represent the relationship between them. The left graph is for the center piece, the right is for the corner piece.To tell if two pieces are the same, we need to know if the two graphs are equivalent. This problem is know as graph isomorphism. Looks like I may have tacked on the -C a little too hastily. This problem is only known to be NP atm. Actually, our problem has a few more restrictions because the edges have lengths (angels between the two vertices).
I'm hoping there's a better/simpler way to look at this.
I have understood you problem. I doubt that it is really that complex. Although it could be treated as graph isomorphism we have a special case which makes it easier. All you have to do is to mark all the sides/slices which keep the piece fixed in space (better to use these sides not the others because there are less of them when changing to the higher orders) and then make up all 48 (for hexahedral symmetries) or 120 (for dodecahedral symmetries) symmetries of this configuration. If one of these symmetries is a duplicate to another piece they are of identical type.

I think the problem is not identifying types of pieces which could act as holdingPoints but to filter those with NonZeroVolume out of them. Maybe your graphical representations could help here.


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