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 Post subject: Analysis of twistability and virtual pieces
PostPosted: Thu Dec 17, 2009 1:11 pm 
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These are the results of a work I did this together with Carl Hoff, aka wwwmwww.
In this thread we want to introduce a scheme to analyze a doctrinaire twistypuzzle. By doing so, we will discover something we call "virtual pieces". We need some ideas about how to interpret them and want to know if they could be made real and/or used for future puzzles.

At first we will describe (roughly) our analyzing method and their results. Afterwards we try/ask for an interpretation.
We use the deepcut puzzles 2x2x2, Skewb, LittleChop and Pentultimate as examples. These deepcut puzzles have only one planar cut on every axis.

Lets take a 2x2x2 and consider, we allow only one moveable side per axis.
There are 8 (2 raised to the power 3 as there are 2 sides to choose from for 3 axes) possibilities to restrict twistability to one side per axis. Every possibility describes one corner of the 2x2x2 which stays fixed in space.
Here is a table to illustrate this:
http://wwwmwww.com/Puzzle/2x2x2Detail.htm

Lets do the same for the Skewb:
There are two halfs to choose from for 4 axes so there are 2^4=16 possibilities to restrict twistability to one side per axis.
These 16 possibilities can be separated into 3 sets:
Allowed shall be twists around these corners: URF, URB, ULF, ULB (6 equivalent possibilities) => A square of the Skewb stays fixed in space.
Allowed shall be twists around these corners: URF, URB, ULF, DRF (8 equivalent possibilities) => A corner of the Skewb stays fixed in space.
Allowed shall be twists around these corners: URF, ULB, DLF, DRB (2 equivalent possibilities) => ???
Here is a table for the Skewb:
http://wwwmwww.com/Puzzle/SkewbDetail.htm

Lets do the same for the LittleChop:
There are now two halfs to choose from for 6 axes so there are 2^6=64 possibilities to restrict twistability to one side per axis.
These 64 possibilities can be seperated into 3 sets:
Allowed shall be twists around these edges: UF, UR, UL, UB, FR, FL (24 equivalent possibilities) => A visible piece of the LittleChop stays fixed in space.
Allowed shall be twists around these edges: UF, UR, UL, RF, RB, DF (16 equivalent possibilities) => ???
Allowed shall be twists around these edges: UF, UR, UL, RB, LB, DF (24 equivalent possibilities) => ???
http://wwwmwww.com/Puzzle/LittleChopDetail.htm

Lets do the same for the Pentultimate:
There are two halfs to choose from for 6 axes so there are 2^6=64 possibilities to restrict twistability to one side per axis.
To make clear which faces are meant, please refer to this image:
http://wwwmwww.com/Puzzle/VPentultimate1a.png
Allowed shall be twists around these faces: 1,2,3,4,5,6 (12 equivalent possibilities) => A pentagon of the Pentultimate stays fixed in space.
Allowed shall be twists around these faces: 1,2,3,4,6,8 (20 equivalent possibilities) => A corner of the Pentultimate stays fixed in space.
Allowed shall be twists around these faces: 2,3,4,5,6,C (12 equivalent possibilities) => ???
Allowed shall be twists around these faces: 1,2,3,5,7,9 (20 equivalent possibilities) => ???
http://wwwmwww.com/Puzzle/PentultimateDetail.htm

We have done the same analysis (in a more condensed form) for the Skewbic Dodecahedron (aka deepcut cornerturning dodecahedron; Gelatibrain 1.2.9) and for the BigChop (aka deepcut edgeturning dodecahedron; Gelatibrain 1.4.3) but we will skip those due to the size of the tables. I hope everybody believes us, that something similar happens there.
From now on we want to call the pieces which are visible as "real pieces" and the others cases as "virtual pieces".

I have made up a similar analysis for puzzles with two cuts per axis (e.g. Megaminx). There I found representations for every real piece of Carl's animations and several representations for virtual ones similar to those above.
Only the NxNxN doesn't contain virtual pieces with zero-volume.

What happens above? We think we have found that the Skewb, LittleChop, etc. contain some kind of virtual pieces.
I almost dropped these virtual pieces as "irrelevant anomaly" but then I discovered that they could be made real: The Halpern-Meier-Tetrahedron is known to be a tetrahedral equivalent of the Skewb. All cutting planes of this puzzle go through its center.
Now lets move all these planes an equal distance towards the tips of the tetrahedron. The triangular faces shrink, the other pieces grow and within the tetrahedron there is now a non-zero-volume which stays fixed if only the tips are allowed to twist.
If you move the planes further to the tips you reach the Pyraminx (ignoring trivial tips) and even later we get Gelatibrain 5.1.1 where this non-zero-volume becomes visible.

We could even go the other way and move the cutting planes to the faces and again inside of the tetrahedron a non-zero-volume comes to life, a volume which is fixed in space if only the faces of the tetrahedron are allowed to be twisted. In the theory above there are two of these virtual pieces and the corners of the Skewb seem to be attached to one of these virtual pieces. This makes sense since the corners of the Skewb can be divided into two subsets in which the corners never change there position to each other.
In the Pyraminx and in Gelatibrain 5.1.1 only one of these subsets is visible.

We have found some kind of explanation for the virtual pieces of the Skewb. We still have problems to interpret the virtual pieces of LittleChop and Pentultimate.

Tell us, what you think.

Carl Hoff (aka wwwmwww) and Andreas Nortmann.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Fri Dec 18, 2009 9:18 am 
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Thanks Andreas. I'll be posting an animation which I think shows the 2 virtual pieces in the Skewb in action soon. I've got it rendering now. Before I say too much and influence how others may see these virtual pieces by letting them know how I see them I'm really curious what others think when first introduced to this concept. My interpretation may not be the best... or even correct for that mater.

These are the questions I'm struggling with and there may be better questions I should be asking too...
(1) Can these virtual pieces always be made physical? I think the 2 in the Skewb can be and I'll show you how soon. Andreas has already told you above.
(2) If we can add these pieces to the puzzle what do they add to the puzzle solving experience? Anything?

Thanks,
Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Fri Dec 18, 2009 5:58 pm 
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Again, nice work you two! :)

I will add some other type of information (not similar to the above) as soon as I am done updating
my new database! (i.e. definitions not only for twistability, but also for slidability, shiftability, etc etc etc)


wwwmwww wrote:
(1) Can these virtual pieces always be made physical? I think the 2 in the Skewb can be and I'll show you how soon. Andreas has already told you above.
(2) If we can add these pieces to the puzzle what do they add to the puzzle solving experience? Anything?


Question (1) should have a "yes" answer (in general). I mean, there can be magnetic puzzles, or other puzzles
which can be made, but making them is not worth financially. And I have not even gone into the area where
new mechanisms (which are not invented yet! ;) ) could provide with all sorts of new options.

Question (2) is easier, in the sense that mathematically, it shouldn't be too hard to find out whether the addition
of new parameters could reinforce the complexity of a puzzle in a non-trivial *and* non-tedious (when compared
to the previous design) way. There is more to this, but I need to shut up myself!

:P

Pantazis

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Fri Dec 18, 2009 6:09 pm 
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I don't understanbd how you came up with virtual pieces... :roll:
The tables don't make sense to me...

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sat Dec 19, 2009 4:00 am 
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elijah wrote:
I don't understanbd how you came up with virtual pieces... :roll:
The tables don't make sense to me...
Argh! We write lines and lines and still miss the point.

Okay. Another try:
Take a Skewb. Consider one face-piece as fixed in space. Twists around which corners are allowed to let the choosen face stay fixed?
Now do the same for a corner of the Skewb.
In both cases you will get 4 allowed twists where none of them are opposite to each other.
Now do it the other way: How many possibilities are there to make up a set of 4 twistable corners where no two corners are opposite to each other? The table for the Skewb given above is the result.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sat Dec 19, 2009 5:02 am 
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Don't worry guys. I got answers to all of your questions :wink:

I started explaining but ended up writing almost 4 pages and still haven't finished. But I will give you the beginning of my response, I bet Carl will get a kick out of it :lol:
Quote:
Wow guys excellent job.

I swear Carl you copy EVERYTHING I do myself, but I beat you on this one too ;)

I have discovered the existence of these "virtual pieces" as well though my approach was vastly different. I do however think my way is an easier way to look at things though....

First off, I should say that tetrahedrons/octahedrons have a strange relationship that cause coincidences that are not true in general. I believe your analysis of the skewb is falling into this trap a little bit when you’re expecting your virtual pieces on the other puzzles to behave similarly. Focus on non-tetrahedral/octahedral puzzles first!

First let me explain what I did....


I'll make a new topic sometime tomorrow with my full explanation, but for now sleep :D

(and don't worry, I am in no way denying the existence of your virtual pieces 8-) )

Peace,
Matt Galla

PS) I am not at all surprised people are lost. You guys are WAY in right field now. But that's ok because I LOVE right field :mrgreen:


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sat Dec 19, 2009 10:33 am 
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Andreas Nortmann wrote:
Argh! We write lines and lines and still miss the point.
Okay. Another try


Let me take a shot... if we say the same thing enough different ways I hope others will SEE it.

Let’s start with the 3x3x3, everyone here should have one. Now let’s look at one axis through the 3x3x3 that allows rotation. You can rotate the left side, you can rotate the right side, or you can rotate the slice layer between the other two. Are these 3 rotations independent? No… rotating the right side one direction is the same as rotating the left and slice layers the other direction. So we can fix one of these layers in place and still be totally general. On the 3x3x3 we can do the same for the other two axes as well. So we have 3 choices as to which layer to fix for each of 3 axes or 3^3 choices. Each of these 3^3 combinations yields a different piece of the 3x3x3. Let’s look at a few and I’ll use the rotation found here:

http://www.randelshofer.ch/cube/notations/combined_eng_3x3.html

R L U D F B - Leaves the core fixed in space. You are in effect holding the 3x3x3 by it’s core.

R MR U D F B - Leaves the face center cubie on the left side untouched. You are now holding the 3x3x3 by that cubie.

R MR U MU F B - Leaves the left/down edge cubie untouched. You are now holding the 3x3x3 by that cubie.

R MR U MU F MF - Leaves the left/down/back corner cubie untouched. You are now holding the 3x3x3 by that cubie.

Now lets apply this to the simplest puzzle that has virtual pieces. For that I’ll use the notation seen here:

http://twistypuzzles.com/solutions/skewb-01/

And if you have a Skewb… pick it up in your hands and follow along. That’s what I do. Having the 3D object in your hands really helps.

We now just have 2 sides to choose from per axis of rotation as to which half we’ll hold fixed. 2^4 choices should result in 16 pieces.

If we make this choice…

DFL, DFR, DBL, DBR ask yourself what piece are you holding onto the Skewb by. It’s the U face.

DFL, DFR, DBL, UFL and we see we are now holding the Skewb by the corner UBR.

DRL, DBR, UFR, UBL now where are you holding onto the Skewb? Where is that piece? Find it… and you will have found a virtual piece.

It’s not easy… if you think it’s on the inside as the core of the 3x3x3 is… do the following. Find a block of wood the same size as your Skewb. Cut it up using the same cuts that allow rotation on your Skewb. Count the pieces… you should have 14 pieces of wood in front of you at the moment. So we are still 2 pieces short…

If you actually try the above with a piece of wood and saw I suspect some of you may actually end up with 15 pieces. I’m almost 100% certain I would as I can’t draw a strait line let alone cut one. Now ask yourself why or how this process might produce 15 pieces? Is that extra piece one of the missing pieces from the analysis above? If yes… where might the 16th piece be?

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sat Dec 19, 2009 10:50 am 
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Allagem wrote:
I bet Carl will get a kick out of it

Nice!
Allagem wrote:
I swear Carl you copy EVERYTHING I do myself, but I beat you on this one too ;)

Maybe you need to change the password on your PC. ;)
Allagem wrote:
I'll make a new topic sometime tomorrow with my full explanation, but for now sleep :D

Are you sure you want to start a new topic? It would be welcome in this thread and this topic is far enough out there as is I don't think we need to split up our audience with two thread.
Allagem wrote:
PS) I am not at all surprised people are lost. You guys are WAY in right field now. But that's ok because I LOVE right field :mrgreen:

Andreas and I have been sitting on this for several months just for that reason. Personally I thought we were in left field. At least that is were I thought I had pitched my tent and set up camp. I'm very eager to see the vastly different approach the right field camp has taken.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sat Dec 19, 2009 11:42 am 
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This is an interesting topic. I would interpret the virtual pieces as a fixed relation between pieces, although some of them can be quite strange. Take skewb again as the example. One of these relations says that the corner is solid, one that the centers are solid, and the other that the four corners that are those connected by a tetrahedron stay in the same place relative to each other. For the 3x3x3, it is already shown that the 4 cases are the three piece types and the core, also more abstractly the relation between the centers staying fixed together. Although in these two cases, they describe a construction method, in general I believe that the relations can be really strange.
Another way of analyzing this would be a suitable application of Burnside's Lemma, to eliminate duplication by symmetries. For pure deep cut puzzles, counting them can be thought of as coloring a polyhedron with two colors such that opposite sides are different colors. For more layered puzzles, the bijection is a bit stranger and I can't come up with it off the top of my head. It is still pretty hard to count them, but that is just visualization. I hope I haven't completely missed the point.

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sat Dec 19, 2009 5:01 pm 
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wwwmwww wrote:
Allagem wrote:
I swear Carl you copy EVERYTHING I do myself, but I beat you on this one too ;)

Maybe you need to change the password on your PC. ;)
Allagem wrote:
I'll make a new topic sometime tomorrow with my full explanation, but for now sleep :D

Are you sure you want to start a new topic? It would be welcome in this thread and this topic is far enough out there as is I don't think we need to split up our audience with two thread.


Ok, all my passwords are changed, let's see you copy me now! :twisted: :lol:

And if you insist I will post it here. WARNING THIS WILL BE LONG!!

First let me explain what I did....

With the new outpouring of "circle" puzzles I of course looked into exactly what was happening. If we define pieces not by their location or shape, but by which rotations move them we get perhaps the best definition of pieces on a twisty puzzle. For simplicity of desribing these rotations, I typical highlight the faces on the core shape of the puzzle that rotates a given piece. For example, an edge on a typical Rubik's cube could be labelled as the piece that comes from a cube with two adjacent faces highlighted. Notice that any one type of piece will always produce the same pattern of highlighting on the core shape. Another neat pattern that emerges (that is quite obvious once you think about it) is that the number of pieces of a specific type is always the number of orientations of the core shape divided by the number of rotational symmetries of the defining pattern. The only exception to this rule isn't really an exception at all, and that's when the pattern does not have reflectional symmetry. In this case there are double the number of pieces as the number of orientations of the core shape. But this makes perfect sense if you consider the reflected pattern and original pattern to be seperate (which makes sense anyway as the pieces that have this property always come in two seperate orbitals anyway) For puzzles with more than one layer, we must have a double highlighting scheme. This still gives each unique piece in a 5x5 for example (including the interior 3x3 pieces!) a unique defining pattern. I should also probably remind everyone that as always I consider deepcut puzzles to not exist. Instead they are just some pieces of the order above it (2x2x2 is is really just a 3x3x3 without edges/centers). Also, again, tetrahedral puzzles can cause problems so I almost always define them as octahedral (so skewb is really a Master Skewb with some pieces missing)

Okay, now that I've explained that concept, I can explain how I look at the "circle" puzzles. Let's start with a typical 3x3x3 circle cube. Beginning with an edge piece within the circle, I check what moves will change one of these pieces (I pick one in particular and focus only on it). For each, only one move actually manipulates the piece, so its defining pattern is just one face of a cube highlighted - note that this is the same as the defining pattern for a standard rubik's cube center piece. Pieces with the exact same defining pattern behave exactly the same way, so I conclude these edge-circle pieces behave like rubik's cube centers, specifically the center next to the face its located on... which is kind of bizarre, but it is what it is. Repeating the process for one of the corner pieces inside the circle I discover it has the same defining pattern as an edge (again one that is located in a rather bizarre location relative to the piece). Repeating this process for many other circle puzzles, I discover a nice pattern that actually makes sense in hindsight: pieces within a circle have the same defining pattern as the piece it would have otherwise been MINUS the face of the circle it is in (note that not all the "circled" pieces in the 2x2x2 circle cube have the same defining pattern, so it's a little tricky and doesn't quite follow this pattern, but that's ok cause I don't consider it a TRUE circle puzzle anyway). For the crazy 4x4 cubes only the outside face is removed from the defining pattern, giving the version 1 circle pieces the same defining pattern as the inner 2x2 pieces, hence how you solve them.

Now so far all of this seems like a way to make something more complicated than this really is because on most of the circle puzzles in existence, every "circled" piece has the same defining pattern as another piece that is already in the mathematical puzzle (you kinda have to imagine it in the crazy 4x4x4's case, but you get the idea...). BUT take a look at this puzzle (this is the first one I discovered this occurence on, but gelatinbrain has since added many circle puzzles that have this same property). The defining pattern for the circled edge pieces is easy enough to deduce: the normal edge piece's defining pattern is two adjacent faces of an octahedron so the circled edge piece is this pattern minus one face, giving the defining pattern of the "center" piece that exists inside the puzzle (same as the center triangles of a rainbow cube or a dino octa). But BAD things happen when we look at the other circled piece. The normal version of this piece has a defining pattern of three adjacent faces of an octahedron that form a letter C. The defining pattern for the circled version of this piece removes the middle face from this C so we end up with a defining pattern of two faces that share only a point. NO OTHER PIECE ON THE PUZZLE HAS THIS DEFINING PATTERN!!! So, I tried to deduce where this piece was "naturally" located. If we extend all the faces of an octahedron into planes, this piece should be located outside of the two planes that came from the faces in its defining pattern, but inside the other 6 relative to the core. A quick 3d drawing will show that there is no such region in space (unless it has 0 volume). WTH? (censored myself a little there :wink: )

So I learned that not every piece in a circled puzzle can be mapped to a piece in the natural form of the same puzzle, but I was still determined to come up with a way of explaining it. I decided that these "imaginary pieces" (what you call virtual) MUST exist mathematically, even if they don't show up in the actual realization of the natural puzzle (just like a parabola ALWAYS has two roots, but they might be imaginary). With this new idea, I looked at just how many pieces are possible with different core structures, that is, how many unique defining patterns exist for a single core shape. This amounted to finding how many unique combinations of sets of faces from a core shape I could come up with. It turns out that a cubical core can produce 10 different pieces, including a trivial piece that no faces turn (defining pattern: 0 highlighted faces -this is essentially the mathematical core of the puzzle and is real for all non-deepcut puzzles though barely worth considering). Only 3 of these are "real" and appear on an actual rubik's cube. When Drew made his Rex Cube long ago, I experimented with curving the cuts of a cube inward and discovered one more of these pieces at the edge when two cuts from opposite faces crossed eachother and made an entirely different edge on this puzzle: Image (if you're still following what I mean here, try turning the middle layer of this puzzle to really blow your mind up!). 5 of these imaginary pieces can be moved by two opposite faces on a rubik's cube that just seems really bizarre, and the 10th piece (of which there is only one!) moves every single time you turn a face. (I am in the middle of writing a program to simulate this puzzle but am having some extreme difficulty determining the easiest way to display all the different pieces.) This opened me up to worlds of new pieces I had never considered before. This more than doubles the number of pieces on EVERY single puzzle (except tetrahedral ones where it only adds one, but again I find it easier to consider all tetrahedral puzzles as the octahedral puzzle one order up minus some pieces, in which case it does). For bonus points someone can figure out how many hundreds of pieces this gives on a rhombic tricontahedral cored puzzle :twisted:

If you've actually read up to here, (Carl, I'm expecting you to be taking notes and already have questions by now! :lol: ) I can finally get to the point of this post, and that was to answer Carl’s questions above:
wwwmwww wrote:
(1) Can these virtual pieces always be made physical? I think the 2 in the Skewb can be and I'll show you how soon. Andreas has already told you above.
(2) If we can add these pieces to the puzzle what do they add to the puzzle solving experience? Anything?

What you have found is a few of these imaginary pieces. For the skewb, notice that you are treating every corner equally so you are therefore treating it as an octahedral puzzle, not a tetrahedral puzzle. However you are limiting yourself to only imaginary pieces that have the same number of faces highlighted in the defining pattern. I hope you have followed everything I've said so far, as I know this is all very strange and I'm not always perfectly clear, but anyways I CAN answer your questions:

1)This is not an easy question to answer. What I can tell you is that if we limit ourselves only to using traditional planar cuts and no circles or anything like that, this answer is a resounding NO. I can also tell you that there is no clean way of making any simulation that includes ALL possible pieces of a puzzle, both real and imaginary simultaneously for the same reason a Venn diagram with more than 3 circles cannot cleanly show all possibilities (some lemma of the 4 color theorem I'm sure, but try it, any workable answer must either break up the circles into multiple parts or double some regions or something messy like that).
I can also tell you that because of the strange tetrahedral/octahedral relationship we can make a traditional puzzle that has some of these imaginary pieces, such as this one:
Image the piece highlighted in green has the same defining pattern as the piece I pointed out on the Circle FTO, but only half of them are present on this puzzle (again tetrahedral/octahedral problems). However most imaginary pieces cannot ever be made real without doing something drastic, such as curving cutting planes in a way that brings the imaginary pieces into existence or adding circles that change the behavior of some pieces. Also the more faces that are highlighted in a defining pattern for a piece (past half the number of faces in the core), the more difficult it is to come up with a reasonable mathematical design that includes any spot for the piece at all. It seems downright impossible to cleanly display the final piece for every core that can be moved by all faces - but this could easily be displayed as a semitransparent piece over the whole puzzle or something like that. I will have to keep looking into more tricks to bring more imaginary pieces into real life, but for now I cannot say yes cleanly as there are still MANY pieces that are eluding all possible clean ways to create them.

2)Absolutely! Every new type of imaginary piece is just the same as every other piece that can be added when deepening the cuts for example. In fact, since most imaginary pieces have more faces in their defining pattern than real pieces, they should get much harder to solve than real pieces (however there is a chance that a backwards effect will begin to happen as you get more and more faces on a defining pattern that might allow for tricks to solving some of the pieces with the greatest number of faces in their defining pattern easier, but I will talk more on this when I understand it more)

Well there ya go! 8-)

That's a BRIEF summary of what I have been working on for the past like 5 months or so :lol:

I'll check back soon to answer all the questions I KNOW I'm gonna get!

Peace,
Matt Galla

PS I don't think that cubical puzzle with the curved cuts is possible to physically build... I'll explain why later :)

PPS and I knew I was gonna get that left field thing wrong. I can never remember the phrase.... :lol:


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sat Dec 19, 2009 9:06 pm 
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Allagem wrote:
And if you insist I will post it here. WARNING THIS WILL BE LONG!!


I'm fine with long. Here are some questions that jumped out at me on the first pass through this…

Allagem wrote:
I typical highlight the faces on the core shape of the puzzle that rotates a given piece.


What if it’s a corner or edge turn puzzle? Ahhh… I see you say “core” shape so the core of a helicopter cube would be a Rhombic Dodecahedron. Correct?

Allagem wrote:
I should also probably remind everyone that as always I consider deepcut puzzles to not exist. Instead they are just some pieces of the order above it (2x2x2 is is really just a 3x3x3 without edges/centers).


Why? And is this needed? You could continue this logic and say a 3x3x3 is just some of the pieces of a 5x5x5.

Allagem wrote:
Let's start with a typical 3x3x3 circle cube. Beginning with an edge piece within the circle, I check what moves will change one of these pieces (I pick one in particular and focus only on it). For each, only one move actually manipulates the piece


We aren’t counting slice moves at this point… correct? I think that makes sense in context as you are talking about face rotations above.

Allagem wrote:
So, I tried to deduce where this piece was "naturally" located. If we extend all the faces of an octahedron into planes, this piece should be located outside of the two planes that came from the faces in its defining pattern, but inside the other 6 relative to the core. A quick 3d drawing will show that there is no such region in space (unless it has 0 volume). WTH?


I was about to ask you to define what the inside and the outside of a plane was but I think I see it. The outside is the part that is rotated and the inside is the part that is stationary when a twist is made on that plane. Correct?

I think I like where this is going…


Allagem wrote:
It turns out that a cubical core can produce 10 different pieces, including a trivial piece that no faces turn (defining pattern: 0 highlighted faces -this is essentially the mathematical core of the puzzle and is real for all non-deepcut puzzles though barely worth considering). Only 3 of these are "real" and appear on an actual rubik's cube.

4 if you count the core… which I consider real. Correct? And if your imaginary pieces are our virtual pieces why didn’t we find 10 piece types in the 3x3x3?

Allagem wrote:
(if you're still following what I mean here, try turning the middle layer of this puzzle to really blow your mind up!).


The middle layer is just those 4 edges that aren’t in the top or bottom later… correct? That seemed easy enough.

Allagem wrote:
5 of these imaginary pieces can be moved by two opposite faces on a rubik's cube that just seems really bizarre, and the 10th piece (of which there is only one!) moves every single time you turn a face. (I am in the middle of writing a program to simulate this puzzle but am having some extreme difficulty determining the easiest way to display all the different pieces.)


Now that is something I’d love to see. Once I get my head around all 10 pieces I might be able to help. Let’s see… the normal 3x3x3 has 4 types covered. This…

Image

Has normal 3x3x3 corners…. But the following are new:
(1) The face centers
(2) The T-centers
(3) The edges
(4) The core which I believe rotates with all faces.

So an exploded view of the 3x3x3 and this puzzle cover 8 types… I think. There may be some interior pieces in picture above that I’m missing.

By the way… I’m assuming a rotation of the top turns the pieces with an x on the front (seen on the left).
Image
I just realized the picture on the right is also a possibility.

Allagem wrote:
This opened me up to worlds of new pieces I had never considered before. This more than doubles the number of pieces on EVERY single puzzle (except tetrahedral ones where it only adds one, but again I find it easier to consider all tetrahedral puzzles as the octahedral puzzle one order up minus some pieces, in which case it does).


Does… more then double the number of pieces? I was asking myself does what? At this point on the first several passes. And the new piece in the tetrahedral puzzles is the one that moves with all faces… correct? Just making sure I’m following.

Allagem wrote:
For bonus points someone can figure out how many hundreds of pieces this gives on a rhombic tricontahedral cored puzzle :twisted:


Andreas has done this for the virtual pieces but I think if the two are related than our virtual pieces are just a subset of your imaginary pieces.

Allagem wrote:
If you've actually read up to here, (Carl, I'm expecting you to be taking notes and already have questions by now! :lol: )


You know me well. I’m still here and eating this stuff up.

Allagem wrote:
What you have found is a few of these imaginary pieces. For the skewb, notice that you are treating every corner equally so you are therefore treating it as an octahedral puzzle, not a tetrahedral puzzle. However you are limiting yourself to only imaginary pieces that have the same number of faces highlighted in the defining pattern.


Let’s check that. A Skewb’s corners should have 4 faces highlighted on an octahedron. The Skewb’s faces also have 4. And checking my animation of the 2 virtual pieces… and yes they have 4 as well. Interesting… not sure what that means… but interesting.

Allagem wrote:
I hope you have followed everything I've said so far, as I know this is all very strange and I'm not always perfectly clear


You haven't lost me yet.

Allagem wrote:
1)This is not an easy question to answer. What I can tell you is that if we limit ourselves only to using traditional planar cuts and no circles or anything like that, this answer is a resounding NO. I can also tell you that there is no clean way of making any simulation that includes ALL possible pieces of a puzzle, both real and imaginary simultaneously for the same reason a Venn diagram with more than 3 circles cannot cleanly show all possibilities (some lemma of the 4 color theorem I'm sure, but try it, any workable answer must either break up the circles into multiple parts or double some regions or something messy like that).


What about just looking at real and the virtual pieces (a subset of the imaginary pieces)?

Allagem wrote:
I will have to keep looking into more tricks to bring more imaginary pieces into real life, but for now I cannot say yes cleanly as there are still MANY pieces that are eluding all possible clean ways to create them.


Hmmm… looking at this puzzle a bit more:
Image

The puzzle on the right is a 3x3x3 with funny new edges on top of the normal edges which now look like T-centers. So it has the following piece types:

(1) Core – normal 0 (colored faces)
(2) Face centers – normal (1 colored face)
(3) Edges – normal (2 adjacent colored faces)
(4) Corners – normal (3 colored faces all adjacent to each other)
(5) Funny Edges – new (4 colored faces with the 2 non-colored faces adjacent)

Now let’s assume we have the puzzle on the left. There we see…

(A) Core – new (6 colored faces)
(B) Face centers – new (5 colored faces)
(C) T-centers – new (4 colored faces with the 2 non-colored faces adjacent)
(D) Corners – normal (3 colored faces all adjacent to each other)
(E) Funny edges – new (2 adjacent colored faces)

But… here I notice something. The puzzle on the left and the puzzle on the right ARE the same puzzle. A top turn on one is exactly the same thing as a bottom turn on the other. This means…

(1) = (A) Core
(2) = (B) Face Center
(3) = (C) = (5) = (E) Edge
(4) = (D) Corner

So this puzzle really is just a 3x3x3 and those funny edges aren’t so funny. They are always opposite the normal edge on the opposite side of the puzzle so once you’ve solved the normal edges these piece should also be solved. What we are still missing is the piece with 3 colored faces, two of which are opposite each other; the piece with 4 colored faces with the 2 non-colored faces opposite each other; and the piece with 2 colored faces opposite each other. I suspect the last two are the same for the same reason the piece with 6 colored faces equaled the piece with zero colored faces. I think this reduces your 10 types to 6. Four are in a normal 3x3x3 and I don’t have a picture in my head for the other two yet. Since the 3x3x3 doesn’t have any virtual pieces I’m hoping those 2 reduce to one of the 4 we already have. In that case the virtual pieces may not be a subset of the imaginary pieces after all and they may in fact be the same.

Allagem wrote:
2)Absolutely! Every new type of imaginary piece is just the same as every other piece that can be added when deepening the cuts for example. In fact, since most imaginary pieces have more faces in their defining pattern than real pieces, they should get much harder to solve than real pieces (however there is a chance that a backwards effect will begin to happen as you get more and more faces on a defining pattern that might allow for tricks to solving some of the pieces with the greatest number of faces in their defining pattern easier, but I will talk more on this when I understand it more)


See above… I’m still not sure ANY of these pieces add anything to the difficulty of solving a normal 3x3x3. The new type of edge you introduced is solved when the normal edges are solved. As for the other pieces in the picture above you can play with them on your normal 3x3x3. How? Well for each face turn, turn the slice layer next to it with it. Doing that alone gives you 7 of your 10 piece types and it adds nothing to the 3x3x3. Still not sure about the other three (which I think are 2 at the most) or even if they can have a physical representation.

Allagem wrote:
Well there ya go! 8-)

THANK YOU. Now this is what I call fun. Yes… I’m odd.

Allagem wrote:
That's a BRIEF summary of what I have been working on for the past like 5 months or so :lol:


And in a couple of hours I’m caught up. ;)

Allagem wrote:
PS I don't think that cubical puzzle with the curved cuts is possible to physically build... I'll explain why later :)


It already has been… it’s your normal 3x3x3 with this mapping…
(using this notation http://www.randelshofer.ch/cube/notations/combined_eng_3x3.html)

R becomes TR
L becomes TL
U becomes TU
D becomes TD
F becomes TF
B becomes TB

Allagem wrote:
PPS and I knew I was gonna get that left field thing wrong. I can never remember the phrase.... :lol:


Left and Right both exist in real space… maybe we are off in an imaginary field. LOL!

Did I live up to expectations?
Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sat Dec 19, 2009 10:16 pm 
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wwwmwww wrote:
So this puzzle really is just a 3x3x3 and those funny edges aren’t so funny. They are always opposite the normal edge on the opposite side of the puzzle so once you’ve solved the normal edges these piece should also be solved.


I think I’ve over simplified the relationship between the two sets of edges. And it’s proving harder to work out then I thought but I still think you can’t solve one without solving the other. Say you have two solved 3x3x3’s at hand. Call one cube 1 and the other cube 2.

Everytime you make a move (x) on cube 1 where x=L, R, U, D, F, or B also make the move T(x) on cube 2. Do this for scrambling the cubes. Now solve cube 1 as you would normally but again mapping the moves to cube 2 using the same mapping used during scrambling. Is cube 2 solved? I think so… but I can’t prove it at the moment.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 20, 2009 2:02 am 
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In general if you cut space into N planes there's a formula for the maximum number of regions they can divide space into and the practical number if they all go through a single point, as is the case in a deep cut puzzle, is less.

Consider dividing a plane with lines. Every time you add a new line it has exactly one point of intersection with each other line (assuming everything is 'in general position', which basically means nothing lines up precisely) and the number of new regions will be the number of points of intersection plus one, so the first line add one region (going from one to two, because the plane is one region when it's undivided), the next one will add two, then three, etc. Even though in principle the number of regions could be 2^N for N lines, the only way for that to happen is for the 'lines' to wriggle enough that two of them can have more than one point of intersection.

Now consider dividing space with planes. Each plane will be criss-crossed with lines of intersection with the other planes, and the number of regions of space each plane adds is equal to the number of regions the lines of intersection on it separate it into. So the first plane adds a number of regions equal to the number of regions if you divide a plane with zero lines, the next one if you divide a plane with one line, then two, three, etc. This is again much less than 2^N, in fact it's just cubic. To get above this number of regions, you'd have to make your 'planes' undulate sufficiently that two of them intersect in something other than a single line. Making them conical-ish so the intersections are a closed loop will increase the number of regions, but still not bring it anywhere near 2^N.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 20, 2009 8:06 am 
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Allagem wrote:
(note that not all the "circled" pieces in the 2x2x2 circle cube have the same defining pattern, so it's a little tricky and doesn't quite follow this pattern, but that's ok cause I don't consider it a TRUE circle puzzle anyway)
That can be solved if you accept deepcut puzzles... :)
Frank Tiex will attest, that on the real Circle2x2x2 he owns the circle pieces of one corner behave as glued to that corner.

wwwmwww wrote:
4 if you count the core… which I consider real. Correct? And if your imaginary pieces are our virtual pieces why didn’t we find 10 piece types in the 3x3x3?

Allagem uses a different approach:
1. He ignores slice moves
2. He considers only faceturns. I wonder what he does with puzzles like the 5x5x5
3. He asks himself the question "Which faceturns move that specific piece?"
Our approach asks: "Which turns (including slices) doesn't affect the piece?"

The 10 defining patterns he means are:
UDLRFB => inverted core; 1 moved in every turn
UDLRF => inverted faces; 5 moved per turn
UDLR => not yet explained; inverted version of UD
UDLF => inverted edges; 8 moved per turn
ULF => normal corners; 4 moved per turn; impossible to be inverted
UDL => not yet explained; impossible to be inverted
UD => not yet explained;
UL => normal edges; 4 moved per turn
U => normal faces; 1 moved per turn
[] => normal core; 0 moved per turn

As Carl discovered the problem with Allagem's non-planar cut puzzle is that there are both types of edges present, the normal ones and the inverted edges. The other seemingly strange pieces could be defined away by redefining what is a turn on this puzzle.

wwwmwww wrote:
Andreas has done this for the virtual pieces but I think if the two are related than our virtual pieces are just a subset of your imaginary pieces.
I did something similar but not exactly that. A list of all of Allagem's designing pattern would have at least 8947849 entries for the Rhombic Triacontahedron and that is just a lower bound.

Seems like Allagem has found a superset of virtual pieces.
But the "CircledCorner" of Gelatibrain 4.1.11 (identical to the green piece in Allagem's offset-octahedron) is included our set of virtual pieces of that class.
The anti-edges are so far the only piece I have seen which can't explained by our subset of virtual pieces. The world of planar cuts is easier than thought.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 20, 2009 8:25 am 
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Bram wrote:
Now consider dividing space with planes. Each plane will be criss-crossed with lines of intersection with the other planes, and the number of regions of space each plane adds is equal to the number of regions the lines of intersection on it separate it into. So the first plane adds a number of regions equal to the number of regions if you divide a plane with zero lines, the next one if you divide a plane with one line, then two, three, etc. This is again much less than 2^N, in fact it's just cubic. To get above this number of regions, you'd have to make your 'planes' undulate sufficiently that two of them intersect in something other than a single line. Making them conical-ish so the intersections are a closed loop will increase the number of regions, but still not bring it anywhere near 2^N.


Let's consider a general set of N-1 planes. Fine the line perpendicular to each plane that runs through the origin. Call the distance along this line from each plane to the origin d(1), d(2), d(3)... d(N-1). Now if we add a Nth plane let's make it's d(N) >> then all the others. If we allow this Nth plane to move along it's perpendicular line through the origin and an equal distance out the other side we can imagine new regions being created and destroyed as it moves.

For example, consider a tetrahedron formed by 4 planes. Now add a 5th plane and allow it to move through the tetrahedron. At first its just one region, let's call it region 1. As soon at the 5th plane enters this region it creates a new region, let's call it region 2. And as the 5th plane leaves the opposite side of the tetrahedron region 1 is destroyed and the entire volume of the tetrahedron is now occupied by region 2.

Now if we go back to the general case and we allow all N planes to move, I think you can find all 2^N regions. Sure, you can't make a physical puzzle out of something with moving cut planes but you may be able to make an applet that allows you to play with one.

Carl

P.S. I don't think you even need to allow all N planes to move. I can find all 16 regions created by 4 planes by just moving 1 plane.

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 20, 2009 9:24 am 
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Andreas Nortmann wrote:
That can be solved if you accept deepcut puzzles... :)
Frank Tiex will attest, that on the real Circle2x2x2 he owns the circle pieces of one corner behave as glued to that corner.

The most work I've done looking at Circle Cubes was done with the Crazy 4x4x4's. From there I see how a Circle 3x3x3 works and I've tried to carry it to higher orders but I have to admit I haven't looked at a Circle 2x2x2 yet. Is Frank's Circle 2x2x2 different from other Circle 2x2x2's? If so how do the other Circle 2x2x2's function? I feel like I'm missing something. What exactly does accepting deepcut puzzles fix?
Andreas Nortmann wrote:
Allagem uses a different approach:
1. He ignores slice moves
2. He considers only faceturns. I wonder what he does with puzzles like the 5x5x5

I think he assigns 2 colors to each face of the core. If that face isn't colored it's in the slice layer. If it just has one color it's only turned by the deeper cut face turn. If it has 2 colors it's turned by both the deeper cut and the shallower cut on that face. At least that is what I think Matt is doing, feel free to correct me if I'm wrong.
Andreas Nortmann wrote:
3. He asks himself the question "Which faceturns move that specific piece?"
Our approach asks: "Which turns (including slices) doesn't affect the piece?"

Yes that is it in a nut shell.
Andreas Nortmann wrote:
The 10 defining patterns he means are:
( 1) UDLRFB => inverted core; 1 moved in every turn
( 2) UDLRF => inverted faces; 5 moved per turn
( 3) UDLR => not yet explained; inverted version of UD
( 4) UDLF => inverted edges; 8 moved per turn
( 5) ULF => normal corners; 4 moved per turn; impossible to be inverted
( 6) UDL => not yet explained; impossible to be inverted
( 7) UD => not yet explained;
( 8) UL => normal edges; 4 moved per turn
( 9) U => normal faces; 1 moved per turn
(10) [] => normal core; 0 moved per turn

I’ve numbered this list above. Here is what I think.

(1) = (10) Both can be a valid interpretation of a normal 3x3x3 core.
(2) = (9) Both can be a valid interpretation of a normal 3x3x3 face center.
(3) = (7) Not sure about this one as I’m not sure what this piece would look like yet.
(4) = (8) Both can be a valid interpretation of a normal 3x3x3 edge piece.
(5) is your normal 3x3x3 corner.
(6) is… I have no idea.

Now the question I’m struggling with to answer… is this. A puzzle can be constructed, at least in applet form, that has 24 edges [12 of type (4) and 12 of type (8)]. But I think that is redundant information. If one set of 12 is solved I believe the other set of 12 will be solved too. How can I prove (or disprove) that?
Andreas Nortmann wrote:
As Carl discovered the problem with Allagem's non-planar cut puzzle is that there are both types of edges present, the normal ones and the inverted edges. The other seemingly strange pieces could be defined away by redefining what is a turn on this puzzle.

Also notice that redefining what is a turn on this puzzle turns what you are calling the normal edges into inverted edges and vice versa.
Andreas Nortmann wrote:
Seems like Allagem has found a superset of virtual pieces.

I’m not sure. I think they may still be one in the same with some redundancies in Matt’s set. I think we need to understand pieces of types (3), (6), and (7) better to be sure. And we need to prove the set of type (4) pieces is (or is not) just a redundant copy of the type (8) pieces before I’ll be sure one way or the other.
Andreas Nortmann wrote:
But the "CircledCorner" of Gelatibrain 4.1.11 (identical to the green piece in Allagem's offset-octahedron) is included in our set of virtual pieces of that class.

Agreed.
Andreas Nortmann wrote:
The anti-edges are so far the only piece I have seen which can't explained by our subset of virtual pieces. The world of planar cuts is easier than thought.

By anti-edges I assume you mean what you called inverted edges above. In that case I think they are covered in the set of real+virtal pieces. Just take a normal 3x3x3 and limit it to turns of the form T(x) where x=R, L, U, D, F, and B and T is defined here:

http://www.randelshofer.ch/cube/notations/combined_eng_3x3.html

As such I think those pieces are equivalent to normal edges and would therefore be considered “real”.

And by only piece you have seen… I assume you aren’t counting types (3), (6), and (7). Correct?

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 20, 2009 2:04 pm 
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wwwmwww wrote:
The most work I've done looking at Circle Cubes was done with the Crazy 4x4x4's. From there I see how a Circle 3x3x3 works and I've tried to carry it to higher orders but I have to admit I haven't looked at a Circle 2x2x2 yet. Is Frank's Circle 2x2x2 different from other Circle 2x2x2's? If so how do the other Circle 2x2x2's function? I feel like I'm missing something. What exactly does accepting deepcut puzzles fix?
Franks Circle 2x2x2 is the only one I had the pleasure to put my hands on. If Matt Galla aka Allgem accepts deepcut puzzles he can introduce the Circle2x2x2 into his system without problem.
wwwmwww wrote:
At least that is what I think Matt is doing, feel free to correct me if I'm wrong.
He is the one who has to do that. If he did, then I apologize for overreading it.
wwwmwww wrote:
Andreas Nortmann wrote:
The 10 defining patterns he means are:
( 1) UDLRFB => inverted core; 1 moved in every turn
( 2) UDLRF => inverted faces; 5 moved per turn
( 3) UDLR => not yet explained; inverted version of UD
( 4) UDLF => inverted edges; 8 moved per turn
( 5) ULF => normal corners; 4 moved per turn; impossible to be inverted
( 6) UDL => not yet explained; impossible to be inverted
( 7) UD => not yet explained;
( 8) UL => normal edges; 4 moved per turn
( 9) U => normal faces; 1 moved per turn
(10) [] => normal core; 0 moved per turn

I’ve numbered this list above. Here is what I think.
(1) = (10) Both can be a valid interpretations of a normal 3x3x3 core.
(2) = (9) Both can be a valid interpretation of a normal 3x3x3 face center.
(3) = (7) Not sure about this one as I’m not sure what this piece would look like yet.
(4) = (8) Both can be a valid interpretation of a normal 3x3x3 edge piece.
(5) is your normal 3x3x3 corner.
(6) is… I have no idea.
They (1,2,4,5 and their invertations) can be interpreted like that. Allagem (if I understood him right) does this interpretation in a specific way. And as you have recognized by yourself, switching (or better: inverting) this interpretation inverts the pieces if possible.
wwwmwww wrote:
Now the question I’m struggling with to answer… is this. A puzzle can be constructed, at least in applet form, that has 24 edges [12 of type (4) and 12 of type (8)]. But I think that is redundant information. If one set of 12 is solved I believe the other set of 12 will be solved too. How can I prove (or disprove) that?
No idea yet. It would be nice to get rid of the inverted pieces, so we can stay with out reduced set of virtual pieces. No pun intended, Matt.
wwwmwww wrote:
And by only piece you have seen… I assume you aren’t counting types (3), (6), and (7). Correct?
Correct. There hasn't been any applet, where this pieces were visible.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 20, 2009 6:06 pm 
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Hey guys, it looks like you've done some excellent thinking here.

Andreas, you nailed the list I was thinking of. Mine was in a slightly different order, but I will just use yours for simplicity.

I would like to point out that (4) does NOT equal (8) I'm not sure about the rest of the pieces, but I can give you an easy enough way to check: I found an easy way to display all pieces, except (6), well at least relatively easy compared to writing a simulation and finding locations for all of the pieces :lol:

Have 3 Rubik's Cubes all side by side with the same starting orientation. One of them behaves as a normal Rubik's cube (only face moves are allowed, no slice moves). The moves you make on it get transferred to the other 2 cubes, but in a specific way. The second cube I will call a double cube. For every move you make on the normal cube, make the same move on the double cube except with two layers (for simplicity's sake never make a two-layered move on the normal cube as this gets confusing). The third cube I will call the slice cube. For every move you make on the normal cube, make the same move on the slice cube except one layer inwards (so only the middle slices ever move). Considering all 3 cubes simultaneously, we have a representation for 9 of the 10 pieces (note that in order to get the full representation of these 9 pieces all 3 cubes must be supercubes).
In order of the core piece, center, edge, then corner, each of the three puzzles has the following pieces:
normal: 10, 9, 8, 5
double: 1, 2, 4, 5
slice: 1, 3, 7, 10

Of course in practicality, we can't really observe the core very well, but (I'll explain in a second) we do know that just as piece type 9 can never be permutated relative to piece type 10, piece type 2 can never be permutated relative to piece type 1, therefore it is true to say if all pieces of type 2 are solved, piece type 1 is also solved. Therefore these 3 puzzles are enough to simulate 9 of the 10 pieces. Piece type (6) is proving very difficult to simulate.....

Also, your analysis of some of the pieces is falling into a trap. You (both?) state that piece type 4 moves exactly like piece type 8 when you consider every move to actually be moving the rest of the puzzle the opposite direction. Ok I know that explanation failed, but it makes sense in my mind :) This is true, but now you've flipped the definitions and piece 8 no longer moves like an edge :lol: .
You have to consider both simultaneously. You have correctly realized that piece 4's movement is RELATIVE to piece 1 in the same manner that piece 8's movement is RELATIVE to piece 10. Does that make sense?

BTW what Carl said defining the two cubes (really similair to what I did above as well... yet again!! :lol: ) is true:
wwwmwww wrote:
I think I’ve over simplified the relationship between the two sets of edges. And it’s proving harder to work out then I thought but I still think you can’t solve one without solving the other. Say you have two solved 3x3x3’s at hand. Call one cube 1 and the other cube 2.

Everytime you make a move (x) on cube 1 where x=L, R, U, D, F, or B also make the move T(x) on cube 2. Do this for scrambling the cubes. Now solve cube 1 as you would normally but again mapping the moves to cube 2 using the same mapping used during scrambling. Is cube 2 solved? I think so… but I can’t prove it at the moment.
Try it out, Cube 2 is NOT necessarily solved :wink:

I don't have the time to try out the rest of the pieces, (I'm really busy as usual :roll: ) but I think you guys can figure it out for sure. What I can say is that certain pieces are related in terms of how their movement is restricted. It is very similar to the corners of a skewb. Upon close inspection we find that 4 of the corners cannot be permutated relative to one another and the other 4 corners cannot be permutated relative to eachother either. But the two systems can be permutated relative to eachother. These two systems are analagous to the the following pairs of pieces: (1,10) , (2,9) , (3,7) , and (4,8). Pieces 5 and 6 have no real opposites, and they are not opposite eachother by any means.

I also tested out a complete solve and complete scramble for a normal Rubik's Cube applied to piece type 2 and they came out solved (but I couldn't see orientation, so maybe not) I am not yet sure if this is a coincidence or caused by some mathematical relationship I haven't had enough time to sit down and try to work out. But the same process did not solve piece type 4!!!!!!

I'll let you guys start with that as I need to go :wink: :lol:
Peace,
Matt Galla

PS As Carl correctly discovered with my 3x3x3 cube with curved cuts, every first ordered puzzle can be thought of now in two ways. Basically using either single layered rotations or double layered rotations. But which one is which is not neccesarily well defined. We should probably use the convention that the mathematical core of the puzzle (physically located in the real center) behaves as a core, that is, it does not move ever using the moveset defined for the puzzle. In this case a move on my curvey-cut 3x3x3 is the second interpretation.

PPS Sorry, catching things on the fly here. Yes for puzzles with a second order of rotations (by my definitions this means 4x4x4 or 5x5x5) pieces can be moved by either the first layer out, the second layer out, both, or neither for any given face of the defining pattern. So we need a way to cover all 4 scenarios. A simple enough way is to use two colors: 1 color means the first layer only, the other means the second layer out. If the colors are mixed (or checkerboarded or something like that) it means both layers and an unhighlighted face means neither. Or any equivalent, it doesn't really matter :P


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 20, 2009 6:49 pm 
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Allagem wrote:
Have 3 Rubik's Cubes all side by side with the same starting orientation. One of them behaves as a normal Rubik's cube (only face moves are allowed, no slice moves). The moves you make on it get transferred to the other 2 cubes, but in a specific way. The second cube I will call a double cube. For every move you make on the normal cube, make the same move on the double cube except with two layers (for simplicity's sake never make a two-layered move on the normal cube as this gets confusing). The third cube I will call the slice cube. For every move you make on the normal cube, make the same move on the slice cube except one layer inwards (so only the middle slices ever move). Considering all 3 cubes simultaneously, we have a representation for 9 of the 10 pieces (note that in order to get the full representation of these 9 pieces all 3 cubes must be supercubes).


This line of reasoning leads to an actual construction, which I think would work well.

The 'double cube' has ordinary corners, but its edges and face centers are in two layers - the outer being mostly normal shaped piece with a window into the inner piece, which is normal, but maybe rounded a bit. When you play with it the two layers mostly turn in tandem, acting as a regular rubik's cube, but there's an added move, where a slice can be done of just the outer layer pieces.

Coming up with a mechanism for this would be reasonably straightforward, at it's a novel concept. It would probably be about like a 5x5x5 to solve - a rubik's cube plus a bit more, unless you made the centers have orientations, which might make it quite a bit more difficult.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 3:28 am 
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Hey guys, I just did a sample scramble/solve on my simulation with three cubes as I defined above, and based on that I discovered that solving all the real pieces guarantees NOTHING about any of the other pieces. Piece 10 is automatically solved and I manually scrambled and solved pieces 9,8, and 5 on the normal Rubik's Cube, keeping track of all my moves on the other two cubes. In the end, NONE of the simulated imaginary pieces were solved (except one of type 7, which I think it is safe to be accounted to random chance). I should state that I did this by hand on three cubes with unmarked center orientation, so I'm assuming two things: 1. I never screwed up and spun one of the cubes a different way than the other 2 and 2. Center orientation does not affect anything between the relationship of the piece types. Both of which are LARGE assumptions. If anyone has three supercubes and wants to confirm this. Scramble and solve one and keep track of the moves on the other two according to the rules I posted above and let us know which pieces get solved on the other two puzzles if any. If any piece type does not get solved on any one attempt, this proves they are not directly related to the pieces on the normal cube. I'm still working on a decently nice way of displaying piece type 6. It seems like no matter what, some pieces will have to literally cut through one another in order to equally display all 12 of them (they should really exist somehow lined up with the centers, except there would be two type 6 pieces existing in each center spot. I guess the best way on a computer simulator is to elongate each in a different direction parallel to the face of the cube and literally have them slice through one another at 90 degrees. Using this method no two will never occupy the entire same spot.

Peace,
Matt Galla

PS Good to see you following this Bram. If someone can come up with a mechanism for something like this I have no doubt it will be you. However, don't oversimplify things too much. Remember that some of the piece types (4 for example) can be moved by two opposite faces from a single position. Not only does this have to be true on a physical puzzle, the physical mechanism must force this to happen. I cannot see how this is possible using any kind of 5x5x5 mech-mod.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 9:44 am 
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Well I know I've been thinking about this problem too much. The last post I read last night was Bram's and I couldn't think of anything to say as I knew I needed to think about this for a bit. I gave up and went to bed. However my mind didn't want to stop working on the problem apparently so when I woke up this morning it's like my brain taps me on the shoulder and says here’s what I came up with and suddenly there are a ton of new ideas that weren't there last night. So that sort of put me in the spot of messenger here I guess. My conscious brain still needs to sort through these and see if any of them make sense but here are some ideas in random.

(A) Matt’s notion of the normal 3x3x3, the double 3x3x3, and the slice 3x3x3 tells me that:
(1) and (10) can be a valid interpretation of a normal 3x3x3 core.
(2), (9), and (3) can be a valid interpretation of a normal 3x3x3 face center.
(4), (8), and (7) can be a valid interpretation of a normal 3x3x3 edge piece.
(5) and (10) can be a valid interpretation of a normal 3x3x3 corner.
I still can’t see (6)…

But this clearly breaks (3) = (7) and as Matt has seen it appears to break all the others assumptions I was making.

(B) So I’m still asking myself why is the set of imaginary pieces apparently bigger then the set of virtual pieces. Is it possible we aren’t dealing with a normal order=2 puzzle anymore? I’m thinking this may now be a degenerate higher order puzzle. Just looking at types 4 and 8 for a second… I’m thinking those may be both sets of 4x4x4 edges on a 3x3x3. Is that even possible? If so we may be back to the imaginary piece set and the virtual piece set being the same but just classified differently. By degenerate… think about this… a 5x5x5 has 4 cut planes per axis of rotation. We typically think of this as two twistable cuts per face. However Matt’s normal/double/slice 3x3x3 also has two twistable cuts per face… it’s just that those cuts are coincident with the cuts of the opposite face. Types 4, 7, and 8 may account for all the edges on a 5x5x5. So here is an idea.

Next to the normal 3x3x3 sit a normal 5x5x5. Limit yourself to face turns on these puzzles. Whenever you turn R on the 3x3x3, turn R on the 5x5x5 etc.

Next to the double 3x3x3 sit a double 5x5x5. On this set the R turn from the normal cubes becomes TR on the 3x3x3 and NR or M1R on the 5x5x5. Using the notation found here:
http://www.randelshofer.ch/cube/notations/combined_eng_5x5.html

Next to the slice 3x3x3 sit a slice 5x5x5. On this set the R turn from the normal cubes becomes MR on both the 3x3x3 and the 5x5x5.
http://www.randelshofer.ch/cube/notations/combined_eng_3x3.html

Doing this can we match up the 9 types represented with the following on the Multi5x5x5?
(a) Core
(b) Inner 3x3x3 Face Center
(c) Inner 3x3x3 Edge
(d) Inner 3x3x3 Corner
(e) 5x5x5 Face Center
(f) 5x5x5 T-Centers
(g) 5x5x5 X-Centers
(h) 5x5x5 Middle Edges
(i) 5x5x5 Wings

And note… the wings exist in two sets of 12 so we might be able to break them up and have all 10 types represented.

(C) Matt’s way of dealing with the 4x4x4 may not be adequate, I can think of this puzzle as having 3 layers that can be turned per face not just 2. Ok… my conscious brain is starting to catch up. We just dealt with the 3x3x3 with one color per face so it should be fine to deal with the 4x4x4 with just 2 colors. Forget this one…

(D) And yesterday I started some test rendering in POV-Ray to see if I could play with these weird cubes there. Well my first test render is still rendering… this is far too slow to actually play with these. So… again I woke up remembering something this morning. Check out my conversation with Waran and rawcoder that starts here:

http://twistypuzzles.com/forum/viewtopic.php?p=126892#p126892

Anyone here know enough to know how to put 3 of these:
http://www.randelshofer.ch/cube/professor/?
And 3 of these:
http://www.randelshofer.ch/cube/rubik/?
On the same page with some JavaScript as rawcoder mentions here:
http://twistypuzzles.com/forum/viewtopic.php?p=126987#p126987

I think it shouldn’t be too hard to make a page that does exactly what I describe above but I know next to nothing about applets and JavaScript. If this could be done it would be thousands of times faster then POV-Ray and much more user friendly. If someone can make the page I have a site I’d be happy to host it at.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 11:33 am 
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wwwmwww wrote:
(B) So I’m still asking myself why is the set of imaginary pieces apparently bigger then the set of virtual pieces. Is it possible we aren’t dealing with a normal order=2 puzzle anymore?

I would say, that Matts imaginary pieces are unnecessary if you restrict yourself to planar cuts. That was an implicit assumption within our analysis in the first post. And by doing so, we missed Matts imaginary pieces.
I would really like to see what my enumeration of 10 pieces above would look like if we switch over to the 5x5x5. I am still puzzled :) by Matts approach to not consider deepcut puzzles.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 11:47 am 
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Wow guys! I go off to take my finals and go to a few parties and come back to all this! :)

Very interesting. I'll try to read it all the way through later today.
However, I had an interesting development this morning in the area I've been working. I think we may be able to use maps that look something like metal alloy phase diagrams to categorize 2nd order (what Carl calls 4th order) puzzles, and I just did it for Cube V and there were not nearly as many "new puzzles" as I expected to find (that's good! :) )

Let me just say that Andreas's last post makes a very good point :)
I think when you're doing something like this, you have to concentrate on a manageable subset of puzzles at a time. However, it's easy to get sidetracked and step outside that subset once you get going :)

For me, I am still working down in the "lower dimension" of "surface puzzles", where the only pieces that "exist" are the ones that you can put a sticker on ;)
Feel free to call me a neanderthal now :p

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 12:21 pm 
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Allagem wrote:
PS Good to see you following this Bram. If someone can come up with a mechanism for something like this I have no doubt it will be you. However, don't oversimplify things too much. Remember that some of the piece types (4 for example) can be moved by two opposite faces from a single position. Not only does this have to be true on a physical puzzle, the physical mechanism must force this to happen. I cannot see how this is possible using any kind of 5x5x5 mech-mod.


I already suggested a novel and interesting mechanism for a puzzle based on the ideas in this thread. Since the usual number of puzzle ideas to come out of a discussion is zero, I think one is pretty good, especially for something which is completely actionable as far as building it is concerned.

Similar ideas can be applied to puck puzzles. For example, you can have several 12-piece puck puzzles next to each other, and they all flip the right half in tandem, but when you rotate the first one one unit clockwise it rotates the second one two, third one three, etc. I thought of this idea several years ago and have never come up with a mechanism for it which I particularly liked, but it is similar to Bram's Cube, which is a way of tying to unrelated puzzles together which actually works.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 12:52 pm 
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Andreas Nortmann wrote:
wwwmwww wrote:
(B) So I’m still asking myself why is the set of imaginary pieces apparently bigger then the set of virtual pieces. Is it possible we aren’t dealing with a normal order=2 puzzle anymore?

I would say, that Matts imaginary pieces are unnecessary if you restrict yourself to planar cuts. That was an implicit assumption within our analysis in the first post. And by doing so, we missed Matts imaginary pieces.


I’m not sure. Matt talks about faces on the core shape. These faces are all planar. And now that my conscious brain has had a chance to mull over the musings of my unconscious brain from last night I see something…

Take Matt’s concept of a normal/double/slice 3x3x3 and add ONE puzzle to the picture. Make it a 5x5x5. For each of these moves on the normal 3x3x3 perform the following on the 5x5x5.

L maps to LML
R maps to RMR
U maps to UMU
D maps to DMD
F maps to FMF
B maps to BMB

Using this:
The 10 defining patterns:
( 1) UDLRFB => inverted core; 1 moved in every turn
( 2) UDLRF => inverted faces; 5 moved per turn
( 3) UDLR => not yet explained; inverted version of UD
( 4) UDLF => inverted edges; 8 moved per turn
( 5) ULF => normal corners; 4 moved per turn; impossible to be inverted
( 6) UDL => not yet explained; impossible to be inverted
( 7) UD => not yet explained;
( 8) UL => normal edges; 4 moved per turn
( 9) U => normal faces; 1 moved per turn
(10) [] => normal core; 0 moved per turn

We see…

The central core is type (1)
The inner 3x3x3 corners are collectively one piece, type (10)
The inner 3x3x3 face centers are type (3)
The inner 3x3x3 edges are type (7)
The 5x5x5 face centers are type (2)
The 5x5x5 X-centers move in groups of 4 pieces at a time which are the type (9) pieces.
The 5x5x5 T-centers are our elusive type (6) pieces in groups of 2 pieces.
The 5x5x5 middle edges are our type (4) pieces
The 5x5x5 corners are our type (5) pieces
The 5x5x5 wings are our type (8) pieces again in groups of 2 pieces

I noticed my unconscious brain that posted above left off the 5x5x5 corners. So there we go… we have all 10 pieces in one puzzle. And I think I’d call this a degenerate 5x5x5 so I think we can call these order=4 virtual/real pieces or order=2 imaginary pieces. Maybe…

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 1:15 pm 
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Bram wrote:
I already suggested a novel and interesting mechanism for a puzzle based on the ideas in this thread.


You are talking about your first post above... correct? I need to sit down and read that one a few more times as I haven't got a picture of the mechanism in my head yet. Once I get it I'll see if I can render it... a picture goes a long ways.

Bram wrote:
Since the usual number of puzzle ideas to come out of a discussion is zero,


I'm not sure how to take that. The intent here was first to gain an understanding of what we were dealing with. Not to just produce a bunch of hot air. Though maybe I'm reading this the wrong way.

Bram wrote:
I think one is pretty good, especially for something which is completely actionable as far as building it is concerned.


On that much we agree. I'll be looking for it on Shapeways probably before I figure out how to render it. :) To be honest the most I expected here was to be able to to get the idea for several applets out in the public where someone that might know how to make them would see them. A real physical puzzle I can hold in my hands... WOW!!!

Bram wrote:
Similar ideas can be applied to puck puzzles. For example, you can have several 12-piece puck puzzles next to each other, and they all flip the right half in tandem, but when you rotate the first one one unit clockwise it rotates the second one two, third one three, etc. I thought of this idea several years ago and have never come up with a mechanism for it which I particularly liked, but it is similar to Bram's Cube, which is a way of tying to unrelated puzzles together which actually works.


Nice ideas... but I'm missing the connection to virtual or imaginary pieces. Maybe after I get the above mechanism figured out in my head I'll see the connection.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 2:02 pm 
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Bram wrote:
The 'double cube' has ordinary corners, but its edges and face centers are in two layers - the outer being mostly normal shaped piece with a window into the inner piece, which is normal, but maybe rounded a bit. When you play with it the two layers mostly turn in tandem, acting as a regular rubik's cube, but there's an added move, where a slice can be done of just the outer layer pieces.


Ok... not sure why I had to read this so many times but I think I see it now. You have a 3x3x3 which contains:

These piece types:
(10) Core... not seen.
(9) Face center fixed to core (this would be your inner layer face center)
(8) Edges held in place by the fixed face centers (this would be your inner layer edge)
(5) Your normal 3x3x3 corners
(2) These are your floating face centers. They don't need to be fixed to any face as each face turn can be thought of as a double layer turn of the opposite face. (this would be your outer layer face center with the window)
(4) These are the floating edges which you can think of as being held in place by the floating faces. More likely its the other way around. The floating edges I would assume run in tracks in the corners and they hold the floating faces. (this would be your outer layer edge with the window)

Now that I see it I'm not sure a picture would help too much. The assembled puzzle would look like a 3x3x3 with a window in each face center and each edge that allowed you to see the piece under it. I got an idea... off to render a picture.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 5:02 pm 
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wwwmwww wrote:
Take Matt’s concept of a normal/double/slice 3x3x3 and add ONE puzzle to the picture. Make it a 5x5x5. For each of these moves on the normal 3x3x3 perform the following on the 5x5x5.

L maps to LML
R maps to RMR
U maps to UMU
D maps to DMD
F maps to FMF
B maps to BMB

Using this:
The 10 defining patterns:
( 1) UDLRFB => inverted core; 1 moved in every turn
( 2) UDLRF => inverted faces; 5 moved per turn
( 3) UDLR => not yet explained; inverted version of UD
( 4) UDLF => inverted edges; 8 moved per turn
( 5) ULF => normal corners; 4 moved per turn; impossible to be inverted
( 6) UDL => not yet explained; impossible to be inverted
( 7) UD => not yet explained;
( 8) UL => normal edges; 4 moved per turn
( 9) U => normal faces; 1 moved per turn
(10) [] => normal core; 0 moved per turn

We see…

The central core is type (1)
The inner 3x3x3 corners are collectively one piece, type (10)
The inner 3x3x3 face centers are type (3)
The inner 3x3x3 edges are type (7)
The 5x5x5 face centers are type (2)
The 5x5x5 X-centers move in groups of 4 pieces at a time which are the type (9) pieces.
The 5x5x5 T-centers are our elusive type (6) pieces in groups of 2 pieces.
The 5x5x5 middle edges are our type (4) pieces
The 5x5x5 corners are our type (5) pieces
The 5x5x5 wings are our type (8) pieces again in groups of 2 pieces

I noticed my unconscious brain that posted above left off the 5x5x5 corners. So there we go… we have all 10 pieces in one puzzle. And I think I’d call this a degenerate 5x5x5 so I think we can call these order=4 virtual/real pieces or order=2 imaginary pieces. Maybe…

Carl
Bravo Carl! This works beautifully. How the heck did you come up with this???
You are absolutely correct. A 5x5x5 has 10 unique pieces (including the inner 3x3x3 pieces and inner core). Restrict moves to your moveset, where the only moves allowed are a single layer face move combined with a middle slice move underneath and you give all 10 types of pieces a different defining pattern, one of each of the possible defining patterns of a 3x3x3. Notice that since only one type of move is allowed in each direction from the core this is in one sense still defined as a 3x3x3 (that is there are only 6*3=18 moves available, 6 faces each of which can spin 90, 180, or 270 degrees)

I went ahead and simulated this in a program I've been working on and right off the bat noticed how incredibly confusing it was. The "core" piece (type 10) is not in the center as you would expect but the inner 3x3x3 corners, so it's like a core that has been sliced down the middle 3 ways. This trend continues through the rest of the real 3x3x3 pieces. The "center" pieces have been split down the middle twice and the edge pieces have been sliced down the middle once as well. Several other pieces have this odd effect, multiple pieces on this weird 5x5x5 map to the same imaginary 3x3x3 piece. So on the 5x5x5 these pieces always follow eachother. It's actually very counterintuitive.
Each type 10 piece is simulated by a group of 8 pieces (there's only one type 10 piece)
Each type 3 piece is simulated by a group of 2 pieces
Each type 7 piece is simulated by a group of 4 pieces
Each type 9 piece is simulated by a group of 4 pieces
Each type 6 piece is simulated by a group of 2 pieces
Each type 8 piece is simulated by a group of 2 pieces

So the 1+6+3+12+8+12+3+12+6+1=64 pieces of a "complex" 3x3x3 is simulated
by the 1+6+12+8+6+24+24+12+24+8=5^3=125 pieces of a super super 5x5x5.
Interseting......

BTW results of me solving a super 3x3x3 on this simulation: only one type of piece was solved outside types 5,8,9, and 10 and that was piece type 7. In hindsight this is rather obvious. There is only one way to spin piece type 9 and that only changes the orientation of the piece. Since this does not change what face spins the piece, we can simply look at the orientation of a center on a 3x3x3 to find out how many degrees the sum of the spins of the face its on makes during the solution. Example if a scrambled U face center is rotated 180 degrees, then we know overall the U face must rotate either some multiple of 360 degrees + 180. However piece 7 almost falls under the same restriction. Piece 7 can move two ways, neither of which change the permutation of the piece (the same two moves will ALWAYS affect the same type 7 piece). So for the type 7 piece that is affect by the U and D faces, for example, we know that it is only rotated by U or D so if it starts rotated 90 degrees we know that the sum of the rotations of U and D in the solution must add up to a multiple of 360 + 270 degrees. Since this is already guaranteed by the piece types 9 in faces U and D respectfully, that particular piece type 7 is guaranteed to be solved if the two relevant pieces of type 9 are solved. Since this is true for every type 7 piece, we can see that if piece type 9 has orientations visible, piece 7 IS redundant. If the orientation is not visible, well piece type 7 can be used as a hint to what the orientations really are (but multiple piece type 9 configurations will solve piece type 7, so type 7 alone is not enough to uniquely define orientations).

So I have learned that of my 6 imaginary piece types on a 3x3x3 one is sadly redundant, the one you guys have labelled as type 7. This bums me out a little because I thought every imaginary piece added something... :cry:

BUT the other 5 types are NOT redundant and they are not guaranteed to be solved if just the real pieces are solved and that I'm ecstatic about :mrgreen:

I've attached screenshots of my simulation (yeah I know the graphics are a little squirrely, some lines seem a little crooked a such, but this was made 100% from scratch, I even wrote the 3D engine myself without ever seeing the code to a real 3D engine so I'm very proud of it :mrgreen: )

Peace,
Matt Galla


Attachments:
File comment: The "real" 3x3x3 is solved here. Can you see it? The only other piece type that is solved is type 7, which is simulated by the inner 3x3x3 edges.
Complex3x3x3SimulationResults.jpg
Complex3x3x3SimulationResults.jpg [ 74.39 KiB | Viewed 10247 times ]
File comment: Here is my simulation in midturn. I didn't have to hold anything down to make this move, it is restricted so this is the only move that is allowed.
Complex3x3x3SimulationMidTurn.jpg
Complex3x3x3SimulationMidTurn.jpg [ 79.85 KiB | Viewed 10247 times ]
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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 10:50 pm 
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Allagem wrote:
Bravo Carl! This works beautifully. How the heck did you come up with this???


Well I went to bed last night thinking there may be something higher order then a 3x3x3 involved in the big picture. The slice cube of your normal/double/slice combo showed how the corners of the 3x3x3 could collectively act as a single piece so I know that helped push me in that direction. When I woke up this morning I was just about certain I was looking at some sort of 5x5x5 and I just had to put a few pieces together. It's odd... the problem solving process. I love to dabble with POV-Ray and I'm far from an expert but I swear my brain can run POV-Ray code in my sleep. I can struggle with a problem where I try to figure out how to do something in POV-Ray all day and I give up and go to bed and the next morning the answer is just sitting there in my head when I wake up.

Another problem I solved where I really liked the solving process was this one:
http://www.smart-kit.com/s1512/untouchable-11-visual-spatial-puzzle-contest/
Yes, the posts by Carl H. are mine. The problem as presented is one of the hardest problems to solve by hand that I've ever seen. And there are no programs easily available to solve them for you... so I tried to write my own. After it was written and I realized that it would need longer then I'm likely to live to completely search the problem space it dawned on me while I was off to lunch with some co-workers not even thinking of this problem that it could be mapped to another problem for which solvers are a dime-a-dozen and with basically no additional programming on my part I was able to do a complete search of the solution space of the hard problem and find all 7 solutions in under 2 hours. Nice... The creator of the puzzle was only aware of two solutions himself. He had found 1 by hand and someone else had found another by hand and emailed it in.

Allagem wrote:
You are absolutely correct. A 5x5x5 has 10 unique pieces (including the inner 3x3x3 pieces and inner core). Restrict moves to your moveset, where the only moves allowed are a single layer face move combined with a middle slice move underneath and you give all 10 types of pieces a different defining pattern, one of each of the possible defining patterns of a 3x3x3. Notice that since only one type of move is allowed in each direction from the core this is in one sense still defined as a 3x3x3 (that is there are only 6*3=18 moves available, 6 faces each of which can spin 90, 180, or 270 degrees)


Agreed... one very odd 3x3x3. So can you solve it yet? I think I'll address the rest of your post tomorrow.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Mon Dec 21, 2009 11:44 pm 
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By the way, here is another valid picture. I'm not sure if one is any better then the other.

L maps to LML . . . . . . . . . . . . . . . . . . . . . . . L maps to VL
R maps to RMR . . . . . . . . . . . . . . . . . . . . . . .R maps to VR
U maps to UMU . . . . . . . . . . . . . . . . . . . . . . U maps to VU
D maps to DMD . . . . . . . . . . . . . . . . . . . . . . D maps to VD
F maps to FMF . . . . . . . . . . . . . . . . . . . . . . . F maps to VF
B maps to BMB . . . . . . . . . . . . . . . . . . . . . . . B maps to VB

The central core was type (1) . . . . . . . . . . . . [same as before]
The inner 3x3x3 corners were type (10) . . . . [becomes our type 5 pieces]
The inner 3x3x3 face centers were type (3) . [becomes our type 2 pieces]
The inner 3x3x3 edges were type (7) . . . . . . [becomes our type 4 pieces]
The 5x5x5 face centers were type (2) . . . . . . [becomes our type 3 pieces]
The 5x5x5 X-centers were type (9) . . . . . . . . [becomes our type 8 pieces]
The 5x5x5 T-centers were type (6) . . . . . . . . [same as before]
The 5x5x5 middle edges were type (4) . . . . . [becomes our type 7 pieces]
The 5x5x5 corners were type (5) . . . . . . . . . . [becomes our type 10 pieces]
The 5x5x5 wings were type (8) . . . . . . . . . . . [becomes our type 9 pieces]

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 22, 2009 1:05 am 
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The sad thing is I want to understand, but, even looking at the picture above didn't help, and I don't see the solved 3x3...
Plus, these long posts are impossible to read, and I'm just lost...

Is there any way you could make an animation or something to explain Carl?
I think this is a case where a picture is worth thousands of words, and I'll never be able to take all this information in words...

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 22, 2009 9:17 am 
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Prompted by elijah's post I gave a more thorough reading ot the first part of Andreas's post.

I understand the analysis method and the conclusion reached for the skewb, but to understand this development for others puzzles I need a better definition of what it means to:

restrict twistability to one side per axis

After that, i understand that you are -doing whatever that means- and then seeing what piece is left fixed, and when there isn't a physical one, you call it a virtual piece.

Quick question, have we already checked to verify that the virtual pieces do not appear on other puzzles? That is, are the little chop's virtual pieces present on the Toru or Rua cubes?

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 22, 2009 10:17 am 
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AndrewG wrote:
I understand the analysis method and the conclusion reached for the skewb, but to understand this development for others puzzles I need a better definition of what it means to: restrict twistability to one side per axis

I tried to address that point here:
wwwmwww wrote:
Let’s start with the 3x3x3, everyone here should have one. Now let’s look at one axis through the 3x3x3 that allows rotation. You can rotate the left side, you can rotate the right side, or you can rotate the slice layer between the other two. Are these 3 rotations independent? No… rotating the right side one direction is the same as rotating the left and slice layers the other direction. So we can fix one of these layers in place and still be totally general.

Think of it as restricting the puzzle to only one set of twists that are all independent.
AndrewG wrote:
After that, i understand that you are -doing whatever that means- and then seeing what piece is left fixed, and when there isn't a physical one, you call it a virtual piece.

Correct.
AndrewG wrote:
Quick question, have we already checked to verify that the virtual pieces do not appear on other puzzles? That is, are the little chop's virtual pieces present on the Toru or Rua cubes?

Short answer is yes. Longer answer... it depend on what you consider other puzzles. For example some of the virtual pieces can "appear" on the circle puzzles. The Circle FTO is one such puzzle discussed above.

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 22, 2009 11:03 am 
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Is another way of explaining "restrict twistability to one side per axis" to say that:

"For each axis you can turn either the 'left side' or the 'right side', so for each axis just pick one, and the set that you end up picking defines which piece is fixed"

?

That makes sense to me.
Thank you

I will mull over what you say about circle FTO...

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 22, 2009 11:41 am 
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AndrewG wrote:
Is another way of explaining "restrict twistability to one side per axis" to say that:
"For each axis you can turn either the 'left side' or the 'right side', so for each axis just pick one, and the set that you end up picking defines which piece is fixed"?


Yes, that works for what I've called the order=1 puzzles. For the order=2 puzzles there is also a slice layer. In those cases you can pick two layers to turn or pick one layer to hold fixed... two different ways of saying the same thing.

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 22, 2009 2:56 pm 
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elijah wrote:
Is there any way you could make an animation or something to explain Carl?
I think this is a case where a picture is worth thousands of words, and I'll never be able to take all this information in words...


I’ll try to kill 2 birds with one stone here and also present yet a 3rd view of this complex 3x3x3 with 10 piece types that I like even more then the two views presented above.

Let’s first start with the 10 piece types. They are defined above as…

The 10 defining patterns:
( 1) UDLRFB => inverted core; 1 moved in every turn
( 2) UDLRF => inverted faces; 5 moved per turn
( 3) UDLR => not yet explained; inverted version of UD
( 4) UDLF => inverted edges; 8 moved per turn
( 5) ULF => normal corners; 4 moved per turn; impossible to be inverted
( 6) UDL => not yet explained; impossible to be inverted
( 7) UD => not yet explained;
( 8) UL => normal edges; 4 moved per turn
( 9) U => normal faces; 1 moved per turn
(10) [] => normal core; 0 moved per turn

But what does this mean…
A 3x3x3 has 2 face layers and a slice layer along each axis of rotation. A rotation of the slice layer can be thought of as a rotation of both faces so let’s drop that rotation so we just have the two face layers that are independent. These face turns are U-up, D-down, L-left, R-right, F-front, and B-Back. The notation above describes the piece type that is moved with each listed face turn.

So let’s start with the pieces that are present on a normal 3x3x3.

The core. Which face turns affect the core’s position or orientation? None… So it is piece type (10) from the list above.

A face center. Which face turns affect the position or orientation of any given face center? It’s just the rotation of that given face turn. So that makes a face center an example of a type (9) piece.

The best animation I can think to show this is here:
http://www.randelshofer.ch/cube/rubik/?U
Click on the gear thing at the right end of the control bar and set Twist Speed to Slow and make sure the Show Rear box is checked. Then hit play.

Notice how the yellow face center is rotated by the above face turn but isn’t by any of these:
http://www.randelshofer.ch/cube/rubik/?D
http://www.randelshofer.ch/cube/rubik/?L
http://www.randelshofer.ch/cube/rubik/?R
http://www.randelshofer.ch/cube/rubik/?F
http://www.randelshofer.ch/cube/rubik/?B

An edge piece. Which face turns affect the position or orientation of any given edge piece? It’s just the rotation of the two faces that share that edge piece. So that makes an edge piece an example of a type (8) piece.

Notice these animations move the yellow/red edge:
http://www.randelshofer.ch/cube/rubik/?U
http://www.randelshofer.ch/cube/rubik/?L
But that edge isn’t moved in these:
http://www.randelshofer.ch/cube/rubik/?D
http://www.randelshofer.ch/cube/rubik/?R
http://www.randelshofer.ch/cube/rubik/?F
http://www.randelshofer.ch/cube/rubik/?B

A corner piece. Which face turns affect the position or orientation of any given corner? It’s just the rotation of the three faces that share that corner. So that makes an edge piece an example of a type (5) piece.

Notice these animations move the yellow/red/blue corner:
http://www.randelshofer.ch/cube/rubik/?U
http://www.randelshofer.ch/cube/rubik/?L
http://www.randelshofer.ch/cube/rubik/?F
But that corner isn’t moved in these:
http://www.randelshofer.ch/cube/rubik/?D
http://www.randelshofer.ch/cube/rubik/?R
http://www.randelshofer.ch/cube/rubik/?B

So in a normal 3x3x3 we’ve found only 4 of the 10 types of pieces. Those being types (10), (9), (8), and (5). Where are the other 6 types? To find them we need to put this 3x3x3 inside a 5x5x5. But as this is a 3x3x3 puzzle you are only allowed to manipulate the 3x3x3 in the center, you have no direct control over the 5x5x5 layers independently. You are still just limited to your 6 basic face rotations U, D, L, R, F, and B. However we need to define what these moves mean on the 5x5x5. Remember we are moving the 3x3x3 on the inside.

To do that look at this image.
Image

The slice layer is the normal slice layer but we are holding those fixed, remember? So we won’t discuss those. The right layer of the 3x3x3 can be viewed as starting at the cut that allows the rotation of the right face of the 3x3x3 and continuing to infinity off toward the right. If we consider space as closed we can even continue past that point and continue until the right layer approaches the surface of the left face of the 3x3x3 from the left. That’s a difficult concept for some to grasp but it’s not needed. You can just accept this definition of a turn of the right face as applied to the 3x3x3 on the inside.
http://www.randelshofer.ch/cube/professor/?NRSR
This animation will show this as two separate moves but consider them as occurring simultaneously.

Do the same for all the other faces. You can see this applied to the left face in the image above.

Now that we have defined the affect of the 6 face turns of the 3x3x3 on the exterior 5x5x5 let’s do this same analysis to the 5x5x5 pieces.

But first look at these pages:
http://www.speedsolving.com/wiki/index.php/Center
http://www.speedsolving.com/wiki/index.php/Edge
If you aren’t familiar with terms like X-Center, or Wings.

A 5x5x5 face center. Is a type (7)”UD” piece as seen in the below animations.

Notice these animations move the yellow and white 5x5x5 face centers:
http://www.randelshofer.ch/cube/professor/?NUSU
http://www.randelshofer.ch/cube/professor/?NDSD
But those same 2 face centers aren’t moved in these:
http://www.randelshofer.ch/cube/professor/?NLSL
http://www.randelshofer.ch/cube/professor/?NRSR
http://www.randelshofer.ch/cube/professor/?NFSF
http://www.randelshofer.ch/cube/professor/?NBSB

A 5x5x5 X-center. Is a type (4)”UDLF” piece as seen in the below animations.

Notice these animations move a yellow and a white 5x5x5 X-center:
http://www.randelshofer.ch/cube/professor/?NUSU
http://www.randelshofer.ch/cube/professor/?NDSD
http://www.randelshofer.ch/cube/professor/?NLSL
http://www.randelshofer.ch/cube/professor/?NFSF
But those same 2 X-centers aren’t moved in these:
http://www.randelshofer.ch/cube/professor/?NRSR
http://www.randelshofer.ch/cube/professor/?NBSB

A 5x5x5 T-center. Is a type (6)”UDL” piece as seen in the below animations.

Notice these animations move a yellow and a white 5x5x5 T-center:
http://www.randelshofer.ch/cube/professor/?NUSU
http://www.randelshofer.ch/cube/professor/?NDSD
http://www.randelshofer.ch/cube/professor/?NLSL
But those same 2 T-centers aren’t moved in these:
http://www.randelshofer.ch/cube/professor/?NFSF
http://www.randelshofer.ch/cube/professor/?NRSR
http://www.randelshofer.ch/cube/professor/?NBSB

A 5x5x5 middle edge. Is a type (3)”UDLR” piece as seen in the below animations.

Notice these animations move the yellow/orange, orange/white, white/red and red/yellow 5x5x5 middle edges:
http://www.randelshofer.ch/cube/professor/?NUSU
http://www.randelshofer.ch/cube/professor/?NDSD
http://www.randelshofer.ch/cube/professor/?NLSL
http://www.randelshofer.ch/cube/professor/?NRSR
But those same 4 middle edges aren’t moved in these:
http://www.randelshofer.ch/cube/professor/?NFSF
http://www.randelshofer.ch/cube/professor/?NBSB

A 5x5x5 wing. Is a type (2)”UDLRF” piece as seen in the below animations.

Notice these animations move one set (there is another… as Yoda would say) of yellow/orange, orange/white, white/red and red/yellow 5x5x5 wings:
http://www.randelshofer.ch/cube/professor/?NUSU
http://www.randelshofer.ch/cube/professor/?NDSD
http://www.randelshofer.ch/cube/professor/?NLSL
http://www.randelshofer.ch/cube/professor/?NRSR
http://www.randelshofer.ch/cube/professor/?NFSF
But those same 4 wings aren’t moved in these:
http://www.randelshofer.ch/cube/professor/?NBSB

A 5x5x5 corner. Is a type (1)”UDLRFB” piece as seen in the below animations.

Notice all 8 of the 5x5x5 corners are moved in these:
http://www.randelshofer.ch/cube/professor/?NUSU
http://www.randelshofer.ch/cube/professor/?NDSD
http://www.randelshofer.ch/cube/professor/?NLSL
http://www.randelshofer.ch/cube/professor/?NRSR
http://www.randelshofer.ch/cube/professor/?NFSF
http://www.randelshofer.ch/cube/professor/?NBSB

So there you have a normal or “real” 3x3x3 surrounded by the “imaginary” pieces needed to make it what Matt calls a “complex” 3x3x3 just as a real number added to an imaginary number equals a complex number. And it all fits inside a very odd 5x5x5.

Does that help? I like the way this interpretation goes together better then the two I presented above but I think all 3 generate very confusing puzzles to play with.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 22, 2009 3:02 pm 
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Sorry for not answering this promptly. I was on a journey home to family. You know why...

@ AndrewG:
Carl explained it and you seem to have understood it. For puzzles with more than one cut per axis we need to start working with slices.
I gave a not perfectly explained but fully implemented analysis for the 3x3x3 here:
viewtopic.php?f=14&t=15016

AndrewG wrote:
Quick question, have we already checked to verify that the virtual pieces do not appear on other puzzles? That is, are the little chop's virtual pieces present on the Toru or Rua cubes?

As far as I know our virtual pieces (subset of Matt's imaginary pieces) are present on:
  • Circle FTO
  • Gelatibrain 5.1.1
  • Some cornerturning tetrahedrons with two cuts
  • The "offset-octahedron" Matt presented above

TORU and RUA are both EH2 while the LittleChop is EH1. I classified them in you-know-which-thread.
Anyway: Both do not have any visible pieces of EH2 other than the ones n Carls animation. That is not surprising since in both cases the cuts are perfectly symmetrically aligned with respect to the center.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Wed Dec 23, 2009 2:45 pm 
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I'll be on the road for a few days. I just wanted to let you guys know in case I can't get to a PC till I get back.

Happy Holidays,
Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Wed Dec 23, 2009 2:48 pm 
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Aww, you're going to miss me post the essay I've been working on for three days then! :P

Regardless, have a merry christmas!

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Wed Dec 23, 2009 6:21 pm 
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Finally, I can post the response I was working on at the time of the forum crash.

So, If I am understanding everything correctly:

The Complex 3*3*3 is equivalent to a Super 5*5*5 Mulitcube with restricted movement.
These restricted movements lock the 1st, 2nd, and 5th layers together during twists.
These restricted movements also lock the 3rd and 4th layers together during twists.
The Inner 3*3*3 represents the four types of real piece: Core, Face, Edge, and Corner.
The Outer 5*5*5 represents the six types of imaginary piece.
The Eight 5*5*5 corners represent a single piece: the Inverted Core.
X-centers = Inverted Edges
Wings = Inverted Faces
Midges and Face-Centers = Fourth invertible pair
T-centers = Elusive non-invertible imaginary piece.
The inverted core is not unique in being represented by multiple physical pieces. Could someone explain this part again?

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 27, 2009 6:16 pm 
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AndrewG wrote:
Aww, you're going to miss me post the essay I've been working on for three days then! :P


I'm back... Did you post the essay in another thread? I just got back and haven't gone looking for it yet.

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 27, 2009 6:25 pm 
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Jeffery Mewtamer wrote:
The Complex 3*3*3 is equivalent to a Super 5*5*5 Mulitcube with restricted movement.
These restricted movements lock the 1st, 2nd, and 5th layers together during twists.
These restricted movements also lock the 3rd and 4th layers together during twists.
The Inner 3*3*3 represents the four types of real piece: Core, Face, Edge, and Corner.
The Outer 5*5*5 represents the six types of imaginary piece.
The Eight 5*5*5 corners represent a single piece: the Inverted Core.
X-centers = Inverted Edges
Wings = Inverted Faces
Midges and Face-Centers = Fourth invertible pair
T-centers = Elusive non-invertible imaginary piece.


Looks correct to me.

Jeffery Mewtamer wrote:
The inverted core is not unique in being represented by multiple physical pieces. Could someone explain this part again?


There are 8 physical corners on the 5x5x5 but they are all looked together and behave as a single piece... the imaginary core. I think that is all that was meant there. However the way you pick to restrict the movement of the 5x5x5 isn't unique. There have been 3 ways it can be done presented in this thread and the effect of one way over another is to just move the 10 types of pieces around. The one you go through above is the one I like the most because it leaves the "real" pieces intact in the 3x3x3 core.

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 27, 2009 6:41 pm 
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Bram wrote:
The 'double cube' has ordinary corners, but its edges and face centers are in two layers - the outer being mostly normal shaped piece with a window into the inner piece, which is normal, but maybe rounded a bit. When you play with it the two layers mostly turn in tandem, acting as a regular rubik's cube, but there's an added move, where a slice can be done of just the outer layer pieces.


Ok... Here is a picture of the puzzle I believe Bram is talking about above.
Image

The puzzle on the left is a normal 3x3x3. On the right I picture a few of the floating face centers and floating edges. Now that I look at it I see a few problems that Bram may already know how to fix.

(1) Assemby may be an issue as I have it pictured.
(2) How does one control the inner 3x3x3 slice layers during a face turn? Maybe the holes I've made for the windows need to be big enough that you can hold this layer with your fingers.
(3) And I'm now thinking these floating face centers and edges aren't quite the same as Matt's imaginary inverted edges and face centers. Their movement should be tied to the face turns. That restriction I believe is gone here making this an easier puzzle then it would be if these were Matt's imaginary pieces. Not to take anything away from Bram's puzzle as I still think its a great puzzle in its own right.

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Sun Dec 27, 2009 8:52 pm 
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wwwmwww wrote:
(2) How does one control the inner 3x3x3 slice layers during a face turn? Maybe the holes I've made for the windows need to be big enough that you can hold this layer with your fingers.


I think the best way to deal with this problem is to make the center slices wider than the outer two, and make the interior sphere big enough that it actually bulges through the face centers and has to be cut off to make a smooth cube, without any need for windows at the face centers. That way the interior is directly manipulated by your fingers. Ideally the friction of the puzzle is such that if you push the center slice at the edges it just pushes the outer part.

wwwmwww wrote:
(3) And I'm now thinking these floating face centers and edges aren't quite the same as Matt's imaginary inverted edges and face centers.


It's fairly close. If you do RL- and then rotate the cube as a whole down, it does something very similar to moving just the center slice. In fact I think you need some of the other virtual pieces to differentiate. Even with the other pieces added in, for example the 5x5x5 mod discussed above, it would be very hard to keep the center slices from rotating independently. The problem is that the center slice can move relative to one side because of R, and relative no the other because of L, and slice don't particularly like letting you move independently of either A or B but not both at once.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 29, 2009 10:26 am 
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Bram wrote:
I think the best way to deal with this problem is to make the center slices wider than the outer two, and make the interior sphere big enough that it actually bulges through the face centers and has to be cut off to make a smooth cube, without any need for windows at the face centers. That way the interior is directly manipulated by your fingers. Ideally the friction of the puzzle is such that if you push the center slice at the edges it just pushes the outer part.

Ok... I did a little re-designing of the parts and here is what I got.
ImageImage
You start to cut into the mechanism of the puzzle a bit but I actually like the look of that.

By the way... this got me thinking. With the wider slice layers this is starting to look like the Mixup Cube. However even if my cuts were correctly spaced the dovetails on the edges that enter the groves on the corners are at the same radius as the face center dovetails. I think if you set these two types of dovetails at different radii and had two seperate sets of groves to accommodate them you could put a Mixup Cube with windows on top of a normal 3x3x3. As we are already cutting into the mech with just one set of dovetails and groves I'm not sure there is room for a second. I took a peek at the first picture at Shapeways and I think there it has the dovetail on the corners and just groves in the face centers and edges, with no dovetail between those two piece types. Is that correct? If that works as it seems some pieces might get loose mid turn then I'm pretty sure you could do the same here. Though I'm not sure I understand the mech of the Mixup Cube so I may be off base.

Bram wrote:
It's fairly close. If you do RL- and then rotate the cube as a whole down, it does something very similar to moving just the center slice. In fact I think you need some of the other virtual pieces to differentiate. Even with the other pieces added in, for example the 5x5x5 mod discussed above, it would be very hard to keep the center slices from rotating independently. The problem is that the center slice can move relative to one side because of R, and relative no the other because of L, and slice don't particularly like letting you move independently of either A or B but not both at once.

You are right... they may be close enought that you couldn't differentiate them from Matt's immaginary pieces. Hmmm... I'm just not sure, if you do R, L, and rotate the whole cube you've moved the outer slice 90 degrees one direction at the same time the inner slice has moved 90 degrees in the opposite direction.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 29, 2009 1:03 pm 
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wwwmwww wrote:
I think I'll address the rest of your post tomorrow.

Ok… that took me longer then I wanted to take but I’m finally getting caught back up.

Allagem wrote:
I went ahead and simulated this in a program I've been working on and right off the bat noticed how incredibly confusing it was. The "core" piece (type 10) is not in the center as you would expect but the inner 3x3x3 corners, so it's like a core that has been sliced down the middle 3 ways. This trend continues through the rest of the real 3x3x3 pieces. The "center" pieces have been split down the middle twice and the edge pieces have been sliced down the middle once as well. Several other pieces have this odd effect, multiple pieces on this weird 5x5x5 map to the same imaginary 3x3x3 piece. So on the 5x5x5 these pieces always follow eachother. It's actually very counterintuitive.

Did you find either of the other ways of grouping the moves on the 5x5x5 any better?
Allagem wrote:
So I have learned that of my 6 imaginary piece types on a 3x3x3 one is sadly redundant, the one you guys have labelled as type 7. This bums me out a little because I thought every imaginary piece added something... :cry:

I follow that and agree.
Allagem wrote:
BUT the other 5 types are NOT redundant and they are not guaranteed to be solved if just the real pieces are solved and that I'm ecstatic about :mrgreen:

But there is some redundancy among the imaginary pieces themselves. If the inverted face centers (type 2) are in the correct position, regardless of orientation, you know the inverted core (type 1) is solved as well. And is there a similar relationship between the type 2 pieces and the type 3 pieces as you are finding between the type 9 and type 7? It seems like there would be but I haven’t tried to work it out in detail yet.
Allagem wrote:
I've attached screenshots of my simulation (yeah I know the graphics are a little squirrely, some lines seem a little crooked a such, but this was made 100% from scratch, I even wrote the 3D engine myself without ever seeing the code to a real 3D engine so I'm very proud of it :mrgreen: )

Nice!!! I do everything in POV-Ray and wouldn’t know were to start making my own 3D engine. By the way, I like this exploded view. I copied it when making the pictures explaining how the Crazy 4x4x4’s were really 4x4x4 Multicubes.

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 29, 2009 6:43 pm 
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Yep, that's basically right. You should probably make the central spherical puzzle just a bit larger though, so that it's possible to shave down the parts bulging above the outer pieces to be flat. That way it can be easily aligned properly. It does wind up being a bit thin, but your clever trick of making the outer centers be round in the visible part helps a lot.

It's getting a bit ahead of yourself to start going mixup cube with this one. It might even me a less interesting puzzle that way.


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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 29, 2009 7:29 pm 
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wait, so if I'm following this correctly, your putting a 3x3 inside a 3x3?

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 29, 2009 7:46 pm 
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Location: Missouri
elijah wrote:
wait, so if I'm following this correctly, your putting a 3x3 inside a 3x3?

That is certainly one way to look at it. The inside and the outside 3x3x3 share the same corners though. I'm just adding a second set of face centers and edges.

Carl

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 Post subject: Re: Analysis of twistability and virtual pieces
PostPosted: Tue Dec 29, 2009 8:07 pm 
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and this is what virtual (or imaginary) pieces are?
Or... how exactly does this relate to imaginary pieces?

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