I try not to answer without doing the resaerch, but.. it is a long slog through most of that, and I only understand some of it. I wonder, can you give me the short of it?
I'll take a shot at this. This topic started off as an "Analysis of twistability" which was a way of looking at the pieces in a twisty puzzle.
The first approach was taken by Andreas and went something like this. Let's use a 3x3x3 for our first example. In a 3x3x3 there are 2 independant layers of rotation on each of 3 axes. On each axis we have 3 choices of the 2 layers we can pick as the independant ones. The 3rd layer is fixed and will never rotate. We can do this for each of the 3 axes. So lets pick the 6 face layers and ask yourself what is the piece that never moves. I've called this a "holding point" its a point fixed in space with this choice of turnable layers. With that pick we can see the "holding point" is the core cubie. It never moves. Let's pick 3 slice layers and 3 face layers. There are 8 ways to do this and each way maps to one of the 3x3x3 corners. In fact there are 3^3 or 27 different selections of layers which can be made and each maps to one of the 27 cubies in a 3x3x3. All these are REAL pieces, i.e. they have real volume.
The second approach was taken by Matt Galla and instead of looking at the pieces which didn't move it looked at the way pieces moved with the face layers. In effect the core is always considered as the holding point. There are 6 face layers and any piece can move with ANY combination of all 6 layers. For example, it allows for the existance of a piece which turns with both the left and the right faces for example. Each piece moves with a given layer or it doesn't, a binary choice. And there are 6 layers. So this method defines 2^6 pieces or 64 pieces. Of these 64 pieces, 27 are real and the same 27 pieces identified with the first method above. In general this method is always a superset of the first method. The other 37 pieces are imaginary. The 3x3x3 doesn't contain any virtual pieces.
The above two approaches are detailed in this thread
. There is also a third approach I've taken which I view as a more general use of the second approach which allows for an arbitrary pick of a holding point. I go into more detail about it here
. And Matthew S may have yet a 4th approach he's detailing here
which appears to be yet an even bigger superset. I have yet to fully understand that one myself.
All that said lets focus on just the first two methods and look at the dodecahedron. I AM trying to keep this as simple as I can. First let's count the pieces in Tom's MultiDodecahedron
1 = Core (Piece Type 0)
12 = Megaminx Face Centers (Piece Type 1)
30 = Megaminx Edges (Piece Type 2)
20 = Megaminx Corners (Piece Type 3)
30 = Master Pentultimate Edges (Piece Type 4)
60 = Master Pentultimate X-Centers (Piece Type 5)
12 = Master Pentultimate Face Centers (Piece Type 6)
20 = Master Pentultimate Corners (Piece Type 7)
You can see all the piece types labeled in this animation.
This is a total of 1+12+30+20+30+60+12+20 or 185 pieces. All of which are playable and solveable in Tom's puzzle. The only one which isn't stickered directly is the core but you DO know its orientation as its directly attached to the Megaminx Face Centers.
Now lets use the first method above to look at the face turn dodecahedron. It, just like the 3x3x3, has 2 independant layers per axis of rotation. On each axis we have 3 choices of the 2 layers we can pick as the independant ones. However we now have 6 axes of rotation so this leads to 3^6 possible holding points. That's 729 pieces. Of these we know 185 are real. The other 544 are virtual. I still like that name. It lends itself to nice naming. Tom's puzzle is the Face Turn MultiDodecahedron. A puzzle with all 729 of these pieces I'd call the Face Turn Augmented Dodecahedron as real + virtual = augmented.
Now let's use the second method to look at the face turn dodecahedron. We hold on to the core so its our holding point. There are 12 faces which can turn. For each face a given piece is either in it or it isn't. Again a binary choice so this method defines 2^12 pieces. That's 4096. Of these 185 are real, 544 are virtual, and the remaining 3367 are imaginary. Or you could include the virtual pieces as a subset of the imaginary if you like. And a puzzle which had all 4096 of these pieces I'd call the Face Turn Complex Dodecahedron as real + imaginary = complex.
The virtual pieces can and do turn up in quite a few puzzles. The one you found above is a great example. The "plus 1" I found in this puzzle
is another. They are much easier to bring into existance (be given a real positive volume) then the other imaginary pieces. Here is my Augmented Skewb
idea which could be made as a real physical puzzle. Note it has the corners and face centers from a normal skewb (the real pieces) plus both of the virtual cores. To my knowledge there are no imaginary pieces in any of the playable puzzles you can currently find on line. The most talked about complex puzzle is the Complex 3x3x3 and someday I believe we will have an app which will allow us to play with it. Andreas has already calculated its number of permutation (states) here
. And I have ideas on how to make it a real playable puzzle. However its starts with first making my Real5x5x5
a working real puzzle which I have yet to do. And it gets even more complicated from there. In short you really have to jump though hoops to give the imaginary pieces a real positive volume that playable... even in an app. I believe this is due to the fact they can't serve as holding points so in that way they are even less real then the virtual pieces.
I hope you could follow that. Basically I just went though counting the pieces. Knowing how to group them into types, corners, face centers, cores, etc. is detailed in all the above threads.
I hope this helps,
Now if this is a megaminx and a pyraminx crystal, there is nothing else `in there` that they could be, right? Because a pentultimate etc, is on the `outside`. So
Well if you go IN far enough you come out the other side so yes I could see Pentultimate like pieces turning up in something like this. However there is a whole slew of virtual pieces (FAR more then the real ones in there) that just don't have any volume in a puzzle cut with planes. These circle puzzles aren't simply planes and that does allow some of the more exotic pieces to "become real" and turn up in puzzles like this. Pick one of them and pretend you are holding the puzzle by that point. Now ask yourself which two layers are free to turn on each axis without this piece moving. You can now see its a virtual piece (not imaginary) and you have its signature. As for Andreas' naming scheme which names this piece Tz15 I'm not sure about that myself at the moment. I suspect its some encoding of this signature.