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 Post subject: Classification of Cubes
PostPosted: Thu Dec 10, 2009 6:38 pm 
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AndrewG posted a very nice essay here:
http://twistypuzzles.com/forum/viewtopic.php?p=138786#p138786

This thread is about that essay and its a very good read.

AndrewG wrote:
Quote:
I would also call those the canonical elementary cubes. I don't see the logic to adding in the helicopter and dino without also adding the 3x3x3.


The short answer is that the helicopter and dino contain completely different set of pieces while the 3x3x3 is just a higher order 2x2x2. The three you consider are the deep-cut fundamentals.

The long answer of why there are eight fundamentals is because with these eight you can form (and therefore, classify) all cubic non-shapeshifting (exlcuding jumbling) puzzles with planar cuts, through ordering, combination and hybridization. For example the skewb-by-2 is Cube FA2VC2 (face-turning, type A, 2nd order + vertex-turning, type C, 2nd order)

Of course I'm far from authoritative on this subject, but I did write an essay on it a while back which I posted in the GelatinBrain thread. In the interest of not hijacking this thread too badly, here is a link to the correct page of that thread. The essay is an attachment.
viewtopic.php?f=8&t=7830&p=138786


Wow! That is alot of nice work. However I'm not sure I agree with all of it. I'm not sure there is necessarily a right or wrong way to classify these puzzles but I see a few issues with where you started.

I was initially going to reply to your comment "The short answer is that the helicopter and dino contain completely different set of pieces while the 3x3x3 is just a higher order 2x2x2" by saying I view the dino as a higher order Skewb and the helicopter as a higher order little chop (24-Cube). And the 3x3x3 contains pieces that aren't found on a 2x2x2. The 2x2x2 doesn't have edges, faces centers, or a core. To be even more specific I consider the skewb the order=1 (as it has one cut plane per axis of rotation) Corner-Turn Multicube. The dino cube is just one of the puzzles in the order=2 Corner-Turn Multicube.

Image

The helicopter is also just one of the puzzles that make up the order=2 Edge-Turn Multicube. The order=1 Edge-Turn Multicube being little chop aka 24-Cube.

Image

Where I see an issue is where you count the number of pieces per cut depth. The way you did this its clear that you are just counting the pieces on the surface. As such I think you are missing something... its hard to put my finger on it but it is creating some issues with your naming scheme. For example look at the shallow cut helicopter puzzle. Using your naming scheme I believe that would be a Combination, as such it should bear the two letters of the fundamentals around it but you've only named the fundamental on one side of it so the shallow cut helicopter is "Cube E?A". I guess you could count the local minima that occures at cut depth 0 as your 9th fundamental. That just being the uncut cube. In my view that is the order=0 Multicube... zero cut planes per axis of rotation.

Also you make the comment that the Skewb is the 2nd order Cube VC so its named VC2. Why? What makes it order 2 while VA1 and VB1 are order 1? I'm missing something. Same for EA2, EB1, and EC1 why are these different orders? Personally using order to equal the number of cut planes per axis I'd call FA2 as order 1 as well but at least there you give a reason why you call it order 2. I'm missing the reason for the others.

Your monofeatric hybrids I'd also deal with differently. What you call Cube VB1VC2 (Dino + Skewb) is just one of the puzzles in the order=3 Corner-Turn Multicube.

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 Post subject: Re: Classification of Cubes
PostPosted: Thu Dec 10, 2009 7:59 pm 
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Hi!
I agree, a very interesting read from Andrew.
Carl, I also subscribe to your use of the term "order" to mean the number of cuts between pairs of opposing features. A while back I created a series of diagrams summarizing my thoughts about classification of tetrahedral, cubic, octahedral, and dodecahedral puzzles (I have yet to complete the icosahedral diagram), and I, too, think of the Dino as a descendant of the Skewb.

I think of the "fundamental" set as consisting of only 3 members - the order-1 face, vertex, and edge-turning puzzles, namely the 2x2x2, Skewb, and 24 Cube (for cubic puzzles - similar arguments apply for all the platonic solids). I view all the others as arising either as hybrids containing combined face/vertex/edge cuts (e.g. the Super-X) or as higher-orders.
(I note that now I'll have to revise the chart to include Adam's new Skewb by 2, GB 3.4.1!)

Where things get interesting, and IMHO resist easy classification, is at order 2 and higher the cuts can be moved around from a symmetric/equi-spaced distribution, resulting in, as you and Andrew (and others) have observed, different piece sets, different puzzles, different solution algorithms. I tried to show that in the central part of the diagram, moving the 2 order-2 cuts out from the (imaginary) Skewb cut.

One thing I did think of but didn't draw out, is that many order-3 species of a given cut type can be derived combinatorially by hybridizing all order-1 and order-2 cuts of the same type to see what results - for instance the Skewb + Dino yields an order-3 vertex-turning puzzle.

Here is the cubic chart I made:
(The others can be seen on my website http://robspuzzlepage.com/rearrangement.htm)

Attachment:
cubic-puzzles4.png
cubic-puzzles4.png [ 150.08 KiB | Viewed 8841 times ]

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 Post subject: Re: Classification of Cubes
PostPosted: Thu Dec 10, 2009 8:43 pm 
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@Carl: I like AndrewG's paper too. Regarding classification by order, I see things a bit differently from both of you and Rob. If you took a knife and lined up each face/corner/edge in turn, making the same number of cuts for each, how many of those cuts per face/corner/edge would you make (never cutting away more than half the puzzle) to produce the final puzzle? That's my definition of the puzzle's order. Thus I regard a deep cut puzzle as a special case 1st order puzzle, not as belonging in an order by itself. In my view:

2x2x2 cube and 3x3x3 cube = single cut puzzle, 1st order.
4x4x4 cube and 5x5x5 cube = double cut puzzle, 2nd order.
6x6x6 cube and 7x7x7 cube = triple cut puzzle, 3rd order.

Several people commented with surprise on learning that the V-Cube 6 had the same number of parts as the V-Cube 7, with all those hidden parts. As I view the 6x6x6 and 7x7x7 cube as both triple cut or 3rd order puzzles, with their shared "triple-cuttiness", it was no surprise that they had huge similarities inside them. With the 6x6x6, the outermost cuts from opposite faces happen to coincide; with the 7x7x7, they don't.

By the same logic, looking inward towards the center of the puzzle from the corners, I consider the Skewb a 1st order puzzle (along with the Dino Cube and Master Skewb).

I also find it disconcerting that if a regular puzzle were Oscar-and-Bram-ified in some way to make it irregular and no longer have parallel cutting planes, your scheme would presumably double the number of axes and change the classification of the puzzle, even though the same number of cuts at approximately the same relative depth have simply been skewed a bit!

Edit: Today I've realized that my way of ordering doesn't work well with tetrahedra. :? So I consider the Master Skewb a 1st order puzzle, but its mathematical equivalent the Master Tetrahedron (Gelatinbrain 5.1.9) would either be a 1st order hybrid (face and corner cuts) or a 2nd order pure puzzle? I don't like that inconsistency, and I see the advantage of counting cuts along entire axes now.


Last edited by Julian on Fri Dec 11, 2009 7:45 am, edited 1 time in total.

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 Post subject: Re: Classification of Cubes
PostPosted: Thu Dec 10, 2009 10:10 pm 
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Thanks for making the thread! I was beginning to regret not putting it in its own thread last year, but I felt like it would be awkward to make a thread for it now unless people actually wanted to discuss it more. I am glad that seems to be the case.

When I wrote that essay, I definitely patched up a few details in convenient ways. I think that ultimately, there isn't a perfect way to classify these things. The fact that so many of us have tried before and never quite hit the nail on the head seems to indicate this. But it's also probably like Linnaeus classifying living organisms; we can still come up with something that works well.

I like your animations. By the way, I like to refer to pieces by greek letters. For instance, saying the Helicopter cube contains alpha and beta pieces, and on Cube EB the beta pieces vanish.

The big reason I don't like call puzzles purely by the number of cuts is that we end up with multiple puzzles with the same name. (You mentioned this). EA (Heli), EB (Rua), and EC (Toru) are the best example. I didn't mention this in the essay, but I call all the puzzles on that # pcs vs. depth graph as "one cut puzzles" because they have one cut below each feature. So a one-cut cube, having 8 corners, has 8 cuts (like the dino). At depth 180, pairs of cuts happen to coincide. I didn't differentiate these deep-cut puzzles because it was convenient, however you can argue that they are fundamentally quite different from other puzzles (for one, all three are fundamentals :) )

I also have the feeling that there is something just a bit wrong with that essay I wrote, but it's hard to put a finger to. I'm going to double post here, the second post will be about a lot of the nitpicky things that are a bit weird in my classification system (several of which you identified in your first post). Like I said, I patched over a lot of annoying little details for teh sake of an simplier, more easy-to-use final product :D

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 Post subject: Re: Classification of Cubes
PostPosted: Thu Dec 10, 2009 10:31 pm 
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Quote:
Also you make the comment that the Skewb is the 2nd order Cube VC so its named VC2. Why? What makes it order 2 while VA1 and VB1 are order 1? I'm missing something. Same for EA2, EB1, and EC1 why are these different orders? Personally using order to equal the number of cut planes per axis I'd call FA2 as order 1 as well but at least there you give a reason why you call it order 2. I'm missing the reason for the others.


Honestly, I did that pretty arbitrarily. The reason why I do not start at 1 with everything is because obviously we want the 2x2x2 to be Cube FA2 and the Megaminx to be Dodeca FA3, etc. Making the Skewb, Bevel, and Little Chop start at 2 was likewise arbitrary but rooted in intuition. If you look at the EA sequence, EA4 definitely says "4" to me, although of course that's not objective.
I would support changing the Dino sequence to be VB3,5,etc instead of VB1,2,3,etc if people "feel" like that's better. I used 1,2,3 at first because the middle layer never gets any thinner on higher order puzzles.

Calling puzzles like VB1VC2 "monofeatric hybrids" is definitely one of those things I "patched up" in the essay instead of delving into more detail. :) That's actually one of the weakest spots of my system; the monfeatric Tetra V hyrbids are the best example. Also, you could say that Cube FA4 = Cube FA3FA2... (I just noticed that Rob mentioned this previously. It's a point to consider).

Quote:
For example look at the shallow cut helicopter puzzle. Using your naming scheme I believe that would be a Combination, as such it should bear the two letters of the fundamentals around it but you've only named the fundamental on one side of it so the shallow cut helicopter is "Cube E?A". I guess you could count the local minima that occures at cut depth 0 as your 9th fundamental. That just being the uncut cube. In my view that is the order=0 Multicube... zero cut planes per axis of rotation.


At first I was thinking this was another patched up issue, but then I remember why I did it that way.
I put the shallow-cut helicopter cube as EA3. Also, it's not a combination with the uncut cube because it has edge pieces which appear of neither the uncut cube or the helicopter. It is very similar to how FA3 (3x3x3) is a shallow-cut 2x2x2, it just so happens that Rubik's cube center aren't totally stationary. Cube VA1 also has some of the "core" showing. VA1 is really the weirdest fundamental because it doesn't lie at a single cut depth like all the others do, but VAB isn't at a single depth either!

I'll reattach that essay so other looking at this thread don't have to scroll through the other one to get it. I'll also attach this hybrid list that I made a while ago. It demonstrates one of the things I like best about my system: it handles hybrids quite well. There is also a list of Octahedral hybrids; the other solids do not hybridize nearly as nicely, and a very large amount of the interesting-looking puzzles involve the half chop.
It's an xlsx file but the forum doesn't allow us to upload those, so you'll have to change the extension manually.


Attachments:
Hybrid List - change to xlsx.txt [11.78 KiB]
Downloaded 295 times
Classification_of_Cubes.pdf [265.21 KiB]
Downloaded 130 times

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 Post subject: Re: Classification of Cubes
PostPosted: Fri Dec 11, 2009 1:57 am 
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Seems like a dictionary entry for order might look like this:

Or-der
Adjective
1. Number of layers between opposing features.
2. Number of pieces along an edge.
3. Number of cuts per axis.
4. Number of cuts per feature.

Unless stated otherwise, I will use' definition 3 of order below.

I will agree that Order-1, aka Deep Cut, puzzles form the most restrictive set of puzzles and are the most worthy of being called 'fundamental' or 'elementary'.

As for order-2 puzzles, while some are most naturally constructed as higher-order variants of the Order-1, there are others that have a strong case for being considered fundamental in their own right. Perhaps their is grounds to consider these as an adjunct group to the order-1 puzzles.

To make an analogy with polyhedra and colors
Order-1 is to Order-2 as Platonic is to Archimedean as Primaries are to secondaries.

Interestingly, these two groups are unified by Definition 4 above.

Also, another property that seems to arise with Order-2 puzzles is that some are fixed depth and others are variable depth. Fixed depth puzzles have a sole cut depth that produces that particular puzzle while variable depth puzzles have a range of values that produce mathematically identical puzzles, though aesthetics or mechanism might force a particular depth.

Order-1 puzzles are trivially fixed depth due to the nature of deep-cuts. For shallow cut cubes, we have the following:

Fixed:
Face: None
Vertex: Dino
Edge: Heli, GB3.3.3, GB3.3.5

Variable:
Face: 3*3*3
Vertex: Trivial Tips, Shallow Dino, Master Skewb
Edge: Shallow Heli, GB3.3.4, GB3.3.6

Is the fact that, counting order-1 puzzles within the group of fixed cuts, the two groups are of equal size a coincidence or an underlying point of mathematical theory?

When I referred to the 2*2*2, Dino, Skewb, Heli, and Little Chop as the five "elementary" cubes in the Skew-by-2 thread, the inclusion of the Heli and Dino was influenced by them being the most obvious of the Order-2 cubes that is not obviously a 'Master' version of a order-1. Also, until I saw wwwmwww's EXCELLENT animation of the Heli-Chop family, I thought them the only shallow fixed cut cubes.

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 Post subject: Re: Classification of Cubes
PostPosted: Fri Dec 11, 2009 11:30 am 
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Rob wrote:
Where things get interesting, and IMHO resist easy classification, is at order 2 and higher the cuts can be moved around from a symmetric/equi-spaced distribution, resulting in, as you and Andrew (and others) have observed, different piece sets, different puzzles, different solution algorithms. I tried to show that in the central part of the diagram, moving the 2 order-2 cuts out from the (imaginary) Skewb cut.
That is no problem. We just need some good animations (thanks Carl) and some diligence. I have made a suggestion here:
viewtopic.php?f=1&t=15258
To sum it up:
We take the Multihedron (read this for the concept: viewtopic.php?f=1&t=14875) of the 6 (!) classes of puzzles with 2 cuts per axis. Then we count/list the existing pieces. And then we enumerate those which are visible, just like Rob did on his website and I redid here: viewtopic.php?f=14&t=15541
The last step would be to add a set which tells which visible pieces are orientatable.
By doing so, we become independent from shape.
Example:
The Master Skewb and the FaceTurningOctahedron belong into the same class and have an identical number of cuts per axis. The Master Skewb has one type of piece more and shows no orientations of its faces.
The same is true for the cases AndrewG mentioned: HelicopterCube, Toru, RUA, EitansRhombicDodecahedron, OscarsCrazyComet (if jumbling moves are forbidden), Gelatibrain 3.3.X and 4.3.X differ from each other only by pieces sets and in some cases (like 3.3.8 and 4.3.1) only by visible orientations.

The only problem with this scheme is the number of different piece types (hence the diligence needed) but that is a necessary tribute to the complexity of these puzzles. Imagine someone would want to classify edgeturning dodecahedrons just by cutting depth => not less complicated than counting the piece types of that beast. And that last job has been done by Carl, lately.


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 Post subject: Re: Classification of Cubes
PostPosted: Fri Dec 11, 2009 4:03 pm 
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Quote:
I have thought about classification as well and came up with this:
1. A "symmetry symbol" although different named than quickfur does. Here I have left out the TWO tetrahedral symmetries. But if you want them you have to include the edgeturning tetrahedron (or Rhombic hexahedron to name it consistently) with 6 "sides" and 180°-turns-only.
2. An order number where I count the number of cuts per throughgoing axis => 2x2x2, Pentultimate, BigChop etc. would have order=1. Megaminx would have order=2
3. A piece set. Look a the animation in this thread (Thanks again to Carl Hoff!):
viewtopic.php?f=1&t=14875
You see there that the faceturning dodecahedron of order=2 can have at most 8 different pieces. Each of them could be made visible in an appropriatly designed puzzle. Sadly not all at the same time.
4. Orientations needed:
All visible pieces not directly attached to the core need to be positioned correctly. But some have to be oriented, too.

The advantage I see in using piece sets (and orientation sets) is that you can cover easily every "shape mod" (compare 3.3.1 and 3.3.8 from Gelatibrain). Even non-planar cuts are no problem: The Rex cube differs from the Master Skewb by its piece set. Axis-configuration and order are the same.

A Megaminx therefore is a faceturning dodecahedron of second order with piece set [1,2,3] and orientations for [2,3].
The Pyraminx Chrystal is a faceturning dodecahedron of second order with piece set [3,4] and orientations for [3,4].
The Icosaminx would be a faceturning dodecahedron of second order with piece set [1,2,3] and orientations for [1,2].

In addition to that 6 (or 8) classes there are the hybrid puzzles like the SuperX. Think about faceturning archimedian solids and you get some inspirations.

This is pretty brilliant.
We could collapse this into an easy-to-write form with:
"A Megaminx therefore is a faceturning dodecahedron of second order with piece set [1,2,3] and orientations for [2,3]" --> Dodeca F2[1,2*,3*] or Dodeca F{2}[1,2,3]+O[2,3] or something

This is a bit unweildy, BUT if you know what all the pieces number are (ok, that would be a bit hard to memorize) then it gives you a lot of information in a very organized, condensed form.

Part of the reason I started saying things like Dodeca FA3 (Megaminx), Cube FA2VB1 (SuperX) is because I felt like those names were just readable enough to have a chance at becoming commonly used, but also formal enough that they give good information at a glance, and compact enough to be easily typable.

I now understand better what wwwmwww was doing with those animations (before that, I hadn't realized the cutting planes were stationary).

I wonder how this will transfer to hybrids. I think that we may run into the following problem:
In Carl's animations, the puzzle changes by changing the size of the solid, keeping the cutting planes in the same place. You could say that there is 1 "degree of freedom" (1 DoF) here: the size of the solid.
By this thinking, a hybrid is a 2 DoF puzzle. That is, one parent has a certain relative size to its cutting planes, and the other parrent may have a different relative size to its cutting planes. Therefore, I don't expect that we will find a similar phenomenon among hybrids, where you eventually have a puzzle with ALL the pieces from all the same-order puzzles inside of it.

And if this doesn't happen, it might become very hard to number hybrid pieces in a meaningful way.

Carl, is there a change you could make us a similar animation for a hybrid?
For instance, I predict that if we make an animation for Cube FxxVxx, we won't see all five of: FA2VB1, FA2VC2, FA3VB1, FA3VC3_A and FA3VC3_B (see that hybrid list I posted). I'm not totally sure about that though, and I'm not sure which ones we will see and which ones we won't. I think it would be impossible to see BOTH FA2BVB1 (SuperX) and FA2VC2 (SuperO/Skew-by-2)...

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 Post subject: Re: Classification of Cubes
PostPosted: Fri Dec 11, 2009 5:57 pm 
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AndrewG wrote:
Carl, is there a change you could make us a similar animation for a hybrid?


I'll get there eventually. An order=2 hybrid with an order=1 puzzle should be easy as there is still just one degree of freedom. But you are correct, things get complicated when dealing with order=4 puzzles or in this case the hybrid of two order=2 puzzles. I talk about this problem a bit here:

http://twistypuzzles.com/forum/viewtopic.php?p=189739#p189739

To capture all piece types present in a hybrid of the order=2 corner-turn multicube and the order=2 edge-turn multicube one needs to create a Multicube that can't be expressed as a series of puzzles as the order=2 edge-turn Multidodecahedron can in the thread above. That Multicube would be a set of Multicubes where the relative spacing of the corner cuts and the edge cuts is changed. Each of those Multicubes is then in turn a set of puzzles. To specify one of these puzzles you could do something like C2E2TC-M1-P3.5 this could be read as Corner Order=2-Edge Order=2 Turn Hybrid Multicube #1 Puzzle #3.5. I haven't worked out the number of Multihedra needed for each Hybrid but it is doable.

By the way... in this thread:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=14875

I talk about Type-I and Type-II Multihedrons. I now see that Type-I is just a subset of the more general type-II so I think I'll drop that distinction. The gigaminx is but one puzzle in the Order=4 Face-Turn MultiDodecahedron. What ever we name this puzzle... lets call is F4TD-M1-P1 then if we want to talk about the puzzle which is a megaminx inside a gigaminx we could call it F4TD-M1-P1M where the last M tells us its a Multidodecahedron and that we are counting the pieces inside as well.

Carl

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 Post subject: Re: Classification of Cubes
PostPosted: Fri Dec 11, 2009 6:11 pm 
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Jeffery Mewtamer wrote:
Also, another property that seems to arise with Order-2 puzzles is that some are fixed depth and others are variable depth.


I sort of took that into account in my naming of the puzzles in my Edge-Turn Multidodecahedron seen here:
http://twistypuzzles.com/forum/viewtopic.php?p=189739#p189739

All the fixed puzzles are ETD followed by an integer. All the variable puzzles I named as the previous (shallower) fixed puzzle followed by ".5". The implication I wanted to make was that anything between ETD1.1 and ETD1.9 were in effect all the same puzzle.

Carl

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 Post subject: Re: Classification of Cubes
PostPosted: Fri Dec 11, 2009 6:29 pm 
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I too think Andreas is on to something. Look at Jason's Radiolarian (Face Turning Icosahedron) seen here:
http://twistypuzzles.com/forum/viewtopic.php?f=15&t=15597

The pieces for that puzzle would be in a Corner-Turn MultiDedecahedron which shares it's pieces with the Face-Turn MultiIcosahedron. However this puzzle as presented wouldn't be in any of my MultiIcosahedron animations. It's missing pieces that would be present in anything I could make with planar cuts. However once you start talking about the pieces its fair game just to leave them out.

A similiar issue is had with AndrewG's naming scheme. I see how it names the 4x4x4 but as it just looks at surface pieces how could you tell the 4x4x4 apart from some of the Crazy 4x4x4's. To me one is the 4x4x4 Cube and the other is the 4x4x4 Multicube.

Here are a few pictures I made to show that the Crazy 4x4x4's are really planar cut puzzles.

Image
and
Image

These are really both the same puzzle. The image on the right just shows you which surfaces of the 4x4x4 Multicube are stickered in each of the Crazy 4x4x4's on the left.

Carl

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 Post subject: Re: Classification of Cubes
PostPosted: Sat Dec 12, 2009 1:08 am 
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Interestingly, I also made the graphics in that essay with POV-Ray, and i did an interview with Texas Instruments a few months ago. (I saw you had worked for TI while reading through some of the other threads). Now I remember the guy mentioning that they had some things in Mississippi (my family moved to mississippi recently too actually, I'll be going home for the holidays after this semester is over).

overall, I feel like I missed quite I bit in my several months off of the forum! I haven't been on much since early 2009 due to school, work, and all that. When I made that comment in the skew-by-2 thread I had no idea that all of this had developed.

I'm really impressed that you made it all the way through with the edge-turning dodecahedron! A while back I went through most of the puzzles and assigned letters to the fundamentals, but I never look at all the ege turning dodecahedra.
There's something nice and unifying about the whole multi____hedron development, however the major hurdle I see is that it doesn't move up to higher orders very well, and we seem to like to build higher order puzzles.

In the morning maybe I'll have time to make a more useful comment.

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 Post subject: Re: Classification of Cubes
PostPosted: Sat Dec 12, 2009 9:47 am 
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Ok, so I gave all this some more thought. One of the things I constantly reminded myself when writing that essay was that "A problem well stated is half solved." After reading the through work the forum has collectively put into this idea, I think we have a good amount of research done, and what we need right now is a synthesis of what we've come up with.

The purpose I had when I developed my classification system was to give people an easy, organized way to talk about a subset of puzzles that was most interesting to me personally (I picked the polyhedra with planar cuts which do not change shape when scrambled, excluding jumbling). My hope was that, if these puzzles were cataloged and organized well, more people would be interested in solving and building them.

For instance, the Vestar (Dino+Heli) is an attractive, fun puzzle to solve, and it could probably be built with a mechanism paradigm like the SuperX. However it isn't on GelatinBrain, and the most publicity it's gotten was that solution page I wrote a while back on my website that doesn't work anymore. But, it is one of the four clear, non-trivial, uniform hybrids of the one-cut cubes, and the other three are well known (SuperX, SuperO, 2x2x2+Lil Chop).

With the Multihedron development, Carl discovered a totally new type of puzzle which unified things in an interesting way.

So we both basically face the same task of organization so that more really cool puzzles can get more recognition.
I thought of a way we could synthesize the two systems.

Frankly, my system doesn't include pieces below the surface at all. So it makes sense to call these surface puzzles. To contrast with that, the Multihedra would be body puzzles or bulk puzzles.

Right now surface puzzles are made by slicing at a certain cut depth. Higher orders simply have multiple cut depths. Cut depths range from 0 to 180 (or 0% to 50%).
Right now body puzzles are made by with a solid of a certain relative size to its cutting planes. Higher orders simply have two relative sizes (The initial size specifies where the cutting planes are, so if you start with two different size solids and expand them both to the same size you get a higher order puzzle). Relative sizes range from 1 to infinity.

At this point I really like saying that starting with 1 set of cutting planes means order 1 (not order 2), and what we called the "order 1" puzzles are the puzzles at infinite relative sizes. But that's just me (and Julian).

There's got to be an analytical function that maps the domain [0,180] to [1,infinity]. 180/(180-x) works but I'm not sure if its the correct one or not. This is basically like having two different units of measure, like how we have several units to measure viscosity. One uses only finite numbers, the other is more "mathematically beautiful"

The way that I classified puzzles when my system "stopped working" was to say things like, "Cube F120V160 and Cube F90V150 are the two reduced forms of FA3VC3." Using just the cut type and depth definitely "works". The use of letters and sequence numbers just relates the classification to common names.

So now we have three FORMAL ways to refer to puzzles:
1) [Multi?][Solid] [Cut Type][Depth]... Examples: Cube F120 (3x3), Cube F90,180 (4x4), Cube F180V120 (SuperX), MCube E120 (Multicube)
2) [Multi?][Solid] [Cut Type][RelSize]... Examples: Cube F3 (3x3), Cube F1.5,inf (4x4), Cube FinfV1.5 (SuperX), MCube E2 (Multicube) I don't know what the correct relative sizes are so I just guessed.
3) [Solid] [Piece Set, *if orientable]... Examples: Cube F[1,2*,3*] (3x3), Dodeca F[1,2*,3*] (Megaminx)

I'm not sure if 3 will work well with higher order puzzles, but it is more powerful for low-order puzzles. We can resolve this by saying that 1 & 2 are meant to apply to a different subset from 3. 1 & 2 are meant to refer to surface or body puzzles with ALL the generated pieces "in play", and, if it is natural, orientable (puzzles can also be made "super" by sticker variation, but this is a minor issue). 3 is meant to give a higher level of specification to lower order puzzles; specifically recognizing that it is possible to use various methods to include any combination piece types on one puzzle.

And also this INFORMAL way:
1) [Multi?][Solid] [Cut Type][Sequence][Term]... Examples: Cube FA3, Cube FA2VB1. MCube EB1.

These are more suitable for everyday conversation.

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 Post subject: Re: Classification of Cubes
PostPosted: Sat Dec 12, 2009 10:43 am 
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AndrewG wrote:
Right now surface puzzles are made by slicing at a certain cut depth. Higher orders simply have multiple cut depths. Cut depths range from 0 to 180 (or 0% to 50%).
Right now body puzzles are made by with a solid of a certain relative size to its cutting planes. Higher orders simply have two relative sizes (The initial size specifies where the cutting planes are, so if you start with two different size solids and expand them both to the same size you get a higher order puzzle). Relative sizes range from 1 to infinity.


I could have moved the cutting planes from 0 to 180 to make the Multiherda too. My choice to fix the planes and allow the puzzle to grow is equivant and was made simply because it made it easier to visualize that all the shallower cut puzzles really do exist inside the deeper cut puzzles.

Hmmm... thinking about this a bit more. If you measure the cut depth in degrees doesn't that imply the puzzle has to be a sphere? What does a cut depth of 10 degrees mean on a face-turn cube? If its a sphere I belive it has 7 pieces yet your graph jumps to 27 just after 0 degrees. I think it would look something like this:

http://www.jaapsch.net/puzzles/sphere.htm?blue=270&sym=2&angle=330,105,355

The 0% to 50% sounds better if you are measureing from the surface of the solid where the axis of rotation intersects to the core. But in the case of tetahedra where an axis enters on a corner and exits on a face center 1% on the corner side doesn't equal the same distance as 1% on the face center side... does it?

Carl

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 Post subject: Re: Classification of Cubes
PostPosted: Sat Dec 12, 2009 11:15 am 
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Thanky you, AndrewG for your motivating words.
Most of this has already been presented. What I want to present here are some details and I want to apply it to some existing puzzles.

Classification of doctrinaire puzzles:
1. The puzzles axis configuration:
  • HF - faceturning hexahedrons
  • HC - cornerturning hexahedrons
  • HE - edgetruning hexahedrons
  • DF - faceturning dodecahedrons
  • DC - cornertruning dodecahedrons
  • DE - edgeturning dodecahedrons
2. The number of cuts per axis. This definition fits in most cases.
or: The number of logical layers minus one. This definition is more universal and fits even in complicated cases like CirclePuzzles.
3. Pieces of the appropriate Multihedron, which are visible.
4. Subset of pieces which orientation is visible.

As mentioned by AndrewG the enumeration of pieces can be unintuitive but with Carls great animations and this naming scheme it could be handled.
On every puzzle of the 6 axis configurations mentioned above we get these kinds of pieces:
O => A core (1 piece) (or 0)
C => Corners (8 / 20 pieces)
F => Faces (6 / 12 pieces)
E => Edges (12 / 30 pieces)
X => X-Faces (24 / 60 pieces)
T => T-Faces (24 / 60 pieces)
W => Wings (24 / 60 pieces)
L => Obliques (48 / 120 pieces)
The number of pieces are valid for the hexahedrons / dodecahedrons.
Every piecetype is followed by a number how many of this pieces are moved in one (non-slice-)twist.

HF - faceturning hexahedrons
No animation necessary.
There are 4 pieces:
O (Core)
F[1] (Faces); 1/6 per turn
E[4] (Edges); 4/12 per turn
C[4] (Corners); 4/8 per turn

Puzzles with this axis-configuration are:
The 3x3x3 is therefore: HF2 [C4 E4 F1] [C4 E4]
I think the numbers behind the pieces types can be left away when it is unambigous: HF2 [C E F] [C E]
The Magic Octahedron is: HF2 [E F] [E F]
The EdgesOnlyCube is: HF2 [E F] [E]
The Void Cube would be: HF2 [C E] [C E]
Alehs Brillicube would be: HF2 [C E] [C E]
2x2x2: HF1[C][C]
Pyramorphix: HF1[C][C/2] (see further down)
....
Gelatibrain 5.2.1: HF2[C/2, E, F][F] (Only HalfTurns!)
Gelatibrain 5.2.3: HF2[C, E, F][F] (Only HalfTurns!)
Gelatibrain 5.2.2: HF2[O, C/2, E, F][F] (Only HalfTurns!)
There are 12 wannabe-Wings. They could be glued to the edges without changing anything and therefore belong to the edges.

HC - cornerturning hexahedrons
At first I present one piece of Carls great work:
Image
According to the naming scheme above:
Index 0: O (Core)
Index 1: C1 (Inner Corners)
Index 5: C4 (Outer Corners)
Index 2: E[3] (Edges)
Index 3: F[3] (Faces)
Index 4: X[9] (X-Faces)

Puzzles with this axis-configuration are:
DinoCube: HC2 [E][]
Rainbow Cube: HC2 [C1, E][]
ShallowCutDino: HC2 [O, C1, E][C1]
FaceTurningOctahedron: HC2 [E, F, X][F]
RexCube: HC2 [E, F, X][]
Master Skewb or Gelatibrain 4.1.4: HC2 [E, F, X, C4][C4]
DinoOcta HC2 [C1, E, F, X][F]
Skewb: HC1 [C4, F] [C4]
Skewb Diamond: HC1 [C4, F][F]
Skewb Ultimate: HC1[C4, F][C4, F]

DC - cornerturning dodecahedrons
Now, lets get to a tougher one. At first, again Carl Hoffs great work:
Image
That means the puzzles of DC2 are made out of:
Core (1 piece)
Index 0: O
Corners (20 pieces)
Index 1: C1 (Inner Corners)
Index 4: C4 (Middle Corners)
Index 14: C10 (Outer Corners)
Edges (30 pieces)
Index 2: E3 (Inner Edges)
Index 8: E9 (Middle Edges)
Index 12: E12 (Outer Edges)
Faces (12 pieces)
Index 6: F3 (Inner Faces)
Index 16: F6 (Outer Faces)
X-Faces (60 pieces)
Index 3: X9 (Inner X-Faces)
Index 9: X18 (Middle X-Faces)
Index 11: X24 (Outer X-Faces)
T-Faces (60 pieces)
Index 5: T12 (Inner T-Faces)
Index 10: T21 (Middle T-faces)
Index 15: T27 (Outer T-Faces)
Index 17: T30 (Far Outer T-Faces)
Obliques (120 pieces)
Index 7: L30 (Inner Obliques)
Index 13: L54 (Outer Obliques)

Puzzles with this axis-configuration are:
Dodecahedral Skewb (a.k.a. 1.2.9) DC1 [C10, F6, T30] [C10]
Gelatibrain 2.1.5: DC1 [C10, F6, T30] [F6]
Radiolarian: DC2 [C1, E3, X9][E3]
Dino Dodecahedron (aka 1.2.1): DC2 [O, C1, E3][C1, E3]
Gelatibrain 1.2.2: DC2 [E3, X9, C4][E3, C4]
Gelatibrain 1.2.3: DC2 [C4, F3, X9, T12][C4]
Gelatibrain 1.2.4: DC2 [C4, F3, E9, T12, L30][C4, E9]
Gelatibrain 1.2.5: DC2 [C4, F3, E9, X18, L30][C4, E9]
Gelatibrain 1.2.6: DC2 [F3, E9, X18][E9]
Gelatibrain 1.2.7: DC2 [C10, E12, X24, T21, L54][E12, C10]
Gelatibrain 1.2.8: DC2 [C10, F6, E12, T27, X24, L54][E12, C10]
Gelatibrain 2.1.1: DC2 [F3, E3, X9, T12][E3, F3]
Gelatibrain 2.1.2: DC2 [C4, F3, E3, X9, T12][E3, F3]
Gelatibrain 2.1.3: DC2 [C4, F3, X9, T12][F3]
Gelatibrain 2.1.4: DC2 [C4, F3, E9, L30, X18][F3, E9]

HE - edgeturning hexahedrons.
At first Carls animation
Image
Index 0: O (Core); 0/1 per turn
Index 3: C[2] (Corners); 2/8 per turn
Index 1: E1 (Inner Edges); 1/8 per turn
Index 7: E5 (Outer Edges); 5/12 per turn
Index 5: F[2] (Faces); 2/6 per turn
Index 2: X4 (Inner X-Faces); 4/24 per turn
Index 8: X10 (Outer X-Faces); 10/24 per turn
Index 4: T6 (Inner T-Faces); 6/24 per turn
Index 9: T12 (Outer T-Faces); 12/24 per turn
Index 6: L16 (Obliques); 16/48 per turn

Puzzles with this axis-configuration are:
LittleChop: HE1[T12][]
Helicopter-Cube: HE2[X4, C2][C2]
VertexTurningCuboctahedron (made by gingervergo): HE2[X4, C2][]
ShallowCut Helicopter-Cube: HE2[O, E1, X4, C2][E1]
Curvy Copter: HE2[E1, X4, C2][E1]
TORU: HE2[C2, F2, T6][C2, F2]
RUA: HE2[C2, F2, L16, E5, X10][C2, F2]
RhombicDod. (made by Eitan): HE2[E1, X4, C2, T6, F2][C2, F2]
Crazy Comet (without jumbling): HE2[E1, X4, C2][C2]
Gelatibrain 4.3.1: HE2[X4, C2, T6, F2][F2]
Gelatibrain 3.3.2: HE2[X4, C2, T6, F2, L16, E5][E5]
Gelatibrain 3.3.3: HE2[C2, T6, F2, L16, E5][E5]
Gelatibrain 3.3.4: HE2[C2, T6, F2, L16, E5, X10][C2, E5]
Gelatibrain 3.3.5: HE2[C2, F2, L16, E5, X10][C2, E5]
Gelatibrain 3.3.6: HE2[C2, F2, L16, E5, X10, T12][C2, E5]
Gelatibrain 3.3.8: HE2[X4, C2, T6, F2][C2]
Gelatibrain 4.3.2: HE2[C2, F2, L16, E5, X10][F2, E5]
Gelatibrain 4.3.4: HE2[C2, F2, L16, E5, X10, T12][F2, E5]

DF - faceturning dodecahedrons.
At first Carls animation
Image
Index 0: O (Core): 0/1 per turn
Index 3: C5 (Inner Corners); 5/20 per turn
Index 7: C10 (Outer Corners); 10/20 per turn
Index 2: E5 (Inner Edges); 5/30 per turn
Index 4: E10 (Outer Edges); 10/30 per turn
Index 1: F1 (Inner Faces); 1/12 per turn
Index 6: F6 (Outer Faces); 6/12 per turn
Index 5: X[25] (X-Faces); 25/60 per turn

Puzzles with this axis-configuration are:
Pentultimate: DF1[F6, C10][C10]
Gelatibrain 2.2.6: DF1[F6, C10][F6]
Starminx: DF2[E10, X25, F6][E10]
Megaminx: DF2[F1, E5, C5][E5, C5]
Kilominx (aka Impossiball) DF2[C5][C5]
Pyraminx Crystal: DF2[C5, E10][C5, E10]
Alexander Star: DF2[E5][E5]
Pyracosaminx (aka Gelatibrain 2.2.1): DF2[F1, E5][F1, E5]
Gelatibrain 1.1.6: DF2[E10, X25, F6, C10][E10, C10]
Gelatibrain 1.1.2: DF2[F, E, C, E10][E, C, E10]
Gelatibrain 1.1.4: DF2[E10, X25, F6, C10][E10, C10]
Gelatibrain 2.2.2: DF2[F, E, C][F, E]
Gelatibrain 2.2.3: DF2[C, E10, X25, F6][E10, F6]
Gelatibrain 2.2.4: DF2[E10, X25, F6][E10, F6]
Gelatibrain 2.2.5: DF2[E10, X25, F6, C10][E10, F6, C10]

What to do with the tetrahedrons?
I want to include them in the CH-class. This is supported by the behaviour of the FaceTurningOctahedron: The 24 Triangles (or X-Faces to stay in terminology) can be separated into two subsets which can't be intermingled with each other.
So my suggestion for these puzzles are:

HMT: HC1[C4, F][C4/2, F] (Only one half of the C4-pieces have visible orientation)
Pyraminx: HC1[C4/2, F][C4/2, F] (Only one half of the C4-pieces are visible)
EyeSkewb: HC1[C4/2, F][C4/2]
Gelatibrain 5.1.1: HC1 [O, C4/2, F][C4/2, F]
...
Please note that this submethod is not limited to tetrahedrons.
Shim's F-Skewb: HC2 [C4/2, E, X/2][C4/2]

The edgeturning tetrahedrons can be treated similarly as puzzles with axis=HF but restricted to halfturns.

In the last part I have left out the MasterPyraminx on purpose because there I have to extend this classification scheme with another concept which is to complex to introduce it in this single posting. But let me assure you that it can be done. These additions are needed to include the CirclePuzzles as well.

Remarks
  • The system of differentiating Obliques/T-Faces/... from each other by the number of pieces in a turn can be appliead to every class of cuts=2-puzzles. Only for the DE2 it fails in three cases (see kappa and lambda on Carls animation).
  • The notation can't reflect things like the EightColor-4x4x4 (with several identical faces) yet. Another example would be the Mastermorphix.
    My suggestion would be: FH2 [C, E(3+3+3+3), F][C/2, E, F] (there are 4 sets of 3 edges which look identical; one half of the corners have visible orientations)
  • I prefer "cornerturning" above "vertexturning". Just a matter of taste.
  • Since we speak about "dodecahedrons" it seems more consistent to use "hexahedron" instead of "cube". Again a matter of taste.
  • The abbreviations for the 6 axis-configurations are just suggestions.
  • I like AndrewG's idea to fuse orientations into the piece-set but that wouldn't allow to describe the HMT. Maybe somebody has a better idea for that case.
  • I can't classify up to now:
    • Hybrid puzzles with cuts<=2
    • Puzzles with cuts>=4
    • Prisms (with odd and even number of edges) could be done, but there is no animation, yet.

EDIT: corrected spelling and added some puzzles
EDIT2: Updated the concept a bit. Added F-Skewb. Deleted MasterPyraminx.
EDIT3: Refined some definitions.


Last edited by Andreas Nortmann on Tue Jul 19, 2011 11:36 am, edited 11 times in total.

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 Post subject: Re: Classification of Cubes
PostPosted: Sat Dec 12, 2009 12:40 pm 
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wwwmwww wrote:
Hmmm... thinking about this a bit more. If you measure the cut depth in degrees doesn't that imply the puzzle has to be a sphere? What does a cut depth of 10 degrees mean on a face-turn cube?


Quick comment: The 0-180 do NOT refer to degrees. They just linearly go from 0-180 in increasing depth below the given feature. Wouter Meesen used this instead of 0-50% for the UMC program, and I think this was wise because by using 0-180, more common puzzles are at integer depths. (e.g. its much nicer to have the 3x3x3 at depth 120 than at 33.3%)

Andreas I will actually read your post when I have more time.

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 Post subject: Re: Classification of Cubes
PostPosted: Sat Dec 12, 2009 7:42 pm 
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Andreas... boy you have been busy. But I'm confused on a few of the puzzles. Take this one for example.

DinoCube: CH2 [E][]

Why isn't the DinoCube CH2 [E][E]? The edges are 2 color at least on the 6-color DinoCube.

And to avoid listing the pieces twice maybe you could use something like this:

3x3x3 = FH2[C, E, f]

where upper case is used if orientation is visible and lower case if not. And in cases where the orientation is seen on some of the pieces and not others we could use both.

HMT: CH1[C4, c4, F]
Pyraminx: CH1 [C4/2, F]

The "/2" works well when only half are visible.

Carl

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 Post subject: Re: Classification of Cubes
PostPosted: Sun Dec 13, 2009 4:49 am 
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wwwmwww wrote:
Why isn't the DinoCube CH2 [E][E]? The edges are 2 color at least on the 6-color DinoCube. I made an error in
The E-pieces cannot be oriented. See Jaaps page for clarification.
wwwmwww wrote:
And to avoid listing the pieces twice maybe you could use something like this:
...
I have always hated case sensitive programming languages.
So I suggest a comeback of AndrewG's idea:
HMT: CH1[C4/2*, C4/2, F*]

Now that I have posted that system I have discovered a problem:
My handling of the tetrahedrons doesn't work. HMT and Pyraminx are no problems but MasterPyraminx is.
The Edge-Pieces of the MasterPyraminx are orientable, so they can't be equivalent to the E-pieces of CH2 which are never orientable. And they can't be F-pieces of CH2 because then a slice move of the Pyraminx wouldn't move them. No solution yet.


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 Post subject: Re: Classification of Cubes
PostPosted: Sun Dec 13, 2009 10:12 am 
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Andreas Nortmann wrote:
The E-pieces cannot be oriented. See Jaaps page for clarification.

Ahhh... I see that. http://www.jaapsch.net/puzzles/dinocube.htm

Andreas Nortmann wrote:
I have always hated case sensitive programming languages.
So I suggest a comeback of AndrewG's idea:
HMT: CH1[C4/2*, C4/2, F*]

Looks good.

Andreas Nortmann wrote:
Now that I have posted that system I have discovered a problem:
My handling of the tetrahedrons doesn't work. HMT and Pyraminx are no problems but MasterPyraminx is.
The Edge-Pieces of the MasterPyraminx are orientable, so they can't be equivalent to the E-pieces of CH2 which are never orientable. And they can't be F-pieces of CH2 because then a slice move of the Pyraminx wouldn't move them. No solution yet.

I think the fix there may be to consider the MasterPyraminx an order=3 puzzle. They are built up from Skewb Diamonds which are a deep cut puzzle and the puzzle is really half missing. Think of a Skewb Diamond where all the faces are built up the same way. Someone actually built one once as I remember seeing pictures of it but I can't remember what he named it. Twin Master Pyraminx comes to mind but that must not be it as I still can't find it. Anyways I think you need CH3 for identifing the pieces. And if you get really picky I think the trivial tips are piece C1 from CH5.

Also aren't you missing the trivial tips off the Pyraminx too? And taking a closer look I see you are using CH1 for the Pryaminx. As Ch1 is a deep cut puzzle and the Pyraminx isn't shouldn't you be using CH2? The MultiPryaminx would have a core piece too.

How about this?
Pyraminx: CH2[C1/2*, C4/2*, F]
Tetraminx: CH2[C1/2*, F]

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 Post subject: Re: Classification of Cubes
PostPosted: Sun Dec 13, 2009 10:23 am 
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Andreas Nortmann wrote:
What to do with the tetrahedrons?
I have considered these puzzles as included in the CH-class.


It depends on the puzzle. Here is one that isn't. ;)

Pyramorphix: FH1[C4/2*, C4/2]

And as I didn't see the 2x2x2 on your list.

2x2x2: FH1[CH4*]

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 Post subject: Re: Classification of Cubes
PostPosted: Sun Dec 13, 2009 1:56 pm 
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I added the Pyramorphix and the 2x2x2
and the edgeturning tetrahedrons from Gelatibrain

Considering the Pyraminx with trivial tips as CH2 doesn't work.
To get my way through this "cutting in half" of a tetrahedral (with 2 cuts) puzzle, I did this:
How many pieces of a kind are moved by a turned face?
How many pieces of a kind are moved by the opposite turned tip?
Add these numbers and you should get numbers which you find in my enumerations.
Doing this with the Pyraminx tells you that both the tips and the pieces below them are C4.

I found another way to classify the MasterPyraminx: I introduce virtual pieces.
Think about the "piece" which is left from the MatserSkewb if you turn these sides and slices
Sides: DLB DRF URB ULF DLF DRB
Slices: URF DRF
Lets call this piece VE ("virtual edge")
One the Master Skewb, nothing is left, but you can explain the edges of the MasterPyraminx like this

4 tips = C4/2
4 faces = C1/2
12 wings = X/2
6 Edges = VE/2

And the same is possible on the other tetrahedrons of Gelatibrain:
5.1.5
4 tips = C4/2
12 wings = X/2
12 triangles = X/2
6 edges = VE/2

5.1.6
4 tips = C4/2
4 faces = C1/2
12 wings = X/2
6 edges = VE/2
12 triangles = E

5.1.7
4 tips = C4/2
4 faces = C1/2
12 wings = X/2
6 edges = F
12 triangles= E

5.1.8
4 tips= C4/2
12 wings = X/2
6 edges= F
12 triangles = X/2
12 rhombi = E

5.1.9
4 tips = C4/2
4 faces = C4/2
12 wings = X/2
6 edges = F
12 trapezoids = X/2
12 rhombi = E

The advantage is, that the MasterPyraminx can stay with order=2.
I wanted that badly because 5.1.[7-9] are order=2 too.
Anyway: This is just meant as a suggestion.

I see that I have classified Gelatibrain 5.1.1 as CH1 and mentioned a core, which is not physical on a deepcut-CH-puzzle. But again a virtual piece (the aforementioned core) can be introduced easily.

I have given up my investigations about these virtual pieces. MAYBE I was to fast.


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 Post subject: Re: Classification of Cubes
PostPosted: Sun Dec 13, 2009 4:42 pm 
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Andreas Nortmann wrote:
I found another way to classify the MasterPyraminx: I introduce virtual pieces.
Think about the "piece" which is left from the MatserSkewb if you turn these sides and slices
Sides: DLB DRF URB ULF DLF DRB
Slices: URF DRF
Lets call this piece VE ("virtual edge")

The advantage is, that the MasterPyraminx can stay with order=2.
I wanted that badly because 5.1.[7-9] are order=2 too.
Anyway: This is just meant as a suggestion.

I see that I have classified Gelatibrain 5.1.1 as CH1 and mentioned a core, which is not physical on a deepcut-CH-puzzle. But again a virtual piece (the aforementioned core) can be introduced easily.

I have given up my investigations about these virtual pieces. MAYBE I was to fast.


I reserve the term "virtual" piece for a piece which doesn't physically occupy any volume inside the puzzle but can be matematically shown to be there. Like the virtual Kilominx corners inside a Pentultimant. I still don't quite understand them but you don't need them to to explain the MaterPyraminx. All the pieces in a MasterPyramanix have volume obviously. If you want to think of it as a member of the CH-class its a simple mater of finding the CH puzzle that contains it. Here it is:

Image

It is order 5 and as there are 3 types of corner turns each piece needs 3 numbers. I had to study this animation for a bit before I could name them all.

Image

All cut planes are fixed and the puzzles grows.

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 Post subject: Re: Classification of Cubes
PostPosted: Sun Dec 13, 2009 5:26 pm 
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Andreas Nortmann wrote:
Considering the Pyraminx with trivial tips as CH2 doesn't work.
To get my way through this "cutting in half" of a tetrahedral (with 2 cuts) puzzle, I did this:
How many pieces of a kind are moved by a turned face?
How many pieces of a kind are moved by the opposite turned tip?
Add these numbers and you should get numbers which you find in my enumerations.
Doing this with the Pyraminx tells you that both the tips and the pieces below them are C4.


Ok... I thought this too would be a simple mater if finding the CH-Class puzzle that contained the Pyraminx BUT I now see it isn't that easy. Here is a corner-turn hexahedron that contains it...

Image

But its not in what we've called the CH-Class. It has two cut planes per axis of rotation but for each axis they are on the same side of the core!? If you copy these cuts to all 8 corners of the hexahedron as you do in the CH-Class you get these new blue cuts.

Image

So I guess you could make the Pyraminx from a bandaged CH4-Class puzzle but it may be better to create a new class. Call it Cornerturning Tetrahedra (CT) and this puzzle is then a CT2 puzzle as is the cube that contains it in the first picture. Some of the CT(n) puzzles exist inside the CH(n+2) class as is the case for the MasterPyraminx. It's only the CT(n=odd) puzzles where the second deepest cut is no deeper then a Dino cut that I believe this works for. For the CT(n=even) puzzles it only works if the deepest cut is no deeper then a Dino cut. At least that if how I think it plays out.

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 Post subject: Re: Classification of Cubes
PostPosted: Mon Dec 14, 2009 1:12 am 
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wwwmwww wrote:
Some of the CT(n) puzzles exist inside the CH(n+2) class as is the case for the MasterPyraminx. It's only the CT(n=odd) puzzles where the second deepest cut is no deeper then a Dino cut that I believe this works for. For the CT(n=even) puzzles it only works if the deepest cut is no deeper then a Dino cut. At least that if how I think it plays out.


Well the above is wrong... I've thought about this a bit more.

CT1 = CH1 if its a deep cut. Unlike the cube with has two opposite corners which forces all CH1 puzzles to be deep cut CT1 puzzles don't have this restriction as they only have 1 corner per axis of rotation. CT1 and CH1 both have 4 cut planes per axis of rotation.

CT1 = CH2 if the deepest cut is no deeper then a Dino cut and the puzzle shape is a Tetrahedra. If it is a Cube CT1 ≠ CH2.

CT2 = CH3 if the deepest cut is "deep cut" AND the second deepest cut is no deeper then a Dino Cut AND the puzzle shape is a Tetrahedra. Note: here we have an even order class with a deep cut. That isn't allowed in any other even class.

CT2 = CH4 if the deepest cut is no deeper then a Dino cut and the puzzle shape is a Tetrahedra. If it is a Cube CT2 ≠ CH4. In general CT2 has 8 cut planes while CH4 has 16 cut planes, however if the puzzles is a Tetrahedra and the deepest cut is no deeper then an Dino cut then half those cut planes are outside the surface of the puzzle.

CT3 = CH5 if the deepest cut is "deep cut" AND the second deepest cut is no deeper then a Dino Cut AND the puzzle shape is a Tetrahedra.

Basically the restrictions where a CT class is identical to a CH class get tighter and tighter with the higher orders.

For example, assume we had the next higher order Pyraminx after the Master Pyraminx. I assume we'd call that the Professor Pyraminx? If we look at the CT class as just turns about the 4 corners of the Tetrahedron then the deepest cut in that puzzle is actually deeper then a "deep cut". It's on the other side of the core from the corner about which it can turn. This isn't allowed in the Hexahedra or Dodecahedra classes. So I really think the Cornerturning Tetrahedra need a class of their own. A Faceturning Tetrahedra class isn't need which I hope is obvious and you don't need an Edgeturning Tetrahedra class either. All the Edgeturning Tetrahedra should map to the same order Faceturning Hexahedra as an edge is always opposite an edge and there are 6 edges on a Tetrahedra just as there are 6 faces on a cube.

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 Post subject: Re: Classification of Cubes
PostPosted: Mon Dec 14, 2009 9:59 am 
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By the way... the CT class can and has been used to make non-Tetrahedral puzzles.

Look at the EyeSkewb:
http://twistypuzzles.com/forum/viewtopic.php?f=15&t=13761

This is a CT2 class puzzle while the CH2 class puzzle of the same geometry is the Rex Cube:
http://twistypuzzles.com/forum/viewtopic.php?f=15&t=12659

And something else I noticed... you have to be careful using this naming scheme for the CT class:

Quote:
O => A core (1 piece) (or 0)
C => Corners (8 / 20 pieces)
F => Faces (6 / 12 pieces)
E => Edges (12 / 30 pieces)
X => X-Faces (24 / 60 pieces)
T => T-Faces (24 / 60 pieces)
W => Wings (24 / 60 pieces)
L => Obliques (48 / 120 pieces)


I think when naming the pieces you must think of them as part of a cube and even there you now have 2 types of Corners. When you start thinking about Tetrahedra you must be aware that Faces become Edges; X-Faces become Wings; Corners can remain Corners or become X-Faces or Faces.

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 Post subject: Re: Classification of Cubes
PostPosted: Mon Dec 14, 2009 4:52 pm 
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This thread has degenerated into a dialog. Sad but inevitable.

I have added a signature for the EyeSkewb neighboring to the Pyraminx, it resembles.

Seems like we have two possibilities regarding the cornerturning tetrahedrons:
1. Treat them as members of CH and accept virtual pieces => One for CH1. Three for CH2 and so far one is still unneeded.
2. Make up your proposed CT-class.

The second solutions seems to have the advantage that it doesn't need the halving notation I used for Pyraminx and the like. But we need that notation in every case for the edgeturning tetrahedrons and Pyramorphix, Mastermorphix, ...

I prefer the first way because when I touched the Skewb for the first time and discovered how it works I was remembered to the Pyraminx and was immediately able to solve faces and one subset of corners.
But so far that is just me.

You can see, that there are just 3 types of CT's with one cut per axis: 5.1.1 and lookalikes, Pyraminx (without trivial tips) and HMT with lookalikes. This could be presented in an animation similar to yours I used above. Everything we get to see is:
6 Edges
4 tips
4 faces (in some cases); the "brothers" of the tips
1 core (in some cases)
That are the 2 real piece-types of the Skewb and the third piece type which is non-existent on the Skewb.

Now imagine you would make a similar animation which shows succesively all variants of CT with 2 cuts per axis. I am sure, that we wouldn't see more than the 6 pieces of CH2 and 3 additional pieces.

To be honest I don't understand what is so important on the trivial tips of the Pyraminx. You could easily add trivial tips to the HMT and even the 3x3x3. We could treat this variants as order+2 variants but that doesn't make sense to me. The formalism proposed by me concentrates on which piece types have to be solved (with or without orientation) because they are visible and there is nothing (non-trivial) to be solved on regarding the trivial tips. You could argue that there is nothing to solve if a puzzles core or the directly neighboring pieces are visible. That is true but at least these pieces make the global orientation of the puzzle visible.

BTW: Through this discussion I have found a way to make the two virtual pieces (but just one at a time) of a Pentultimate visible. But that is another discussion.


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 Post subject: Re: Classification of Cubes
PostPosted: Mon Dec 14, 2009 6:06 pm 
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Sorry to break the dialogue guys.
What exactly do you mean by "deeper-than-deep-cut", Carl?
The way I see it, deep cut splits puzzles in half and any less than that is shallower cut. Unless your talking about the fact that pyraminx/master pyraminx etc. type puzzles have cuts that cut the puzzle into assymetric halves...

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 Post subject: Re: Classification of Cubes
PostPosted: Tue Dec 15, 2009 10:23 am 
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elijah wrote:
Sorry to break the dialogue guys.
What exactly do you mean by "deeper-than-deep-cut", Carl?


Maybe this picture will help.
Image

This only applies to the CT group (if we accept that as a group). Since an axis of rotation that goes through the core and a corner of a Tetrahedra only goes through 1 corner the name CT implies we are talking about turns that rotate about that corner. If you measure the deepth of a cut as the distance from the corner it turns around then a cut on the other side of the core from the corner is "deeper" then deep cut. By definition the deep cut goes through the core. If this were a cube you'd be closer to the opposite corner and you'd just measure your distance from there and call it a shallow cut.

elijah wrote:
The way I see it, deep cut splits puzzles in half and any less than that is shallower cut. Unless your talking about the fact that pyraminx/master pyraminx etc. type puzzles have cuts that cut the puzzle into assymetric halves...


This isn't quite what I was talking about but you do mention an interesting point. The Deep Cut on the Master Pryaminx doesn't cut the puzzle into two isomorphic groups. I thought that was one property all deep cuts should have based on the discussion here:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=11001
Hmmm... not sure what to do about that. Should I raise this as a question in that thread? Or has this been addressed already and I missed it?

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 Post subject: Re: Classification of Cubes
PostPosted: Tue Dec 15, 2009 11:56 am 
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I took Andreas’s post the wrong way and took this discussion to email. He’s ok with this being in public as you’ll see below and I know we’d both like to see other views on this subject so I’m taking this back here to the forums…

Oh and I want to add Andreas and I will be starting a thread on “Virtual” pieces soon. It’s a complicated topic so be patient.

Andreas Nortmann wrote:
This thread has degenerated into a dialog. Sad but inevitable.

Carl via email wrote:
Sorry... did you want me to take this to email?

Andreas via email wrote:
No. That was a piece of self-talk and I just wanted to challenge other members to step in again.
I didn't wanted to post this E-Mail in that thread without allowance => forum rules. But if you want to move back into public, please do so. Sorry for that misunderstanding.


Done… and thanks.

Andreas Nortmann wrote:
Seems like we have two possibilities regarding the cornerturning tetrahedrons:
1. Treat them as members of CH and accept virtual pieces => One for CH1. Three for CH2 and so far one is still unneeded.
2. Make up your proposed CT-class.

Carl via email wrote:
I think part of my problem is I'm not seeing what is "virtual" about any of the pieces in a Pyraminx. I feel like I'm missing something obvious.

Andreas via email wrote:
On the Pyraminx is nothing virtual if we ignore the tips. By doing so, we can press it into CH1.


Well the MultiPyraminx does have a core too in addition to the tips.

Andreas via email wrote:
BUT how do we explain Gelatibrain 5.1.1 ?


That is a shallow cut Pyraminx that exposes the core. So far in all the other classes all order=1 puzzles have been deep cut. However in my picture the CT class allows shallow cut puzzles in order=1. Can you treat this as a shallow cut CH order=1 puzzle? Yes... and light bulb goes off... its the virtual core of the skewb that is attached to the 4 corners that are present. The other virtual core is missing as the other 4 corners are missing. I see that much now. So I can now see how you can label the Tetraminx an order=1 CH puzzle with "virtual" pieces. More on the Pryaminx below.

Andreas via email wrote:
There is a core visible, which isn't there on the Skewb.
We have dicovered three types of pieces on the Skewb:
Corners, Faces and a type which we considered virtual back then because on the Skewb it didn't occupy any non-zero volume. But on the Pyraminx it occupies some volume and on Gelatibrain 5.1.1 it is even visible.


Very very interesting. That gives me an idea… I see how you could make an applet that showed all 3 piece types of the Skewb with each type having positive volume, including both of the virtual pieces (there are two of the same type). I’ll save that for the virtual piece thread.

Carl via email wrote:
If you have tied this in with the "virtual" pieces you may very well be on to something I just don't see yet. I have wondered if some of those virtual pieces we were finding were tied to the fact that we were fixing the cut plane to be "deep cut" in the order=1 puzzles. And I've even thought a bit about the fact that we fix all the cuts at the same deepth in order=2 puzzles. Think of a 3x3x3 where the cuts on each of the 3 axes are at different deepths. You still have a puzzle... its a 3x3x3 that's limited to 90degree turns. In the case of the 3x3x3 this doesn't expose (give volume to) any of the virtual pieces as it has none, but could something similiar be done to the other puzzles? If so how... If you try the same to a Megaminx you don't have a functional puzzle any more... I don't think.

Andreas via email wrote:
I think you got it.
Cuts on the same axis at different depths! The cornerturning tetrahedrons allow such cuts without leading to a non-doctrinaire puzzle.


So what about the other virtual pieces that are outside the CH class? I think you touch on this below and it’s probably a topic best saved for the virtual piece thread.

Andreas Nortmann wrote:
To be honest I don't understand what is so important on the trivial tips of the Pyraminx. You could easily add trivial tips to the HMT and even the 3x3x3. We could treat this variants as order+2 variants but that doesn't make sense to me.

Carl via email wrote:
Well I'm just seeing those pieces as trivial because the pieces around them are missing due to the puzzles shape.If it were a cube with the same cuts they wouldn't be trivial. Think of a 3x3x3 like puzzle that just shows the face centers. These pieces are still obviously the face centers of a 3x3x3. Sure they are now trivial to solve but that doesn't really change what the piece is. For this notation you could make a rule that trivial pieces are ignored but I just like systems were the rules are minimized. It's just a personal preference thing.

Andreas via email wrote:
Makes sense. Really.
The problem is, that you would have to consider the Pyraminx (at least) as order=2.
And the Master Pyraminx had to be considered as order=3.
Wouldn't it be nice to reduce the order of a puzzle as far as possible?


I think we are coming at this from different directions. I would sort of prefer a set of rules and definitions that are as simple as possible with a minimum number of exceptions. You want a notation for the puzzles that is as simple as possible. However the price you pay for that is you need to modify the definition of order. I view order as simply the number of cut planes per axis of rotation. Period. With that notation it becomes the number of cut planes per axis of rotation that don’t result in the creation of trivial pieces. And then you need a definition for trivial pieces. I’m ok with calling the Tetraminx order=1 if we allow virtual pieces and you’ve got mean leaning that direction, because I think it may be more general. The CT class only deals with tetrahedral puzzles. As we know dodecahedral puzzles have virtual pieces too.

Carl via email wrote:
Hmmm... Do you consider the Tetraminx as the same puzzle as the Pyraminx? From a solving perspective they are the same... physically they are a bit different. I honestly don't know what is the best way to deal with that. I was just trying to present a consistent way which would show the pieces in common if you listed it next to the cubical version of the same puzzle.

Andreas via email wrote:
Yes. I consider them as equal. As does Jaap. As does Gelatibrain.
And I think I know what you are struggeling with. You haven't insulted me or anything.


Let me ask it a different way… Do the puzzles names Tetraminx and Pyraminx have the same number of permutations? If not how can they be called the same puzzle?

Andreas Nortmann wrote:
BTW: Through this discussion I have found a way to make the two virtual pieces (but just one at a time) of a Pentultimate visible. But that is another discussion.

Carl via email wrote:
Now that is something I HAVE to see? Since we've raised the topic of virtual pieces on the boards would you mind if I started a thread there to talk about the subject. I wanted to use the 2x2x2, the Skewb, and then the Pentultimate as examples. I don't plan on going any deeper then that.

Andreas via email wrote:
Accepted. But this time lets present something finished together.
I will write something up and send you an E-Mail later this day.


There was a bit more to Andreas’s email but I’ll save that for the thread we should have up soonish (a day plus or minus a day).

I hope there are some math geeks out there like me… I’ve been wanting to talk about this topic for a while now.

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 Post subject: Re: Classification of Cubes
PostPosted: Tue Dec 15, 2009 1:17 pm 
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Lets answer this before I get something to eat and some sleep...
wwwmwww wrote:
So what about the other virtual pieces that are outside the CH class?
I need one of the virtual pieces of CH2 to press the MasterPyraminx, Gelatibrain 5.1.5 and Gelatibrain 5.1.6 into CH2, too. I mentioned it above (on 13th Dec) and tried to describe it.
wwwmwww wrote:
Andreas via email wrote:
Wouldn't it be nice to reduce the order of a puzzle as far as possible?
I think we are coming at this from different directions. I would sort of prefer a set of rules and definitions that are as simple as possible with a minimum number of exceptions. You want a notation for the puzzles that is as simple as possible. However the price you pay for that is you need to modify the definition of order. I view order as simply the number of cut planes per axis of rotation. Period. With that notation it becomes the number of cut planes per axis of rotation that don’t result in the creation of trivial pieces. And then you need a definition for trivial pieces. I’m ok with calling the Tetraminx order=1 if we allow virtual pieces and you’ve got mean leaning that direction, because I think it may be more general. The CT class only deals with tetrahedral puzzles. As we know dodecahedral puzzles have virtual pieces too.
A definition of trivial tips should be easy: "A turn is possible which orients these pieces and never moves or orients other pieces."
Another very practical disadvantage of trivial tips is that we haven't analyzed order=4+ yet.
I have reread your posts from 14th Dec. You come to the conclusion that you can't find the appropriate CH-order for the Pyraminx which should be less than 5 because CH5 would contain the MasterPyraminx with trivial tips. To get around this you want to define CT as the seventh class. If we do so, we have to make a seventh analysis of twistability (sorry to the other readers). One year ago I did that for a Tetraminx, discovered the equivalence to the Skewb and deleted that. Analyzing the twistability of CTx we will get the exact results of CHx because of the equivalent configuration of axis.
wwwmwww wrote:
There was a bit more to Andreas’s email but I’ll save that for the thread we should have up soonish (a day plus or minus a day).
I am afraid it will be more than a day. But I will try my best.
BTW: Thank you to elijah and all silent readers we haven't lost yet.


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 Post subject: Re: Classification of Cubes
PostPosted: Tue Dec 15, 2009 6:33 pm 
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I'm still reading too :)
I am right in the middle of final exams right now so I don't have much time to contribute (or comprehend) right now.

This discussion reminds me that I started an essay to try to apply my classifications to tetra and octahedra, but never finished. Hopefully over the winter break I can do some more work on that. I'll also be doing .stl files though... I just finished my first printed & cast Dino cube and I hope to do some more puzzles next semester :)

It sounds like right now our trouble is with Tetrahedra. I'm just going to agree here; there's something very odd about those little buggers.

I actually also had some trouble "picking the fundamentals" for the tetrahedra and octahedra, but I am thinking what ya'll have discovered will resolve that... perhaps.
One result I had was that there are MORE of my so-called "fundamental" edge turning octahedra than edge turning cubes. :?

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 Post subject: Re: Classification of Cubes
PostPosted: Tue Dec 15, 2009 7:54 pm 
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I am still reading, though I have little to comment on. Honestly, I feel like a 1 Dan mathematician reading the musings of 9 Dans.

The tetrahedron is the most peculiar of the platonic solids. It is self-dual, and while the Hexa-Octa Compound and Dodeca-Icosa Compound have an Archimedean solid as their cores* and a Catalan solid as their shells**, the Tetra-Tetra compound has platonic solids as both its core and shell***. The tetrahedron's unusual behavior certainly does not end with the shape itself, but permeates all twisty puzzles based on it.

*By core, I mean the volume of overlap. For the Hexa-Octa compound, this is the Cuboctahedron and for the Dodeca-Icosa compound, the Dodecaicosahedron.
**The shell is the polyhedron that connects all vertexs of the compound. The Hexa-Octa compound gives rise to the Rhombic Dodecahedron and the Dodeca-Icosa compound gives rise to the Rhombic Tricontahedron.
***The core is an Octahedron; the shell is a hexahedron.

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 Post subject: Re: Classification of Cubes
PostPosted: Wed Dec 16, 2009 12:08 pm 
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@Jeffrey Mewtamer:
No more than 5 Dan. :)
Correct observations. LATER we should visit the tetra-tetra compounds. First finish these virtual pieces...
Andreas Nortmann wrote:
One result I had was that there are MORE of my so-called "fundamental" edge turning octahedra than edge turning cubes.
Hrrm. Have you taken a look at faceturning octahedra? Is there a different number of fundamentals then in the cornerturning hexahedra?
Anyway, I am sure, they can all be covered with piece sets.


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 Post subject: Re: Classification of Cubes
PostPosted: Fri Dec 18, 2009 9:00 am 
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Andreas Nortmann wrote:
wwwmwww wrote:
There was a bit more to Andreas’s email but I’ll save that for the thread we should have up soonish (a day plus or minus a day).
I am afraid it will be more than a day. But I will try my best.
BTW: Thank you to elijah and all silent readers we haven't lost yet.


It's up. The thread is here:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=15667

I'm working on an animation that I think shows how the two virtual pieces inside the Skewb behave. It may take POV-Ray a day or so to render all the frames but it will be posted in that thread. Please check it out and let us know what you think.

This topic has been on my mind for a while... its what I meant when I refered to the "abstract analysis of the Pentultimate" here:
http://twistypuzzles.com/forum/viewtopic.php?p=182577#p182577

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 Post subject: Re: Classification of Cubes
PostPosted: Tue Dec 22, 2009 9:26 am 
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Andreas Nortmann wrote:
Hrrm. Have you taken a look at faceturning octahedra? Is there a different number of fundamentals then in the cornerturning hexahedra?
Anyway, I am sure, they can all be covered with piece sets.


You are right that they can all be described with piece sets. I realize now that the reason you get a different number of fundamentals is because, when you use a different shape, some pieces disappear and other appear. This is why there is not face-turning octahedron that corresponds to the dino cube, etc (I'm still only talking about planar cuts).

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 Post subject: Re: Classification of Cubes
PostPosted: Mon Dec 28, 2009 12:04 pm 
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wwwmwww wrote:
I think the fix there may be to consider the MasterPyraminx an order=3 puzzle. They are built up from Skewb Diamonds which are a deep cut puzzle and the puzzle is really half missing. Think of a Skewb Diamond where all the faces are built up the same way. Someone actually built one once as I remember seeing pictures of it but I can't remember what he named it. Twin Master Pyraminx comes to mind but that must not be it as I still can't find it.


I found it.
http://www.shapeways.com/model/48711/tetrahedral_twins.html

This is the puzzle I was looking for:
Image

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 Post subject: Re: Classification of Cubes
PostPosted: Fri Jan 01, 2010 4:48 am 
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Good that you mention this, Carl.
I made up the pieces list for CH3, by splitting up the pices of CH2.
The puzzle you show there is:
CH3 [C13 X39] [C13]

This thread has come to some sort of a dead point.
What is the next step?


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 Post subject: Re: Classification of Cubes
PostPosted: Sun Jan 03, 2010 10:50 am 
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regarding next steps:

Overall, we need to distill what we know and make it relevant.

The three threads we have going now are very good as far as distillation goes. Your post on virtual pieces is good. So if we get to a step of "finalizing" our knowledge in any other areas, I think we should make similarly well-written discussion threads.

Something like this also needs to be made relevant to the community. Obvious, not very many different people have been participating in this thread right now, because not that many people NEED this kind of theory for whatever they do with puzzles. Right now, what we're doing is relevant to:

1) The puzzle builders who are designing new mechanisms (usually printing and casting them). Obviously this is limited by the fact that only so many people have the drive to put in the kind of time required to do something like this. A puzzle builder might set out to build all of a certain subset of puzzles (like Andreas has).

2) The people who solve many different types of puzzles. Most notably the types on GelatinBrain or UMC. There are more people doing that now, but I feel like the crowd for this is relatively small. I tried to start a "weekly random puzzle FMC" a while ago but never got any submissions, but if something like this were to catch on that would be ver good too. I originally got interested in the subject in this way, and I feel like classification gives people a subsets of puzzles that they can "finish". That is, someone could solve all the fundamental cubes (I think I was the first to do so, but I also defined that set :lol: ), or all the face/vertex turning cube hybrids, etc. Part of the problem here is that we don't have the puzzles to play with physically, AND physically they will be considerably harder because there's no undo button in real life :) .

I guess what I mean to say by this is that I don't think people who only speedsolve are going to be very interested here.

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Last edited by AndrewG on Fri Jan 08, 2010 9:58 pm, edited 1 time in total.

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 Post subject: Re: Classification of Cubes
PostPosted: Sun Jan 03, 2010 3:44 pm 
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Okay, the picture of the double master pyramorphix there does make me understand how the 3rd layer is deepcut, because it is based on a skewb diamond, but what about the elite(or proffessor) pyramorphix? Is there a possible octahedron puzzle of that? Maybe if we were to look at that the octahedron puzzle, then break it down to the actual puzzle, we would understand what this "deeper-than-deepcut" really is.

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 Post subject: Re: Classification of Cubes
PostPosted: Mon Jan 04, 2010 11:53 am 
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elijah wrote:
Okay, the picture of the double master pyramorphix there does make me understand how the 3rd layer is deepcut, because it is based on a skewb diamond, but what about the elite(or proffessor) pyramorphix? Is there a possible octahedron puzzle of that? Maybe if we were to look at that the octahedron puzzle, then break it down to the actual puzzle, we would understand what this "deeper-than-deepcut" really is.

Carl always wrote about Pyraminxes, not Pyramorphixes. An important difference.
With "deeper-than-deepcut", Carl meant that the appropriate cut of the ProfessorPyraminx is below the tetrahedrons center.
Honestly I think there is not need to work with "deeper than deepcut" anymore. With the help of virtual pieces we can even classify the ProfessorPyraminx. Now I am sure about that.

AndrewG wrote:
Something like this also needs to be made relevant to the community.
Okay. How?
The point is that Jaap has a lot of very good theory-based subpages on his site and almost never gets feedback for them. If we transform the three threads into an article (or, considering those huge piece sets, into a website of its own) it will suffer from the same fate...
I wouldn't care if that happened. It would help there just for the sake of documenting this.
And here is one additional point. We need an even more sufficient way to eumerate the pieces. Carl discovered ambiguities in ED2. AndrewG found two different X-Faces with identical "turning numbers". We need something to make that unambigous.


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 Post subject: Re: Classification of Cubes
PostPosted: Mon Jan 04, 2010 12:34 pm 
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Well, regarding communal relevance again, it's isn't easy.

One thing I have to say is that right now we're just discovering puzzles. We know WHAT the pieces of the edge turning dodecahedron are now, but I think Julian is the only one who's even come close to solving it. I think he just wrote algorithms for each piece and gave an order in which to do the pieces. I don't blame him for not actually doing any of it, the thing is an absolute monster (can you imagine doing 120 pieces, one or two at a time with long 3-cycles and lots of set-up moves?)

I'm happy that I discovered another Master Skewb/3x3x3 hybrid by making that piece map. But I still haven't solved any of them yet :lol: . So I think one thing we can do is start solving them. GelatinBrain's applet collection has a slowly growing number of people who work on the puzzles and that's good. That's one of the reasons I really like the idea of classification with the aim of finding subset of puzzles that are interesting to people.

Good point about jaap. A while back I had a similar (but much smaller and less well-written) website that didn't get many hits either (I never had a hit counter though)

The other thing we can do is build them. Of course this is very very hard. I built (3D print & cast) a dino cube last month, it was an interesting experience but in the long run I don't know how much I'll be able to do this, it just takes a lot of time. The really advanced puzzles seem to get exponentially harder to make mechanisms for, so we'll have to see where the builders go with that.

A last word though, probably the single biggest thing that needs to be popularized is commutators, because they are the key to really solving new puzzles on your own. Right now I think there's little interest in some puzzles because many people simply can't do them.

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