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 Post subject: Compound Twisty Puzzles
PostPosted: Wed Mar 12, 2014 8:26 pm 
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The recent thread on my collaboration with David Pitcher on Dave's Diamond here, and especially Carl Hoff's comments kicked off some thought on an area we could call compound puzzles.

Dave's Diamond is a compound of two face turning cubes:
Image

It has the interesting property we've been discussing that only one of the cubes seems to be active at the same time. This leads to the interesting question: "What would it take to unbandage it so that all cubes are playable during a scramble and what would you get?"

We can also view Dave's Diamond as the compound of two vertex turning octahedra:
Image

You can see the compound is formed by rotating one octahedron's face halfway relative to the other (60 degrees). On the cube the compound is formed by rotating one cube along the vertex by 60 degrees.

So, what if we did that to every face on the octahedron? We'd need 5 octahedrons. One in the middle, and then 4 that are rotated relative to the base.

Image

Image

Look at that shape and imagine each octahedron cut like Okki's Gem (2x2 cube cuts go through the edges of the octahedrons.)

Don't read this until you look for it.....
.
.
.
.
.
.
.
It makes the BIG CHOP! So Dave's Diamond is the subset of the Big Chop. See the dodecahedron being born in there? And I can see why Dave's Diamond has that unhappy turn that goes nowhere.


Now let's think about Carl's idea of two skewbs which we can think of as two face turning octahedra:
Image

This creates an interesting puzzle with 7 cuts as Carl said. (It would be 8 but they share the equator.)
Unbandaging that, let's look at the 5 compound again:

Image

And now imagine the octahedra all face turning... you'll see that there are 40 faces but only 10 cuts. And their symmetry is indeed the Radio Chop / Chopasaurus!

So Carl's 2 skewb idea is a subset which is 3 cuts short of the Radio Chop.

Adam Cowan has also pointed out that a compound of 4 2x2 cubes where 1 is the base and the other three are each rotated 45 degrees on one face, creates the 24 cube + 2x2 hybrid.

Image

Seems like a rich area to explore indeed

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Wed Mar 12, 2014 8:30 pm 
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Have you seen Nan's Twisty Star? viewtopic.php?f=1&p=281801

Although, I suppose the twisty star is just a shape mod of a Icosahedron.

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Thu Mar 13, 2014 10:14 am 
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I knew FTC is a subset of ETD and that VTC is a subset of of VTD, but its cool to see that these compounds of 2 2*2*2s and 2 skewbs are still subsets of the more complicated geometry.

Which raises questions such as:
Excluding single layer cuboids, is FTC the only doctrinaire subset of ETD?
Is VTC the only doctrinaire subset of VTD?
Excluding single layer cuboids, are there any doctrinair subsets of ETC?
Are there any doctrinaire subsets of FTD?
Are there any compounds of the cube that could generate an interesting superset of the ETC?(What happens if you apply the same compound of five to ETC?)
Are there any Compounds of the Dodecahedron that could generate an interesting superset of FTD, VTD, or ETD?
Do any of these supersets have any doctrinaire subsets which are not subsets of the platonic geometries?

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Thu Mar 13, 2014 11:10 am 
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To our terminology experts:
Does the fact that Dave's Diamond can be unbandaged into the Big Chop mean that is isn't truly a jumbling puzzle? It is simply a bandaged Big Chop?

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Thu Mar 13, 2014 11:12 am 
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Brandon Enright wrote:
Have you seen Nan's Twisty Star? http://twistypuzzles.com/forum/viewtopi ... 1&p=281801

Although, I suppose the twisty star is just a shape mod of a Icosahedron.


Yes, I've even attempted a design at one point.

But yes, what I'm referring to in the original post is the interesting relationships between polyhedron compounds and the common puzzle axis systems.

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Thu Mar 13, 2014 1:22 pm 
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Nan asked me to give the Twisty Star mechanism a go as well. I've tried a few things to keep all the pieces in, but nothing has worked so far. I'm letting it stew for a bit and working on some other projects, but I'll definitely try again sometime.

-Eitan

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Thu Mar 13, 2014 3:06 pm 
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pirsquared wrote:
Nan asked me to give the Twisty Star mechanism a go as well. I've tried a few things to keep all the pieces in, but nothing has worked so far. I'm letting it stew for a bit and working on some other projects, but I'll definitely try again sometime.

It looks like to me that puzzle is a shapemod of the Radio Jam, but with many of the pieces hidden. I might have to try as well.
JasonSmith wrote:
To our terminology experts:
Does the fact that Dave's Diamond can be unbandaged into the Big Chop mean that is isn't truly a jumbling puzzle? It is simply a bandaged Big Chop?

No, it is a jumbling puzzle because the Big Chop alone is a jumbling puzzle. If jumbling moves from the big chop are also possible on that then it would still be jumbling despite the fact that it's bandaged.
This leaves me with a question though: can any jumbling twisty puzzle, be bandaged into a NON-jumbling one? This seems like something interesting to explore...

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Thu Mar 13, 2014 3:13 pm 
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benpuzzles wrote:
JasonSmith wrote:
To our terminology experts:
Does the fact that Dave's Diamond can be unbandaged into the Big Chop mean that is isn't truly a jumbling puzzle? It is simply a bandaged Big Chop?

No, it is a jumbling puzzle because the Big Chop alone is a jumbling puzzle. If jumbling moves from the big chop are also possible on that then it would still be jumbling despite the fact that it's bandaged.
This leaves me with a question though: can any jumbling twisty puzzle, be bandaged into a NON-jumbling one? This seems like something interesting to explore...


Right, of course it still jumbles as the Big Chop. Thanks.

To your question: Can any jumbling twisty puzzle be bandaged into a NON jumbling one:

The Big Chop can be bandaged into a 2x2x2 cube showing there is at least one.
The jumble prism can't be bandaged into a non jumbling puzzle, disregarding a trivial single slice.

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Thu Mar 13, 2014 7:04 pm 
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JasonSmith wrote:
benpuzzles wrote:
can any jumbling twisty puzzle, be bandaged into a NON-jumbling one?
The jumble prism can't be bandaged into a non jumbling puzzle, disregarding a trivial single slice.
I was about to say they could all be bandaged to form a 1x1x2 at a minimum but I think that's what Jason is saying as well.

Lots of interesting questions being asked here. I wish I had more answers. I'll try to make some POV-Ray pictures after G4G11 so I can visualize some of this better.

Carl

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Thu Mar 13, 2014 9:04 pm 
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Quote:
Does the fact that Dave's Diamond can be unbandaged into the Big Chop mean that is isn't truly a jumbling puzzle? It is simply a bandaged Big Chop?
This is a tough question in the general case. You can "unbandage" a Skewb into the Chopasaurus / Radio Chop.

Quote:
To your question: Can any jumbling twisty puzzle be bandaged into a NON jumbling one:

The Big Chop can be bandaged into a 2x2x2 cube showing there is at least one.
The jumble prism can't be bandaged into a non jumbling puzzle, disregarding a trivial single slice.

The Chopasaurus -> Skewb is another.

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Fri Mar 14, 2014 1:37 pm 
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Jeffery Mewtamer wrote:
1. Excluding single layer cuboids, is FTC the only doctrinaire subset of ETD?
2. Is VTC the only doctrinaire subset of VTD?
3. Excluding single layer cuboids, are there any doctrinaire subsets of ETC?
4. Are there any doctrinaire subsets of FTD?
5. Are there any compounds of the cube that could generate an interesting superset of the ETC?(What happens if you apply the same compound of five to ETC?)
6. Are there any Compounds of the Dodecahedron that could generate an interesting superset of FTD, VTD, or ETD?
7. Do any of these supersets have any doctrinaire subsets which are not subsets of the platonic geometries?
Beyond prisms there are only two groups for point symmetries in 3D: Icosahedron and Octahedron. Everything else (like tetrahedron) are isomorphisms and/or subgroups.

The group I (icosahedral point symmetry with rotations only) consists of 24 rotations (explained on a dodecahedron):
I: 6 face rotations + 10 corner rotations + 15 edge rotations
It has 7 non-empty subgroups:
I_1: 1 face rotation
I_2: 1 corner rotation
I_3: 1 edge rotation
I_4: 1 corner rotations + 3 edge rotations
I_5: 1 face rotation + 5 edge rotations
I_6: 3 edge rotations (different configuration than I_4)
I_7: 4 corner rotations + 3 edge rotations
If there is more than one type of rotations you can delete ony type and thereby create more subgroups.

The group O (octahedral point symmetry with rotations only) consists of 13 rotations (explained on a cube):
I: 3 face rotations + 4 croner rotations + 6 edge rotations
It has 9 non-empty subgroups:
O_1: 1 face rotation by 180°
O_2: 1 corner rotation
O_3: 1 edge rotation
O_4: 1 face rotation by 90°
O_5: 3 face rotations by 180°
O_6: 1 face rotation by 180 + 1 edge rotation
O_7: 1 corner rotation + 3 edge rotation
O_8: 1 face rotation by 90° + 2 face rotations by 180°
O_9: 3 face rotations by 180° + 4 corner rotations
If there is more than one type of rotations you can delete one type and thereby create more subgroups.

Therefore I answer:
1. The Rubiks Cube is not a true subgroup of edge-turning dodecahedron. Beyond that: There is only the trivial single-axis variant.
2. There is only the trivial single-axis variant.
3. The trigonal prism (without top and bottom layer)
4. There is only the trivial single-axis variant.
5. Only hybrids which introduce faceturns and/or cornerturns.
6. Only hybrids which introduce the other axis systems.
7. If I understood geometry right you can't go beyond the platonics and prisms.

BTW: O_9 is implemented in the "Offset Skewb 2x2x2".
I asked Tom van der Zanden to designed it just because I considered it as the only subgroup worth to be implemented.


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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Fri Mar 14, 2014 6:53 pm 
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JasonSmith wrote:
You can see the compound is formed by rotating one octahedron's face halfway relative to the other (60 degrees). On the cube the compound is formed by rotating one cube along the vertex by 60 degrees.

So, what if we did that to every face on the octahedron? We'd need 5 octahedrons. One in the middle, and then 4 that are rotated relative to the base.

Am I missing something? The angle of rotation required to create the standard 5-octahedron compound by duplicating a base octahedron and rotating them around the 3-fold axes is not 60 degrees (even visibly so from those pictures). Is there something clever you have left out of your logic that makes this unimportant for being able to unbandage David's Diamond into Big Chop? I can't immediately see how that would be the case.

EDIT: This makes me wonder... if you DID make a puzzle like david's diamond, but offset the 2x2's by the "correct" angle to generate 2 subsets of cuts from big chop... would it jumble more?


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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Fri Mar 14, 2014 8:30 pm 
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DKwan wrote:
JasonSmith wrote:
You can see the compound is formed by rotating one octahedron's face halfway relative to the other (60 degrees). On the cube the compound is formed by rotating one cube along the vertex by 60 degrees.

So, what if we did that to every face on the octahedron? We'd need 5 octahedrons. One in the middle, and then 4 that are rotated relative to the base.

Am I missing something? The angle of rotation required to create the standard 5-octahedron compound by duplicating a base octahedron and rotating them around the 3-fold axes is not 60 degrees (even visibly so from those pictures). Is there something clever you have left out of your logic that makes this unimportant for being able to unbandage David's Diamond into Big Chop? I can't immediately see how that would be the case.

EDIT: This makes me wonder... if you DID make a puzzle like david's diamond, but offset the 2x2's by the "correct" angle to generate 2 subsets of cuts from big chop... would it jumble more?


What do you come up with for the angle? I think the faces (and solids) align every 120 degrees, and the compound we're talking about sits right in between 2 face alignment positions.

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Fri Mar 14, 2014 9:12 pm 
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What I'm saying is that for example in this image, the red and green triangles are not 60 degrees from each other:

Attachment:
5-octahedra.jpg
5-octahedra.jpg [ 19.46 KiB | Viewed 785 times ]

I did some basic trig and get approximately 44.48 degrees as the angle of rotation, but I may have made a mistake along the way, so someone should double check...

You are correct when you said that the Okki's Gem cuts of the standard 5-octahedra compound result in the cuts of Big Chop, but if you change the angle used to generate the compound you no longer get Big Chop.

EDIT: For extra clarity (I hope not more confusion), to get the 5-octahedra compound in this way you need to "checkerboard" the rotations of the original octahedron's faces between CW and CCW... Eg. UFR and UBL clockwise and UFL and UBR counter-clockwise (their respective opposite sides would be the reverse since they are connected, completing the checkerboard)... Not sure if I explained this clearly =/


Last edited by DKwan on Sat Mar 15, 2014 9:17 am, edited 2 times in total.

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Fri Mar 14, 2014 9:50 pm 
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Does this mean you might be able to build up a big chop from a series of 2x2x2 shells?


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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Fri Mar 14, 2014 10:12 pm 
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Jared wrote:
Does this mean you might be able to build up a big chop from a series of 2x2x2 shells?

Yes you can. It's just that the complexity in the shells increases exponentially as you move outwards. By the 5th shell you'd have nearly as many pieces as a normal shell's design, and even less stability. So I don't think it's a workable solution.

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Fri Mar 14, 2014 10:32 pm 
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benpuzzles wrote:
Jared wrote:
Does this mean you might be able to build up a big chop from a series of 2x2x2 shells?

Yes you can. It's just that the complexity in the shells increases exponentially as you move outwards. By the 5th shell you'd have nearly as many pieces as a normal shell's design, and even less stability. So I don't think it's a workable solution.


Just a head's up, if what I'm working on works, a big chop won't be too far behind :D

[edit] Sorry, meant to say Superchop Dodecahedron, or a big chop with no jumblng.


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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Sat Mar 15, 2014 12:22 am 
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garathnor wrote:
Just a head's up, if what I'm working on works, a big chop won't be too far behind :D

[edit] Sorry, meant to say Superchop Dodecahedron, or a big chop with no jumblng.

I believe this until a point... What could you be using in this case? I *thought* that a shells big chop is too impractical. Am I misunderstanding here?

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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Sat Mar 15, 2014 11:43 am 
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NXTgen wrote:
garathnor wrote:
Just a head's up, if what I'm working on works, a big chop won't be too far behind :D

[edit] Sorry, meant to say Superchop Dodecahedron, or a big chop with no jumblng.

I believe this until a point... What could you be using in this case? I *thought* that a shells big chop is too impractical. Am I misunderstanding here?


It's a secret for now :D


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 Post subject: Re: Compound Twisty Puzzles
PostPosted: Sat Mar 15, 2014 5:19 pm 
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Oh darn, you are right about the 5 compound! It was beautiful while it lasted. Sorry I didn't understand your first post.

About building up a Big Chop from 2x2x2 shells--- Good luck. I tried after Schuma brought up the hidden 2x2x2s in the Big Chop at a Bay Area meetup, and you don't get the savings you think, because even the second layer pieces have to be able to reposition over the first 2x2x2 and still allow those cuts. That said, if you do have a workable idea, I'll love to see it, because I've thought about it quite a lot.

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