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 Post subject: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Sat Oct 12, 2013 11:59 pm 
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Just a general question. In other words, I'm asking if it jumbles. I do know that the Battle Gears, a 2D puzzle which rotates in the same 45 degree increments, CAN be fully unbandaged. I would try to find this out myself, but it produces a very large amount of cuts which SW can't handle during the split, even for small numbers of parts. Can anyone provide some insight on this? :?:

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 Post subject: Re: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Sun Oct 13, 2013 1:12 am 
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Well this is a timely question. I just got done with a big update to the Gizmo Gears Jumbling thread, which started with the observation that Battle Gears can be unbandaged.

And looking at GelatinBrain 6.1.2, I suddenly realized that for many 3d twisty puzzles, we can investigate their jumbling behavior just by looking at their Jaap's spheres. (Is this correct?) For some reason I was thinking messy 3d geometry would be required. But maybe I could just adapt my existing program for 2d disk puzzle unbandaging to use projective geometry, mapping the surface of the sphere to the plane. Then we could investigate most 3d puzzles as well.

<--- Like this one.

I won't be able to do this any time soon, but now it seems like this would be a reasonable way to look at 3d jumbling.


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 Post subject: Re: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Sun Oct 13, 2013 10:07 am 
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I made a simple program to repeat 45° rotations by randomly chosen x,y,z axes to one of the circular cut.
Here is the result of 1000 and 10000 times repeat. Only the visible side is drawn.
Impressionning, isn't it? :lol:
Attachment:
45sphere0.gif
45sphere0.gif [ 42.81 KiB | Viewed 1093 times ]

Attachment:
45sphere1.gif
45sphere1.gif [ 3.5 KiB | Viewed 1093 times ]

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 Post subject: Re: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Sun Oct 13, 2013 2:40 pm 
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gelatinbrain wrote:
Impressionning, isn't it? :lol:
Attachment:
45sphere0.gif

Attachment:
45sphere1.gif

I'm not sure what I'm looking at. If we're talking about 6.1.2, then 6 of the "centers" would never become cut up, no matter how many times you unbandaged the puzzle. So there should be 6 areas on the sphere which are intact, but I don't see them. Why is that?

I've also realized that in order for it to be completely unbandaged, any given layer on the puzzle needs to have 8 fold rotational symmetry, where all layers can turn at any permutation.

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 Post subject: Re: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Sun Oct 13, 2013 4:26 pm 
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benpuzzles wrote:
gelatinbrain wrote:
Impressionning, isn't it? :lol:
Attachment:
The attachment 45sphere0.gif is no longer available

Attachment:
The attachment 45sphere1.gif is no longer available

I'm not sure what I'm looking at. If we're talking about 6.1.2, then 6 of the "centers" would never become cut up, no matter how many times you unbandaged the puzzle. So there should be 6 areas on the sphere which are intact, but I don't see them. Why is that?

I've also realized that in order for it to be completely unbandaged, any given layer on the puzzle needs to have 8 fold rotational symmetry, where all layers can turn at any permutation.

OK, so you were meaning that only top layers turn by 45° and not slices, right?
Then it would be like this.
Attachment:
45sphere.gif
45sphere.gif [ 10.11 KiB | Viewed 1041 times ]

Pentagonal pieces on edges become circles.

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 Post subject: Re: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Sun Oct 13, 2013 4:38 pm 
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Cool; you are way ahead of me here in exploring this with software!

But it looks to me like that is non-jumbling; aren't those "circles" actually polygons with not too many sides? This looks a lot like what happens with circle puzzles when R is less than the minimum jumbling radius, like this:



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 Post subject: Re: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Sun Oct 13, 2013 5:11 pm 
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bhearn wrote:
But it looks to me like that is non-jumbling; aren't those "circles" actually polygons with not too many sides?
I think this has been looked at before... not sure if I can find the thread or not but I thought that it had been proven those were in-fact circles. My recollection was that the amount of rotation a few twists applied was irrational so there is never a nice stopping point where you can stop and those pieces are nice polygons.

Carl

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 Post subject: Re: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Mon Oct 14, 2013 12:05 am 
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wwwmwww wrote:
bhearn wrote:
But it looks to me like that is non-jumbling; aren't those "circles" actually polygons with not too many sides?
I think this has been looked at before... not sure if I can find the thread or not but I thought that it had been proven those were in-fact circles.

I think if not a circle, it becomes at lim(&infin;) a smooth pentacular shape. But I'm not sure.
Anyway a composition of two 45° rotations around 2 perpendicular axes doesn't have a cyclic pattern.

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 Post subject: Re: Can GelatinBrain 6.1.2 be Fully Unbandaged?
PostPosted: Mon Oct 14, 2013 4:21 am 
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If we can do such cuts for 6.1.2, we should try the process for a helicopter sphere to see if a 'fudgicopter' can be constructed.

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