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 Post subject: Are the Gizmo Gears jumbling?
PostPosted: Wed Jul 10, 2013 12:25 pm 
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Hi,

Did you know the Gizmo Gears?
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back in this posting GuiltyBystander prooved that the Battles Gears (from the same inventor) are not jumbling.
How about the Gizmo Gears?
Do they jumble? Can anybody demonstrate how an unbandaged version of the Gizmo Gears would look like?

Andreas


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Jul 11, 2013 5:30 pm 
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I'm pretty sure they don't jumble, but I can't prove it. They look like a bandaged version of a puzzle with 12 states per wheel.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Fri Jul 12, 2013 7:41 am 
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Jared wrote:
They look like a bandaged version of a puzzle with 12 states per wheel.

We know puzzles with three, four and six states per wheel which are all doctrinaire.
I would guess Gizmo Gears is doctrinaire but guessing is no proof.
*Curses about math*


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Fri Jul 12, 2013 10:29 am 
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Andreas Nortmann wrote:
back in this posting GuiltyBystander prooved that the Battles Gears (from the same inventor) are not jumbling.
How about the Gizmo Gears?
Do they jumble? Can anybody demonstrate how an unbandaged version of the Gizmo Gears would look like
I should be able to take a shot at this over the weekend. The proof should look just like a more complicated version of GuiltyBystander's unbandaged Battles Gears. So I'm rather certain Gizmo Gears doesn't jumble in principle. In practice I suspect we'll get pieces on the order of (or maybe even smaller) then the size of the gear teeth so if you were to cut up an actually Gizmo Gears puzzle you'd end up with a puzzle that no longer functioned.

I think I know how to make the image you are after in POV-Ray. I'm not certain how GuiltyBystander made his unbandaged Battles Gears image but I suspect he could easily have a much faster way of making the image than I do. Still I'll take a shot as I enjoy the chance to play with POV-Ray when I can.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Jul 13, 2013 6:07 am 
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Thank you Carl, for taking up this challenge.
wwwmwww wrote:
In practice I suspect we'll get pieces on the order of (or maybe even smaller) then the size of the gear teeth so if you were to cut up an actually Gizmo Gears puzzle you'd end up with a puzzle that no longer functioned.
I am sure you are right about this point. The question is still interesting for the sake of the theory.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Jul 13, 2013 9:45 pm 
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Just a little note to support my theory - there are 48 teeth around the outside of each large "gear", which is divisible by 12...


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Jul 13, 2013 10:37 pm 
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Jared wrote:
Just a little note to support my theory - there are 48 teeth around the outside of each large "gear", which is divisible by 12...
To prove the gears have 12 fold symmetry you just need to look at the Mates puzzle. The blue portion of the left gear has 3 fold symmetry and the brown portion of the right gear has 4 fold symmetry. The least common multiple of 3 and 4 is 12. Still un-bandaging the puzzle. With each iteration the render times are getting longer and I now have pieces which are at least an order of magnitude smaller then the gear teeth. I still think the number of cuts needed is finite but I may reach a point where the time to render the solution could take days (or even longer). I'll try to post some progress pics soon.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Jul 13, 2013 11:13 pm 
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Note all the action takes place in the "square" between the two cores. The "triangular" caps can be formed from a rotation out of the "square". So I'll only render this "square" for the time being.

Attachment:
GG0.png
GG0.png [ 12.07 KiB | Viewed 5447 times ]


First iteration (2s):
Attachment:
GG1.png
GG1.png [ 15.73 KiB | Viewed 5447 times ]


Second iteration (7s):
Attachment:
GG2.png
GG2.png [ 17.15 KiB | Viewed 5447 times ]


Third iteration (2m27s):
Attachment:
GG3.png
GG3.png [ 20.01 KiB | Viewed 5447 times ]


And about a quarter of the way to the next iteration (33m39s):
Attachment:
GG3B.png
GG3B.png [ 20.47 KiB | Viewed 5447 times ]


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Jul 14, 2013 12:23 am 
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This get increasingly complicated.
I cant wait to see where this leads us to.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Jul 14, 2013 6:09 pm 
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Andreas Nortmann wrote:
This get increasingly complicated.
It sure does. The render of the full 4th iteration is almost complete. Its been going for over 18 hours so far and it should be done rather soon. And I can tell it still needs more iterations.
Andreas Nortmann wrote:
I cant wait to see where this leads us to.
I'm starting to see symmetry requirements that I don't think I'll be able to satisfy with a finite number of cuts. I'm not sure how to prove that short of making an infinite number of renders and I doubt I'll be able to make many more iterations going at this with POV-Ray. But I'm now almost convinced that this puzzle jumbles. From a practical point of view, even if this puzzle can be unbandaged with a finite number of cuts, some of the pieces I'm currently producing would be on the order of the size of a grain of sand so the difference between jumbling and non-jumbling pretty much loses its meaning at a practical level. I'll continue for the pure theory aspect of this for as long as POV-Ray allows me to but the next full iteration may take months and I suspect there is other software out there that can due this far more efficiently then POV-Ray which is actually performing rotations and ray tracing the pieces produced. Each ray POV-Ray renders basically has to back calculate through the whole iterative process.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Jul 14, 2013 6:45 pm 
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Here is iteration number 4 after 18hours 52minutes and 50seconds of render time. A lot of that is due to me making the image larger to make sure all the pieces were still visible.
Attachment:
GG4.png
GG4.png [ 63.77 KiB | Viewed 5390 times ]
Here is how I tell more iterations are needed. This green triangle needs 3 fold symmetry. The image above needs left/right mirror symmetry and top/down mirror symmetry which it has but that is pretty much guaranteed by the way I'm doing the iterations.
Attachment:
GG4A.png
GG4A.png [ 128.15 KiB | Viewed 5390 times ]
Note the green triangle does NOT have 3 fold symmetry yet.

Also in this image the blue shape and the green shape need to be identical under rotation. This isn't satisfied either.
Attachment:
GG4B.png
GG4B.png [ 132.4 KiB | Viewed 5390 times ]


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Jul 14, 2013 11:52 pm 
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Andreas Nortmann wrote:
I would guess Gizmo Gears is doctrinaire but guessing is no proof.
When do I learn to stand silent and wait for more hints? :roll:

Thank you Carl.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Jul 20, 2013 12:00 pm 
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Here is iteration #5 after 5 days 9 hours 23 minutes and 57 seconds of rendering time.

Attachment:
GG5.png
GG5.png [ 77.99 KiB | Viewed 5267 times ]


Again its obvious more iterations are needed. I'll get iteration #6 started shortly. Of all the pieces seen so far I see two that I feel certain aren't cut up in any future iterations. Beyond that I'm not sure. Want to guess which 2 I'm thinking of?

And with each iteration I feel more sure that this jumbles but I don't believe this is proof of that. Its easy to prove a puzzle doesn't jumble because at some point future iterations don't add any new cuts. But I'm not certain how to prove this jumbles. This process could be finite and stop after 120 iterations... or 12 as far as that goes but I likely won't get that far with POV-Ray.

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Last edited by wwwmwww on Sun Jul 21, 2013 6:51 pm, edited 1 time in total.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Jul 21, 2013 5:23 am 
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Thank you Carl.
I have updated the description of the Gizmo Gears.

Andreas


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Aug 03, 2013 10:08 am 
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After discussion with Carl and Oskar at IPP, I've written a program to investigate unbandaging for this general family of puzzles, parametrized by disk radius and number of turn states.

I don't have an answer yet for N = 12, R = sqrt(2) (Gizmo Gears, I think... does R = sqrt(2)?). I suspect based on behavior for other parameters that all such puzzles can be unbandaged. However, the number of pieces for Gizmo Gears would be astronomical, on the order of hundreds of thousands, at least. I ran it up to 100,000 cut iterations without bottoming out. I am working on a second version of the program, which should give exact answers, but this is much more complicated, and may take me a while.

For any N, there are various critical values of R for which the unbandaging changes dramatically, and you get an extra order of cut interference. For example, here are a few values of R for N = 5:

Image

Image

Image

Where the last one was R = 2. Here, we see it got a lot more messy as the cuts started interfering more, and it's not obviously resolved into pieces. But here is a zoom on the middle that shows that it is in fact resolved.

Image

Finally here is R = sqrt(2), N = 12. It is very dense, even zoomed way in. But I am betting that it bottoms out too, and I should be able to prove it soon.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Aug 03, 2013 8:45 pm 
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So they don't jumble, but they're realistically impossible to unbandage? Awesome!


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Aug 03, 2013 10:10 pm 
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It looks like it should be straightforward to prove that as the radius expands every time there's a crossing the number of pieces remains finite. Proving that the number of such crossings which happen is finite I don't see an obvious approach to.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Aug 04, 2013 3:52 am 
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That's very perceptive. I hadn't looked at it quite that way, but after thinking about it for a while I think I agree with you.

I'm also less sure than I was that everything bottoms out, and less sure than I was that I should be able to prove it numerically for N = 12 even if it does. My original approach is too inefficient, but my new exact approach (which I'm still writing) I think will use too much memory. I've raised my estimate of the number of pieces. I can't even come close to resolving R = 1.7, N = 7 with my original program; there are millions of pieces at least. N = 12, R = sqrt(2) is much worse.

Really, I think this is a very fascinating question! I think this should be a problem of more general mathematical interest, outside of the twisty community.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Aug 04, 2013 4:08 am 
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This is what N = 7, R = 1.7 looks like, zoomed in 100x, at 250,000 cut iterations. The image hasn't changed much as I've zoomed in and cranked up the iterations and the sampling resolution.

Image


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Aug 04, 2013 6:23 am 
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:shock: Now I don't mind assembling puzzles, but this one would be taking it a bit too far :lol:


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Aug 04, 2013 5:21 pm 
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bhearn wrote:
This is what N = 7, R = 1.7 looks like, zoomed in 100x, at 250,000 cut iterations. The image hasn't changed much as I've zoomed in and cranked up the iterations and the sampling resolution.

... and, after a 12-hour run, N = 7, R = 1.7, 200x, 1,000,000 cut iterations (also doubled sampling resolution again):

Image

If that's not jumbling, I doubt I'll ever be able to determine it by resolving the pieces. Better at this point, I think, to fall back to theory, and try to mathematically prove that either it jumbles for some parameters or it doesn't for any. Again, N = 12, R = sqrt(2), as in Gizmo Gears, will be much, much worse than this.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Aug 04, 2013 5:44 pm 
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If any of these are in fact jumbling, there's the very interesting question of for any given N, what's the smallest value of r for which the number of pieces is infinite?

Also note that there are variants here the amount the two circles rotate are different. The same questions apply to those.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Aug 04, 2013 6:12 pm 
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I am not an expert on this, but what is the angle at which Gizmo Gears turns? The octagon centerpiece is not regular. Is it rational?

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Aug 04, 2013 10:52 pm 
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rubikcollector123 wrote:
I am not an expert on this, but what is the angle at which Gizmo Gears turns? The octagon centerpiece is not regular. Is it rational?


Multiples of 30 degrees.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Aug 05, 2013 11:46 am 
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bhearn wrote:
bhearn wrote:
This is what N = 7, R = 1.7 looks like, zoomed in 100x, at 250,000 cut iterations. The image hasn't changed much as I've zoomed in and cranked up the iterations and the sampling resolution.

... and, after a 12-hour run, N = 7, R = 1.7, 200x, 1,000,000 cut iterations (also doubled sampling resolution again):
[...]
If that's not jumbling, I doubt I'll ever be able to determine it by resolving the pieces. Better at this point, I think, to fall back to theory, and try to mathematically prove that either it jumbles for some parameters or it doesn't for any. Again, N = 12, R = sqrt(2), as in Gizmo Gears, will be much, much worse than this.

I was pretty sure that no flat, two-circle puzzle would jumble when N is an integer. This result is troubling. I don't understand why it jumbles.

Is there any chance your simulation is sensitive to floating point errors and that's the source of seemingly infinite cuts?

Bram can you provide some insight into why these jumble?

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Aug 05, 2013 7:03 pm 
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bmenrigh wrote:
Bram can you provide some insight into why these jumble?


Geeze, people expect me to be the oracle of Delphi or something.

I have no idea whether this puzzle actually jumbles. My initial guess was that it does, but I was wrong in guessing/assuming that Battle Gears jumbles, so I'm declining to take any bets.

There is at least one puzzle which has only rational angles but does jumble. It's the one with three axes at 120 degrees to each other, each of which can rotate 90 degrees, where the actual build is a little bit fudged. I'm unfortunately spacing on the name. There was an earlier thread on that one, and the pattern of there being an infinite number of pieces is quite simple and understandable. It isn't a dramatic explosion of pieces at each iteration like these appear to be though.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Aug 05, 2013 7:14 pm 
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I have some new ideas I'm working on, and should have some better (but probably not definitive) answers soon.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Aug 05, 2013 7:18 pm 
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bmenrigh wrote:
Is there any chance your simulation is sensitive to floating point errors and that's the source of seemingly infinite cuts?

Well it's possible, but I don't think so. It's pretty clear that a big combinatorial explosion in number of cuts happens, anyway. Whether those pictures of potential jumble dust have numerical artifacts is not as obvious, but again I don't think so.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Aug 05, 2013 7:37 pm 
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Bram wrote:
bmenrigh wrote:
Bram can you provide some insight into why these jumble?
Geeze, people expect me to be the oracle of Delphi or something.
Not at all! You've proven extraordinarily insightful in the past so I thought I'd try :D

Bram wrote:
I have no idea whether this puzzle actually jumbles. My initial guess was that it does, but I was wrong in guessing/assuming that Battle Gears jumbles, so I'm declining to take any bets.

There is at least one puzzle which has only rational angles but does jumble. It's the one with three axes at 120 degrees to each other, each of which can rotate 90 degrees, where the actual build is a little bit fudged. I'm unfortunately spacing on the name. There was an earlier thread on that one, and the pattern of there being an infinite number of pieces is quite simple and understandable. It isn't a dramatic explosion of pieces at each iteration like these appear to be though.

I think you're referring to the Constellation Six.
I never really felt like I saw or understood why the Constellation Six jumbled. Two circles in the same plane jumbling seems like it'd be simplest base-case though. If we can understand why these puzzles jumble, extending that into understanding more complex (3D) puzzles seems doable.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Aug 05, 2013 8:49 pm 
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bmenrigh wrote:
I think you're referring to the Constellation Six.
I never really felt like I saw or understood why the Constellation Six jumbled. Two circles in the same plane jumbling seems like it'd be simplest base-case though. If we can understand why these puzzles jumble, extending that into understanding more complex (3D) puzzles seems doable.


There are some great pictures and animations later in that thread which explain it. There's also the point that the the Tuttminx jumbles for a similar reason. And there's a calculation of the exact angle between axes which causes the puzzle to start jumbling, with a fairly obvious geometric justification.

The jumbling of those two puzzles is caused by sequences of the form ABCABCABC... When there's only two axes that can't happen, and these rather interesting almost-jumbling phenomena happen.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Aug 06, 2013 3:41 am 
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Bram wrote:
.... There's also the point that the the Tuttminx jumbles for a similar reason. ....
I recollect that there was an agreement that the Tuttminx does not jumble???

E.g. in this thread
Oskar wrote:
PuzzleMaster6262 wrote:
The mass produced tuttminx can rotate like your futtminx. If we think of it as fudged, it can jumble. So should we say a partially fudged tuttminx jumbles?
No, see Tom's response.

Here is my design challenge to you: make a Muttminx (Mike's Tuttminx). Muttminx has the same number of pieces and the geometry of a Tuttminx. All moves are either fluent or blocked, no ambiguous jamming allowed. And unlike Tuttminx or Futtminx, Muttminx has to jumble according to Bram's definition.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Aug 06, 2013 4:00 am 
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Konrad wrote:
I recollect that there was an agreement that the Tuttminx does not jumble???

E.g. in this thread


That thread has agreement that the Futtminx doesn't jumble. The Futtminx is like the Tuttminx, but fudged to get rid of the jumbling :-) It's very analogous to the Constellation Six, but the amount of fudging necessary is even less.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Aug 06, 2013 4:06 am 
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If the fudging in the Futtminx is tiny, and the end results are 3 fold symmetric vertices, Oskar should start to make a face turning series starting with the dino tuttminx. The dual solid is here:

Imagexx


Back on topic, my brain fails to comprehend how a rational angle turning puzzle can jumble. (Does it mean that any puzzle that has irrational turning angles definitely jumbles?)

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 2:19 am 
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Stop the presses!!! I have an answer. :D

N = 7, R = 1.7 definitely jumbles.

This is an image of the rightmost point of the left disk, transformed under all possible rotation sequences, stopped at 5,000,000 distinct points. Shown at 10x relative to my earlier images of the complete puzzle.

Image

So how exactly do I know it jumbles? Let me back up and describe what I've been doing. It started with the observation that the complete set of unbandaged cuts is just the image of the two arcs bounding the disk intersection, under all possible rotation sequences. If this is not obvious, then imagine an unbandaged puzzle, and look at any cut: that cut must have been generated by some particular turn that chopped a larger piece at some point. But that means that undoing some rotation sequence must put it back onto one of those primary arcs, otherwise it couldn't have been cut.

The first version of my program simply sampled those arcs in small steps, mapping each small line segment through some number of iterations of random moves, and plotting the result. That was sufficient for the simpler cases, and to hint at jumbling. But it was very inefficient, because if some small segment has a large orbit under all move sequences, we are just wandering through that space at random, mostly exploring space we've already visited many times before.

So I decided to be clever and record exactly where the cuts were, in terms of mathematical arcs of a given length. Doing this correctly is rather complicated, because of all the math in mapping and chopping the arcs, and more importantly the delicate issue of hashing with floating-point keys. Before I finished this program, it was apparent that it would not be feasible to store all the cuts for a given configuration.

So I went back to playing with the first version. Then I realized -- aha! I don't have to store all the cuts. If a configuration jumbles, then the image of some particular point on a bounding arc must itself have an infinite image under all rotations. If all bounding points have finite images, then so does the entire arc.

So -- back to hashing on cut locations, but this time just for a given point (actually, a tiny sampled segment, to distinguish the possible(?) case where the same point is hit in multiple directions). For a given point, just record where it is, and record whether we have explored (left, right) x (clockwise, couterclockwise) from that point. Keep exploring existing points until all moves have been explored. Then, when we get to the next sampled point along the arc, we can clear the hash table. So we don't have to store all the cuts, just those for a given point.

After nailing down the hashing subtleties... boom! Right off the bat, N = 7, theta = 0 (rightmost point of left disk) generates an uncountable a seemingly infinite set right at R = 1.7 (and not at anything <= 1.69). True, I've only iterated to 10,000,000 distinct points. But there is an apparent complex, perhaps fractal pattern. At R = 1.69, the orbit has 28 elements. It's a sharp transition. I can show samples of R = 1.7 at various smaller numbers of iterations if anyone is interested.

So, OK... this is not a rigorous proof that it jumbles. But it is clear to me now that it does jumble. What comes next is looking at why. Probably, there is some particular generator sequence that does it, that is only possible when R gets to 1.7 (I'm exploring all sequences right now). After finding such a sequence, we can look at the properties of the successive images. It's clear in hindsight what happens at a critical radius that enables the transition to jumbling. If the jumbling generator sequence is e.g. RLLrrl, then it's only when a source point stays within the disk intersection under that sequence that we can jumble, and the intersection grows with increasing radius. (We can always ignore moves that take points outside the disk intersection, because they must re-enter the disk for anything interesting to happen to them, and circling all the way around is equivalent to backing up within the intersection.) It's also conceivable that there is no single generator sequence that does it, but a set of such sequences, one or more of which is enabled at the critical radius.

Also, what about the original problem, Gizmo Gears? Surely that jumbles too, right? Well hold your horses. The image of the same point for R = sqrt(2), N = 12 is actually finite (30 points). My immediate next task is to start stepping and see what happens. I'm sure I'll quickly find a point that generates an infinite set.

Stay tuned!!!

Finally... I have a pretty mathy background, but I'm not an expert on fractals. So I can't say for sure how novel or important a result this is. But I definitely have the sense that this could be interesting mathematics, and important to mathematicians, if it is not already known / obvious. I will start asking people who should know, when I know a little bit more about what we have here.

Andreas, Carl, Oskar, Bram, GuiltyBystander, everyone else who has contributed here -- thank you! Oh, and Scott (VeryWetPaint) -- you suggested at one point that the jumbling cut pattern might be related to Penrose tiles. I was skeptical, but I have to say, the above image certainly resembles a quasicrystal.


Last edited by bhearn on Sat Aug 10, 2013 10:01 pm, edited 3 times in total.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 2:50 am 
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Here's a blowup of the central portion, 100x, this time with 60,000,000 distinct points generated.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 3:29 am 
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Yes, Gizmo Gears jumbles too. Here's an image of theta = 0.1 for N = 12, R = sqrt(2), 25x, 10,000,000 points generated.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 4:40 am 
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Hi Bob,

Those images look fantastic! Now we finally know what the proverbial Bram-Cohen jumble dust looks like.

How do you know for sure that the puzzle jumbles? Do you see scaled self-repetition, like in Fractals? If so, what is its Feigenbaum constant? If not, then how should I read those quasi-crystal images?

Oskar

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 5:00 am 
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Bob, that is some excellent analysis. And although there is no definite proof, it is almost certain
that those structures do jumble. I mean, even if they stopped at a finite point (which they don't)
the moves are so saturated/merged, that constructing it would be a titanic feat.

Now to calculate the corresponding dimension of each of those twisty-fractals!

:-)


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 10:20 am 
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bhearn wrote:
Stop the presses!!! I have an answer. :D

N = 7, R = 1.7 definitely jumbles.

This is an image of the rightmost point of the left disk, transformed under all possible rotation sequences, stopped at 5,000,000 distinct points. Shown at 10x relative to my earlier images of the complete puzzle.

{Really awesome image here.}

Very cool image! The rings in the image remind me a bit of electron orbital probability distributions:
Attachment:
jumble_dust_ring.png
jumble_dust_ring.png [ 25.4 KiB | Viewed 4563 times ]

Do you know if those circles are really "there"? Perhaps they're just a Moiré pattern due to pixel-level sampling of higher frequency data?

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 1:46 pm 
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I think I have a proof that Gizmo Gears doesn't jumble.

The two turning centers A and B are at (0,0) and (1,1) respectively. I'm representing a nonspecific puzzle piece with a cyan point.

Image

Let's perform the move sequence AB.

Image

We can calculate where the cyan puzzle piece ends up by right-multiplying its position column vector [x,y,1] by this matrix:

Image

From right to left, the rightmost matrix rotates the point 30 degrees counterclockwise around A, which lies at the origin. The matrix to the left shifts everything to the left by 1, so B now lies at the origin. The next matrix then rotates everything 30 degrees counterclockwise around B. The leftmost matrix then shifts everything to the right by 1 so A lies at the origin again.
The matrix product can be simplified to:

Image

That is the matrix that, when multiplied with a point's position vector, performs AB on that point.

Applying AB six times gives the identity matrix, so applying AB six times restores the puzzle to its original state. Thus Gizmo Gears is doctrinaire.

I'm fairly certain that it doesn't matter in the puzzle piece doesn't stay within the intersection of the two turning points' radiuses for the whole of the sequence: in which case the piece will just be rotating along one circle until it comes back in. EDIT: I think this is what actually causes the jumbling. Would my reasoning work for puzzles with R infinite?

I believe this reasoning can be applied to all two-circle puzzles that have integer N, regardless of R.

Does this work?

Bram wrote:
There is at least one puzzle which has only rational angles but does jumble.
There's also the unbandaged Bermuda cubes, aka a 3x3x3 whose faces turn in 45 degree increments. I think it can be proven like this: use a point to represent the FUL corner piece. Put a vector starting from that point to identify the piece's orientation. Perform FU. Observe that the point is in the same location, but its orientation has changed by an irrational angle. Therefore no finite amount of repeating FU brings the corner to its original position, and the puzzle jumbles.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 6:54 pm 
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Coaster1235 wrote:
[...] We can calculate where the cyan puzzle piece ends up by right-multiplying its position column vector [x,y,1] by this matrix:

Image

I don't think this captures all of the information about the motion of the piece during a rotation. We know the farther a point is from the center of rotation (the radius R) the more it will move. Your calculation doesn't take R into account. It seems like your calculation just assumes the point is always a constant radius away from both A and B. When you turn A the point stays a constant distance from A but changes in distance to B and I don't see how your matrix calculation takes that into account.

Edit: thinking about this some more... I'm not sure my objection is correct.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Wed Aug 07, 2013 7:52 pm 
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Coaster1235 wrote:
Applying AB six times gives the identity matrix, so applying AB six times restores the puzzle to its original state. Thus Gizmo Gears is doctrinaire.

Without looking in detail at those matrices – this would not at all imply that Gizmo Gears is doctrinaire. You can't generate all the cuts necessary to unbandage just by applying AB repeatedly. I speculated above that there is some generator sequence that produces non-repeating point images, but I have not had a chance to investigate further, and on reflection I think maybe it is unlikely – you might need multiple generators.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Aug 08, 2013 12:35 am 
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bhearn wrote:
After discussion with Carl and Oskar at IPP, I've written a program to investigate unbandaging for this general family of puzzles, parametrized by disk radius and number of turn states.
I LOVED IPP33 it was GREAT!!! And I loved these discussions too. Note I'm also starting to see my role in these things too. When I left for IPP I left my computer running on iteration 6 which I expected to take about 60 days. When I got home I found my PC had crashed and it crashed hard too. I had to do a complete restore and I'm still reinstalling software so sorry for the lack of replies so far. I don't think I'll have my system back to normal till the weekend at the rate I'm going. It turns out to be FAR more productive to get a programmer interested in the problem that is orders of magnitude better then I am. In a fraction of the time that I was planning to do 6 iterations Bob has done 60 MILLION!!!! You should ask Brandon about my stab at the Go First Dice problem after Gathering for Gardner last year. The same basic scenario... in very short order Brandon had a program that was just as many orders of magnitude faster then mine. Though there I think we are still looking for a solution. My role seems to be get excited about a problem. Write a very slow program that allows me to share that excitement and then get a programmer FAR better then I interested in the problem.
bhearn wrote:
Andreas, Carl, Oskar, Bram, GuiltyBystander, everyone else who has contributed here -- thank you! Oh, and Scott (VeryWetPaint) -- you suggested at one point that the jumbling cut pattern might be related to Penrose tiles.
No THANK YOU!!!!! Without you who knows how many times I would have killed my PC before I gave up. I hope you are able to fine a way to share your program as I think it would be very fun to play with.
bhearn wrote:
So how exactly do I know it jumbles?
Still not certain I follow exactly how you know it jumbles.
bhearn wrote:
After nailing down the hashing subtleties... boom! Right off the bat, N = 7, theta = 0 (rightmost point of left disk) generates an uncountable set right at R = 1.7 (and not at anything <= 1.69). True, I've only iterated to 10,000,000 distinct points. But there is an apparent complex, perhaps fractal pattern. At R = 1.69, the orbit has 28 elements. It's a sharp transition. I can show samples of R = 1.7 at various smaller numbers of iterations if anyone is interested.
Yes... I'd love to see more samples. Ideally I think it would be cool to see an animation which zoomed in at a rate which kept the smallest pieces about constant. Zooming in and adding iterations at the same time. Something akin to the animations of flying into the Mandelbrot I think would be fun to watch though maybe painful to generate.

Questions:

(1) What is theta?
(2) What is the uncountable set generated at R=1.7? Is it pieces, cuts, something else? And how can you tell its uncountable? I assume you mean its a set larger then the integers.
(3) Please define orbits and elements in this context? I'm not sure what an orbit having 28 elements means. Going back to question (2) were you saying that orbit had uncountably many elements?
bhearn wrote:
The image of the same point for R = sqrt(2), N = 12 is actually finite (30 points). My immediate next task is to start stepping and see what happens. I'm sure I'll quickly find a point that generates an infinite set.
Are you saying the orbit of that point had 30 elements?
bhearn wrote:
So I can't say for sure how novel or important a result this is. But I definitely have the sense that this could be interesting mathematics, and important to mathematicians, if it is not already known / obvious. I will start asking people who should know, when I know a little bit more about what we have here.
And PLEASE do share what you find... I'm MORE then curious my self.

Thanks again,
Carl

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Aug 08, 2013 2:39 am 
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wwwmwww wrote:
I'm not certain how GuiltyBystander made his unbandaged Battles Gears image
I built that one by hand so I would by no means call it a rigorous proof.

Konrad wrote:
Bram wrote:
.... There's also the point that the the Tuttminx jumbles for a similar reason. ....
I recollect that there was an agreement that the Tuttminx does not jumble???
In the Constellation Six thread I looked at what happens when you don't fudge the corners. The Tuttminx definitely jumbles because those corners would get cut to dust.


I was thinking about building my own version of your program to draw pictures too and I noticed the connection to fractals as well. It most reminds me of the Mandelbrot Set. For those not familiar with it, it is a fractal that contains a set of complex points. A point c is in the fractal if z(n) doesn't tend towards infinity in the function (z(n+1) = z(n)^2 + c | z(0) = 0). Points in the fractal will be stuck in orbits while points outside of the will run away (if anything is more then 2 units away from the origin you can instantly know it will be outside the set). The problem is that you can almost never tell for sure if a point is in the set. You can only know definitively that a point is outside when you find a sufficiently large n so that |z(n)| > 2.
Now in our problem, our z(n) function is going to be a little more complicated for a few reasons:
  • We're using arcs, not point
  • There's multiple rotations that can be applied so it generates a set of arc.
  • Our test of jumble vs doctrinal is about if the set of all z(n) is finite, not whether z(n) tends towards infinity.

There's probably a fractal that's more similar to what we're doing, but it's the first thing that came to mind.

wwwmwww wrote:
I LOVED IPP33 it was GREAT!!!
/jealous

bhearn wrote:
After nailing down the hashing subtleties... boom! Right off the bat, N = 7, theta = 0 (rightmost point of left disk) generates an uncountable set right at R = 1.7 (and not at anything <= 1.69). True, I've only iterated to 10,000,000 distinct points. But there is an apparent complex, perhaps fractal pattern. At R = 1.69, the orbit has 28 elements. It's a sharp transition. I can show samples of R = 1.7 at various smaller numbers of iterations if anyone is interested.
I'd be curious where exactly the transition occurs. Nothing about 1.7 strikes me as an obvious number. I probably need to look at the dodecagons inscribed in the circles and see the relative points where they cross over or something.

wwwmwww wrote:
You should ask Brandon about my stab at the Go First Dice problem after Gathering for Gardner last year. The same basic scenario... in very short order Brandon had a program that was just as many orders of magnitude faster then mine. Though there I think we are still looking for a solution.
Yeah, we're still working on that. I wrote a new version of my program so that we can now search for dice sets where each die can have a different number of sides. Last I heard from Eric, he had spent 81 days of computation to determine that there are no 18-18-18-30-30 sets. We know billions upon billions of ways that these sets don't work.

wwwmwww wrote:
My role seems to be get excited about a problem. Write a very slow program that allows me to share that excitement and then get a programmer FAR better then I interested in the problem.
You do a lot more than just that, but also don't underestimate the value of a seed of an idea.

wwwmwww wrote:
(1) What is theta?
(2) What is the uncountable set generated at R=1.7? Is it pieces, cuts, something else? And how can you tell its uncountable? I assume you mean its a set larger then the integers.
(3) Please define orbits and elements in this context? I'm not sure what an orbit having 28 elements means. Going back to question (2) were you saying that orbit had uncountably many elements?
Correct me if I'm wrong bhearn, but this is what I think he's doing.
  1. Start with a set of points that only includes (x,y). His start list contains the point (r*cos(theta), r*sin(theta)), a point on one of the original circles.
  2. For each point in the set that hasn't been processed, apply all single move rotations of R or L and add these new points to the set.
  3. Repeat step 2 until every point in the set has been processed.
For R<1.7, this set had only 28/30 elements (depending on the other parameters). For R=1.7, the set was continuously growing and not stopping. He terminated the evaluation after 10,000,000 points were added to the set and assumed it would continue to grow. Technically it's not a proof that there's an infinite number of points, but it's highly suggestive of it.
The start point he uses is on one of the active cuts. The rest of the points are going to be on the stored cuts.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Aug 08, 2013 2:59 am 
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GuiltyBystander wrote:
Correct me if I'm wrong bhearn, but this is what I think he's doing.
  1. Start with a set of points that only includes (x,y). His start list contains the point (r*cos(theta), r*sin(theta)), a point on one of the original circles.
  2. For each point in the set that hasn't been processed, apply all single move rotations of R or L and add these new points to the set.
  3. Repeat step 2 until every point in the set has been processed.
For R<1.7, this set had only 28/30 elements (depending on the other parameters). For R=1.7, the set was continuously growing and not stopping. He terminated the evaluation after 10,000,000 points were added to the set and assumed it would continue to grow. Technically it's not a proof that there's an infinite number of points, but it's highly suggestive of it.
The start point he uses is on one of the active cuts. The rest of the points are going to be on the stored cuts.

Yes, that's right. The one thing missing is that we don't allow points to be moved outside of the disk intersection. I explained above why we can ignore such moves. But it's this fact that helps illuminate the jumbling transition – at a critical radius, new move sequences exist which were pruned before because they moved points outside the intersection.

Quote:
I'd be curious where exactly the transition occurs. Nothing about 1.7 strikes me as an obvious number. I probably need to look at the dodecagons inscribed in the circles and see the relative points where they cross over or something.

Actually N = 7 starts jumbling earlier than R = 1.7, around 1.63. But the point at theta = 0 doesn't generate an infinite orbit until about R = 1.6928. I have looked a little bit at where these numbers come from, but I can't say exactly yet. Right now I'm looking for a single generator sequence that works to generate an infinite orbit. Haven't found one yet. I am now printing out the move sequences for each point, so we might gain some insight from analyzing those. I have learned already that it's sufficient to restrict our attention to clockwise moves. Here is a sample set of cut move sequences found:

adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRLRLRLRRLRRLRRLRLRLRLRLRRLRLRRLRLRRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLLRLLRLRLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLLRLRLRLRLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLRLRLRLRLLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLRLRLRLRLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLRLRRLRLRLL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLRLRRLRLRLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLRLRRLRRLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRLRLLRLRLRLRLRLLRLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRRLRLRLRLLRLRLLRLRLL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRRLRLRLRLLRLRLRLRLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRRLRLRLRLLRLRLRLRRLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRLRLRLRLRRLRLRRLRLRRLRLRRLRLRLRLLRLRLRLRRLRLRLRLRLRRLRLRLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLLRLLRLRLRLRLRLLRLRLLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLLRLRLRLRLRLLRLRLRLRRLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLLRLRLRLRLRLLRLRLLRLRLLRLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLRLRLRLRLLRLRLLRLRLLRLRLLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLRLRLRLRLLRLRLLRLRLLRLRLRL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLRLRLRLRLLRLRLRLRRLRLRLRLL
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLRLRLRLRLLRLRLRLRRLRLRLRLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLRLRLRLRLLRLRLLRLRLLRLR
adding cut, got here by LRRLLRLRLLRLRLLRLRLRLRRLRLRLRLRLRRLRRLRRLRRLRLRLRLRLRRLRLRLRLLRLRLRLRLRLLRLRLRLRLRLLRLRLRLRLRLLRL


Last edited by bhearn on Thu Aug 08, 2013 3:35 am, edited 1 time in total.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Aug 08, 2013 3:24 am 
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GuiltyBystander wrote:
I was thinking about building my own version of your program to draw pictures too and I noticed the connection to fractals as well. It most reminds me of the Mandelbrot Set. For those not familiar with it, it is a fractal that contains a set of complex points. A point c is in the fractal if z(n) doesn't tend towards infinity in the function (z(n+1) = z(n)^2 + c | z(0) = 0). Points in the fractal will be stuck in orbits while points outside of the will run away (if anything is more then 2 units away from the origin you can instantly know it will be outside the set). The problem is that you can almost never tell for sure if a point is in the set. You can only know definitively that a point is outside when you find a sufficiently large n so that |z(n)| > 2.
Now in our problem, our z(n) function is going to be a little more complicated for a few reasons:
  • We're using arcs, not point
  • There's multiple rotations that can be applied so it generates a set of arc.
  • Our test of jumble vs doctrinal is about if the set of all z(n) is finite, not whether z(n) tends towards infinity.

Right. Actually I don't think it matters that we're using arcs; all we really care about is all the points along a disk edge arc. A critical possible distinction, I think, is that for the Mandelbrot set – and more generally for other kinds of fractals – there is some particular function we are applying repeatedly. But here, we might need to consider the orbit of a point under a combination of multiple functions. If it turns out that there is in fact a particular generator sequence that works all by itself – repeated applications generate an unbounded orbit – then this distinction will effectively disappear, and we will have something maybe less interesting. Re your last point, yes, that's why we can't directly compare a Mandelbrot set image to the images I've been posting; they're not really showing the same thing, even though both are about images of points under transformations.

Quote:
There's probably a fractal that's more similar to what we're doing, but it's the first thing that came to mind.

Maybe, maybe not – that's what really interests me here; there's a possibility we've run across a new way to generate a fractal. E.g., take a look at the ways fractals are generated on the wikipedia page for fractals. This type of process doesn't really seem to fit any of those categories (assuming there's not a single generator sequence that does it). Correct me if I'm wrong.

Edit – hmm. Actually, this may be an example of an iterated function system.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Aug 08, 2013 3:40 am 
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GuiltyBystander wrote:
I'd be curious where exactly the transition occurs. Nothing about 1.7 strikes me as an obvious number. I probably need to look at the dodecagons inscribed in the circles and see the relative points where they cross over or something.

Oh, and to be clear, R = 1.7 is specific to N = 7, so no need to look at dodecagons. For N = 12, Gizmo Gears, it starts jumbling around R = 1.39. Conveniently just less than sqrt(2), which is where Carl started exploring. So... actually, where did sqrt(2) come from? Is that reflected in Gizmo Gears? N = 12 by itself would not jumble if R < 1.39.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Aug 08, 2013 11:37 am 
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Oskar wrote:
How do you know for sure that the puzzle jumbles? Do you see scaled self-repetition, like in Fractals? If so, what is its Feigenbaum constant? If not, then how should I read those quasi-crystal images?

I see some sort of fractal-like pattern there, yes. I don't know whether it manifests a Feigenbaum constant; it's not clear to me that all fractals must do so. One scaling constant I roughly measured from the first fractal-like image is about 2.24.

Again, having played with this for a while, it is very clear to me that N = 7, R = 1.7 generates an infinite set of cuts when unbandaging. I don't have a formal proof, and might not for a while, but that it happens and how it happens I'm sure of. For now I can't do better than the arguments laid out above, though.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Aug 08, 2013 7:33 pm 
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I've started working on my own version an I've got a couple questions. I have my circles sitting at (0,0) and (1,0) and my R values for what looks like jumbling is half of what you're reporting. Are your circles at (+-1, 0)? Going forward I'm going to assume they are.
I'm also noticing that my R values are slightly less than half of what you said. I think this is because you're seeding your list with 1 value while I start mine with a few hundred or thousand.

In a more general sense, the arcs will get cut to dust, but not every point will have an infinite number of images. I can't prove it, but I have a gut feeling that every point will will have a finite image set even though the line . Is that possible? I think so... Gotta think more about this. Cantor's diagonal argument springs to mind for some reason... My brain confuses me sometimes...

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Aug 08, 2013 7:44 pm 
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GuiltyBystander wrote:
I've started working on my own version an I've got a couple questions. I have my circles sitting at (0,0) and (1,0) and my R values for what looks like jumbling is half of what you're reporting. Are your circles at (+-1, 0)? Going forward I'm going to assume they are.

Yes, that's right. I set it up that way to match what Carl had been doing.

Quote:
I'm also noticing that my R values are slightly less than half of what you said. I think this is because you're seeding your list with 1 value while I start mine with a few hundred or thousand.

I'm not sure what you mean here... the earlier pics above were images of the entire arc, chopped into short segments; the later ones are images of a single point (theta = 0 for N = 7, theta = 0.1 for N = 12).

Quote:
In a more general sense, the arcs will get cut to dust, but not every point will have an infinite number of images. I can't prove it, but I have a gut feeling that every point will will have a finite image set even though the line . Is that possible? I think so... Gotta think more about this. Cantor's diagonal argument springs to mind for some reason... My brain confuses me sometimes...

Typo? I don't understand... if you're saying that every point will have a finite image set, then I disagree. In general, yes, not every point will have an infinite number of images for a jumbling configuration, but I think it is possible for every point to have an infinite number of images, depending on the parameters. I haven't really investigated that.

Looking forward to seeing what you turn up with your program. The thing you have to be most careful with is how to hash the points effectively, so you can tell when you hit the same points. But if you're already getting some results, maybe you've already dealt with that?


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