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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Oct 13, 2013 3:34 am 
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Bram would probably respond: "OK, and how about a 'master' version, with concentric circles".

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Oct 13, 2013 5:38 am 
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wwwmwww wrote:
Very very nice!!! Certainly wall paper worth images. Thanks for all the work you have put into this. Had I stuck with it I might only be 1 or 2 more iterations down my POV-Ray path by now. Did you ever make any headway on Oskar's suggestion above? Are you going to try and publish some of this? A new fractal type certainly seems worthy of publication outside of TwistyPuzzles to me.



Certainly, wall paper worth and published paper worth!
I strongly believe that somehow this could be related to a more generalised and universal jumbling
when we extend the circles to spheres (something which seems very intimidating to me).
And I am convinced that the jumbling we knew for 3D designs would be a small part of this...

And Bob, again, wow!!!


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Oct 13, 2013 5:07 pm 
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I like all of this. There is so much to explore here that is new and interesting even for the non twisty and the non mathematical. The graphics are an instant attraction, and they break new ground, are original, striking, pulling the eye in geometric wonder. This could become a book with twisty puzzle people collaborating, Bob, Bram, Oskar, Carl... with some of the different forum excerpts from the start of the idea of jumbling and its development.

Questions:
This is a kind of war or battle between symmetry and asymmetry. Incredibly it shows nice geometric patterns, along with areas that get cut infinitely into Bram's dust. What conditions or what R would cause the dust to cover the entire area of the two circles? This might exhibit some kind of dust pattern where some of the dust is thicker(as infinities go!). These differential dust areas could be delineated by different colors
producing more interesting graphics to magnetize our gaze. Bob has already done something least similar to this in some of the first graphics he produced that look like Xray images of crystals.

What defines critical R? Is this a rational fixed number or can it vary. Can it be irrational or transcendental? Does it represent some new mathematical constants for various symmetries of two circles? Is there more than one critical R for a given symmetry of two circles? I do not seem to up to par on this, but no doubt it has already been partially or wholly answered above in the explanations.

Pentazis suggestion of trying this with spheres suggests some comments. With two spheres the center of each sphere can turn about an infinite
number of axes. That being the case we must limit the number of axes and only allow rotations about a fixed set of axes. For instance you could restrict one set of axes to be the two axes of a two circle puzzle. Then also allow 90 degree rotations about the axis passing thru the
center of the two spheres. Now do Bob's magic if possible, then take a cross section through the two circles. Would some of these cross sections look exactly the same or similar to some of the existing jumbling patterns so far produced?

Here are some images made with an ion field microscope of atomic surfaces. These are borrowed for research and comparison here. Note that
the patterns have different sized circles due to the wave nature but otherwise some similarity can be seen to Bobs amazing graphics. The rightmost pattern
looks a little like a sphere. It is meant to be the end of a sharp needle.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Nov 04, 2013 11:11 pm 
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Still, I have no answer to many of the very interesting questions posed here. But I can answer this one:

Oskar wrote:
Bram would probably respond: "OK, and how about a 'master' version, with concentric circles".

As far as jumbling goes, I can confidently say that concentric circles will make no difference whatsoever. All we are adding is the orbits of the points on a smaller circle. If the larger circle is at the jumbling threshold, we already know that for smaller R there is no jumbling. And the additional moves (inner vs outer circle) won't really add anything. We are still exploring, for each point in the orbit, all possible moves of rotation by the given angle about each disk center.

So... turns out I am giving a talk on jumbling circle puzzles this weekend. I have a lot of material, and only 15-20 minutes to fill, so I should be fine, but I am kind of stressing out. I would like to have some movies, zooming in and/or changing R, which will be very computationally intensive to generate (but I may try), and I would like to have answers to some of the remaining questions. In particular I'm still unsure about the significance of finding quasicrystals and aperiodic tilings here. It may be that the Penrose tiles for N = 5 are a special case... I am going to try to generate diffractograms for some > critical images for other N. That should disclose quasicrystalline structure definitely. But, again I may not have time for this.

Oh, I do have one new thing to mention here. I talked about two ways of exploring critical R values when we are allowed to vary N and R between the disks: (1) keep R1 = R2, then for any given N1, N2, there is again a critical R we can find, and (2) set R1 = minimum critical R for any R2, and R2 at the minimum critical R for that R1. In both cases, we have a clear definition of a jumbling transition, again generating a discrete family of fractals, two (both conditions) for each N1, N2.

But in general, for any given R1 which admits jumbling at all, there will be a minimum R2 which jumbles. So really we have, for any N1, N2, a 2d plot of what the jumbling transition looks like. Sort of like a pressure-temperature phase diagram in physical chemistry.

Creating such a diagram here is very tedious -- resolving the jumbling transition can take a lot of time and careful adjusting of search parameters. But I have started to fill in one such diagram, for N1 = 3, N2 = 5:

Image

R1 is horizontal, R2 is vertical. Each plotted point is a critical (R1, R2) pair, generating a fractal image. Stepping through the images in quick succession, it looks like seeing successive slices in a 3d structure. The interesting parts are the sharp transitions. I don't totally understand them, but it seems to be that different points on the circle hit the fractal transition at different R. But generally, when you are just past a critical R, the generating point quickly starts to fill the space, and smear out, obscuring any fractal generated by another point. Which suggests there are potentially lots of other fractals hiding in here that I won't find by the methods I've used so far. A little more work here should clarify this important point.

The points define a curve: on one side (left/below) we have non-jumbling, discrete behavior; on the curve we have fractals; and above/right of the curve we have quasicrystalline structure.

Oh, and one more thing. I was discussing this recently with another computer scientist at a Gardner Celebration of Mind event, and he pointed me to the wonderful book Indra's Pearls, by Mumford, Series, and Wright:

Image

I have been reading this for the past few days. This is kind of exciting, because the situations they analyze are sort of similar, but not the same as what we are doing. If our results here were known, I think they would likely have been mentioned in this book. We explore the orbits generated by turning two intersecting disks; they explore the orbits generated by two Möbius transformations of a particular kind. Actually I had already considered generalizing our problem beyond rotating circles to using general Möbius transformations! However, they use continuous transformations, of the entire plane; we use discontinuous transformations, and it is the points on the circle boundaries that are interesting. We can also imagine using Möbius transformations that "slice" along some curves, the points on one side moving, the points on the other not moving; that is what I had been wondering about. What we have now is in fact a special case of this more general problem.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Nov 05, 2013 10:16 am 
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bhearn wrote:
So... turns out I am giving a talk on jumbling circle puzzles this weekend. I have a lot of material, and only 15-20 minutes to fill, so I should be fine, but I am kind of stressing out.
You'll do great!!! Any deatils on the forum? Is this something open to the general public? Would this be something you could record and upload to YouTube for example? I'd LOVE to be able to see it. The movies sound great too but I'd love to see them in the context of the presentation.

Not that you'd want to or that they are particuarly enlightening but you are welcome to use any of my POV-Ray renderings if you'd like to. Totally up to you. Anything to help the cause...

All the best,
Carl

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Nov 05, 2013 12:12 pm 
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wwwmwww wrote:
Any deatils on the forum? Is this something open to the general public? Would this be something you could record and upload to YouTube for example? I'd LOVE to be able to see it. The movies sound great too but I'd love to see them in the context of the presentation.

It's the Hacker's Conference. Unfortunately no, the conference is invitation only, and not only is it not open to the public, but everything that happens there is off the record; no recordings allowed. Alas. I gave the "Fireside Chat" (equivalent of keynote) there a few years ago. I really wish I had a recording of that.

It's not really like it sounds; there's nothing sinister going on. This is about hacking in the original, creative sense, not the computer criminal sense. The conference decided to take a lower profile long ago when CBS news did a segment on them. They were led to believe it would be a positive piece, but it was a hatchet job. So... now it's all "secret". Shhh.

The good news is, this will be the first talk of at least a few; hopefully I can polish it as I go. Probably I'll give a more condensed version at G4G (where likely I will only have five minutes). And I'm giving a CS colloquium at a local college next spring as well, where I may talk about this, depending on where the project sits then. Anyway any movies I make I will post here.

wwwmwww wrote:
Not that you'd want to or that they are particuarly enlightening but you are welcome to use any of my POV-Ray renderings if you'd like to. Totally up to you. Anything to help the cause...

Thanks! I may or may not take you up on that, but regardless I'll credit you for the initial work, and point people to this thread.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Fri Nov 15, 2013 4:36 pm 
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Slides from the talk (warning, 69MB file):

https://dl.dropboxusercontent.com/u/70818776/jumble-circles.pdf

There's not much there beyond images; I did a lot of talking to explain what we were looking at. There are some nice zooms and gradual changes in R.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Fri Nov 15, 2013 9:14 pm 
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bhearn wrote:
Slides from the talk (warning, 69MB file):
Thanks!!! May I ask how the topic/talk was received? Get asked any interesting questions that haven't been asked elsewhere?

Carl

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sat Nov 16, 2013 1:58 am 
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Is there any chance you could record a version of the talk? It looks amazing.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Mar 10, 2014 1:26 am 
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Update, there are a some things going on here.

1. Got my new Mac Pro with 64GB. Now I can get around 500 million segments, and much faster.
2. I'm giving the talk again at G4G in a couple of weeks. Most talks are limited to 6 minutes, but they gave me 17 after I explained the situation. The talk will be recorded; if I can get a public link, I'll post it here.
3. If I can find time over the next couple days before leaving town, I hope to generate some nice movies of deep zooms and parameter sweeps. If so, I'll post them here. I'd wanted to do that before the last talk, but didn't have the time (or horsepower).
4. I'm hoping to turn up experts in dynamical systems, fractals, and quasicrystals (which I am not) at G4G, to help me decide on the venue and focus for a paper.

BTW (mostly unrelated)... by coincidence there is a new video up of a talk I gave at the Sonoma State CS colloquium a couple of weeks ago. It's not about this, but it is about games and puzzles (specifically, their computational complexity, and how they can be viewed as a model of computation). If you are really interested in these topics (a background in theoretical CS helps) and have some time to kill, it's here.

http://www.youtube.com/watch?v=jXrpOwVNHd8&list=FLrZZcM2V1AECXoA6lqjLu7w


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Mar 16, 2014 11:05 pm 
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OK, here is a movie zooming in on N = 7, as R increases from 1 to 1.6234, just short of the critical radius. Maybe not clear from the clip, but as R is changing we are also zooming in to about 1000x. Several hours' processing time on the new Mac Pro. Hmm, I wonder if I can upload in higher quality...

http://youtu.be/6Evk3BbQbi4


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Sun Mar 16, 2014 11:50 pm 
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bhearn wrote:
OK, here is a movie zooming in on N = 7, as R increases from 1 to 1.6234, just short of the critical radius. Maybe not clear from the clip, but as R is changing we are also zooming in to about 1000x. Several hours' processing time on the new Mac Pro. Hmm, I wonder if I can upload in higher quality...

http://youtu.be/6Evk3BbQbi4

Very beautiful. I've watched it about 5 times. I'd like to see it at a lower framerate though. I'm okay with it stuttering, I'd just like to see how it evolves more slowly.

Also, you say this is just below the critical radius. Does that mean that every frame in the video shows a non-jumbling puzzle? Do you have a way of estimating pieces? Any chance you could add a piece counter or even an estimate of the number of pieces?

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Mar 17, 2014 10:12 am 
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Nice!!!! And I watched the video on puzzles and their computational complexity too. That was very nice as well. I'd never heard of the game Amazons before. That sounded very interesting and I want to find the time to explore that a bit more. I'm also not aware of the differences between Chinese and Japanese Go so I wanted to look into that as well.

To add to Brandon's suggestions. I'd be curious to see some estimate of the size of the smallest pieces. Assuming the base circles are say 10cm in diameter at what point do the smallest pieces become of order the cross-sectional area of an atom?

I'm also curious what the transition at the critical radius looks like in an animation like this? Is it obvious once its reached?

I'm really looking forward to seeing the talk at G4G11.

Carl

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Mar 17, 2014 11:48 am 
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That's amazing. Can the frames run independantly? I wish I was going to G4G.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Mar 18, 2014 4:10 pm 
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I'll try to get to your questions soon, swamped right now. But here is another one, a zoom of the N = 7 fractal:

http://www.youtube.com/watch?v=rv2lKlqo_wQ

This is what happens to the previous movie right at the critical radius (1.623579), at least the part of it that goes infinite.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Mar 18, 2014 4:18 pm 
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bhearn wrote:
[...] critical radius (1.623579)
Also, any chance you can determine the critical radius to much more precision? It's a long-shot but I'd like to run it through one of those algebraic number brute forcers to see if it's something like sqrt(2)*sin(1/3) or something simple-ish like that.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Mar 18, 2014 4:31 pm 
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Brandon Enright wrote:
bhearn wrote:
[...] critical radius (1.623579)
Also, any chance you can determine the critical radius to much more precision? It' a long-shot but I'd like to run it through one of those algebraic number brute forces to see if it's something like sqrt(2)*sin(1/3) or something simple-ish like that.

I can possibly improve on it, but I can't easily improve on it by much. Getting that close required generating about 600,000,000 segments (which must all be held in memory at once) on my Mac Pro. Can't do much better than that without more than my current 64GB of RAM; I'd have to improve my memory utilization, which I might be able to do, but not by an order of magnitude. It's also incredibly time-consuming to try to resolve where the threshold is. In principle, to prove that 1.6235789 doesn't jumble, I have to completely resolve it, with a theta step size of 10^-9 (what's required to jumble at 1.623579). I can't do that (it would take too long), and have to rely on testing it around the thetas that explode at 1.623579. And actually, since I am not resolving any configuration exactly, I can never be completely sure that it might not jumble at some smaller R, if my theta step size was small enough. Similarly, I can never be completely sure that it *does* jumble at 1.623579, if I could just resolve enough segments. It takes educated judgment, based on a lot of observation I have made, to make the call as I have. For example, one sure indicator is that if I generate enough segments so that I am able to zoom in far enough to see that the fractal pattern is blurring slightly at high resolution, then R is not quite small enough, and I have to try harder to find jumbling at slightly smaller R. At 1.623579, the fractal looks sharp as well as I can resolve with 600,000,000 segments.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Mar 18, 2014 5:03 pm 
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Not sure if this helps... but to quote Dr. Egon Spengler "Don't cross the streams."

By this I mean the cuts as they cross create new pieces. At a given critical radius there must be two cuts that cross where the threshold from finite piece count to infnite piece count is crossed. If one could identify the cuts which cross and cause the jumbling then finding the exact critical radius should be a geometry problem. One could back calculate from the point where the cuts just touch each other to find it.

Granted that is much easier said then done. The critical cuts could be many iterations deep so the geometry likely isn't trivial. And just the issue of identifying the critical cuts may be a big enough issue in itself to keep this from being a useful approach.

Thinking more... I may be way off base. I'm assuming there is one puzzle with a finite number of pieces in the "last" puzzle before things go infinite. But is that a valid assumption? In other words are there a finite number of different puzzles with finite pieces before everything breaks? What could be happening is that it may be possible to make a finite puzzle with any arbitrarily large (but finite) number of pieces as one gets arbitrarily close to the critical radius. This would mean there are an infinite amount of finite sized puzzles and the critical cuts are likely burried an infinite number of iterations deep.

I hope some of that makes sense to anyone but me...

Carl

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Mar 18, 2014 5:19 pm 
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wwwmwww wrote:
What could be happening is that it may be possible to make a finite puzzle with any arbitrarily large (but finite) number of pieces as one gets arbitrarily close to the critical radius. This would mean there are an infinite amount of finite sized puzzles and the critical cuts are likely burried an infinite number of iterations deep.

That seems most likely to me. For example, in the first video above, I'm approaching the critical radius exponentially slowly, but new structure seems to be appearing linearly.

But I'm not sure I understand what you mean by "critical cuts".


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Mar 18, 2014 5:34 pm 
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bhearn wrote:
But I'm not sure I understand what you mean by "critical cuts".
This was a carry over from my idea that there was one last finite puzzle before things went infinite. I assumed one could zoom in on that puzzle and see two cuts approach each other as critical radius was approached. And if these cuts crossed then things went infinite. So I was wanting to define critical radius as the radius which caused these cuts to just touch each other. As there is likely no last finite puzzle, the term is now basically meaningless as you'd need an infinite resolution to see these "critical cuts" crossing.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Mar 18, 2014 5:54 pm 
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wwwmwww wrote:
bhearn wrote:
But I'm not sure I understand what you mean by "critical cuts".
This was a carry over from my idea that there was one last finite puzzle before things went infinite. I assumed one could zoom in on that puzzle and see two cuts approach each other as critical radius was approached. And if these cuts crossed then things went infinite. So I was wanting to define critical radius as the radius which caused these cuts to just touch each other. As there is likely no last finite puzzle, the term is now basically meaningless as you'd need an infinite resolution to see these "critical cuts" crossing.
At first I had the exact same thought but I don't think it's actually a point where two lines cross but rather where two lines would meet. Take abs(1/x) for example:
Attachment:
crit_line.png
crit_line.png [ 11.15 KiB | Viewed 1037 times ]


There is an asymptote at x=0 and I think in principle you'd have to look for an asymptote to identify where it becomes jumbling. Per the limitations that Bob describes though I don't think it can be done in practice using his method.

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Mon Mar 24, 2014 9:01 pm 
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Please accept my apologies that this video is out of focus. For some reason the auto focus on my camera had stopped working and I hadn't noticed this in time. Still I think most of you will enjoy this so I've uploaded it anyways.

http://youtu.be/PgOewLpMoww

Gathering 4 Gardner had a professional film crew there filming all the talks so I hope a much better version shows up later.

My name was among those he crediting on the title page and it kills me I can't even read it now. And at the 2:45 point I'm indirectly referenced as I was among those who "dragged" Bob into this discussion at the last IPP.

For the record G4G was great and I had a blast. I would hope to see more of you there in 2 years.

Carl

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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Tue Mar 25, 2014 11:54 am 
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Thanks, Carl. Again, apologies for not explicitly mentioning your contribution in the talk. I was a bit flustered, and probably forgot to say a few other things as well. In my run throughs I was consistently at 16-17 minutes, but finished the actual talk in 12:30, talking way too fast. I used to give better talks -- haven't given enough lately; I'm rusty.


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 Post subject: Re: Are the Gizmo Gears jumbling?
PostPosted: Thu Apr 03, 2014 11:59 am 
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Finally had a chance to catch up on the thread recently. You've done some really great work and found lots of cool things. I kind of gave up on my program seeing how much more you've done with yours.

bhearn wrote:
Then, by symmetry, we really only need to care about one quadrant of the intersection.
I was doing the same thing with my program. Looking at some of your more recent pictures, there's actually a few more lines of symmetry we might be able to use. I've drawn them in over your n1=3, n2=5 example. Hopefully it make sense where these come from
Attachment:
35full.png
35full.png [ 599.03 KiB | Viewed 647 times ]

This leaves just a triangle as the critical area. Obviously it loses it's left/right symmetry because n1 != n2
Attachment:
35triangle.png
35triangle.png [ 24.46 KiB | Viewed 647 times ]

It's cool seeing the arc on the right side bounce around. Makes me wonder if you can do all these computations using just mirror symmetry. For some reason, I trust the numerical accuracy of mirroring over rotation but I guess if you combine the matrix transformations they'll end up being the same. There's some more complicated self similarity within the triangle based on what is equivalently rotating alternating circles, but this is the smallest simple shape that doesn't have any redundancy. Granted, it is computationally a harder to detect when you're inside it compared to the quadrants method, but if memory is what's holding you back, reducing the space may help.

I think this shape may help explain the penrose tiling. For the n=5 puzzle, the triangle has angles 36-36-108 which is half of the dart penrose tile. And if you're mirroring and copying this triangle around, it seems almost natural that it would create penrose tiling patterns.

I'd like to map out the intra-triangle self similarity to nail down exactly where/why the jumbling starts, but there's lots of possible mirroring options and you have to worry about staying inside the circles. I think I'll need some kind of hand made tool assistance to help draw them out but I don't know if I have the time for that at the moment.

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