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Bram

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 08, 2013 7:57 pm 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

It is of course possible to generalize the question to the radiuses of the two circles being different and the fraction of a complete rotation they can do being different.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 08, 2013 8:03 pm 

Joined: Tue Aug 11, 2009 2:44 pm

Yes. But we're already seeing interesting stuff just in the case where the disks have the same parameters. I'd like to completely understand the simplest case first.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 08, 2013 8:44 pm 

Joined: Tue Aug 11, 2009 2:44 pm

There's just a whole world of really cool structure here. Here's N = 9, R = 1.42, 20x, theta = 0, 3,000,000 points.


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 08, 2013 9:23 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

bhearn wrote: 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. Sqrt(2) comes from this: Attachment:
Sqrt2.jpg [ 90.54 KiB  Viewed 5150 times ]
The left arc and the top arc don't overlap and the piece between them touches the edge of the puzzle at a point. In other words you CAN produce a puzzle topologically equivalent to Gizmo Gears using a value of R=sqrt(2). However I suspect you could also produce a puzzle topologically equivalent to Gizmo Gears using a value of R just less then 1.39. Visually the piece I've numbered "1" above would now contain an arc segment on the surface of the puzzle but I don't believe that would change the number of states or the solution to the underlying puzzle. Although there is obviously some value of R > 1 which wouldn't allow you to produce the pieces available in Gizmo Gears. Off the top of my head I'm not sure what that value is but it shouldn't be too hard to calculate. However if N=12 and R just less then 1.39 does NOT jumble (Picture PLEASE!!!) then that makes the title of this thread very interesting. It means there are two puzzles topologically equivalent to Gizmo Gears... one jumbles and the other does not. WOW!!! Going back and looking at this picture: Attachment:
GizmoCircles.jpg [ 300.8 KiB  Viewed 5150 times ]
I now have a couple questions... which I suspect aren't trivial to answer but I'll ask them anyways. (1) I believe I see an area of N=12 R=sqrt(2) which contains pieces of measureable area inside the yellow circles. Do the pieces outside these areas (there are more then just the 2 I've circled) contain nonzero area? It appears likely that any piece containing finite area in this region continues to be cut with future iterations and that there may be no islands in this sea of jumble dust. (2) If the pieces with finite area only appear in areas like the yellow circles, are there a finite number of such pieces? If this area is a perfect circle I could see the edge of this area being made of smaller and smaller pieces and there being infinitely many of them. If so there may be some very interesting fractal like behavior on the shore of this sea of jumble dust. (3) If there are a finite number of pieces that contain a positive (nonzero area) then how many are there? Would it make for an interesting doctrinaire puzzle if there was an app that just allowed you to play with these pieces which ignored all the jumble dust? Then again even if there are an infinite number of positive area pieces one could always make a doctrinaire puzzle which ignored the ones below a certain size. bhearn wrote: Yes, that's right. I set it up that way to match what Carl had been doing. Yes that is my fault. That felt more natural to me. But you are of course welcome to define things in what ever way you find easiest. Andreas Nortmann wrote: Thank you Carl, for taking up this challenge. Thank you Andreas for the great challenge and starting this thread. Its been a while since I've had this much fun with a thread here (yes they are all fun, this one just more so) and to be honest this one caught me very much by surprise. Carl P.S. I believe Gizmo Gears is topologically equivalent to N=12, sqrt(2) >or= R > 2/sqrt(3) and yes R = 1.39 falls in this range.
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Last edited by wwwmwww on Thu Aug 08, 2013 9:51 pm, edited 3 times in total.


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Oskar

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 08, 2013 9:38 pm 

Joined: Mon Nov 30, 2009 1:03 pm

bhearn wrote: I see some sort of fractallike 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 fractallike image is about 2.24. Bob, I am fascinated by the qualification "sort of fractallike pattern" that you use. By definition "classic fractals" are self similar, That is, when you zoom in to a specific area, followed by a blowup by a Feigenbaum constant and a rotation+translation, that you see the same image again. However, "Hearn Fractals" seem to be quasicrystalic. What does that mean? Could you illustrate your finding of the "roughly 2.24", please? That is, take an image, identify a 2.24 times smaller box in the image, and show that that smaller cutout is similar to the original. Is that possible? Or how else should I interpret the 2.24 number? bhearn wrote: There's just a whole world of really cool structure here. Here's N = 9, R = 1.42, 20x, theta = 0, 3,000,000 points. I do not understand what I am looking at. I see islands that are not connected by any cuts. Am I missing something? Wow, this all is very interesting Twisty Puzzle science! Oskar
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Last edited by Oskar on Thu Aug 08, 2013 10:07 pm, edited 2 times in total.


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 08, 2013 9:59 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Oskar wrote: I do not understand what I am looking at. I see islands that are not connected by any cuts. Am I missing something? I was confused too when Bob first started just mapping single points. In short he is no longer drawing cuts... he is just picking a single starting point defined by his theta (which I asked about earlier) and then showing us all the spots that point visits in a finite portion of its orbit. I still don't see why he says that orbit is uncountable. Being infinite is enough to mean the puzzle jumbles but being uncountable (to me at least) means something totally different. The integers are countable but infinite for example. The reals are also infinite but they are not countable. Seeing as there is an iterative process being used to generate these orbits the set of points in that orbit would appear countable to me. Carl
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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 08, 2013 10:40 pm 

Joined: Tue Aug 11, 2009 2:44 pm

wwwmwww wrote: I still don't see why he says that orbit is uncountable. Being infinite is enough to mean the puzzle jumbles but being uncountable (to me at least) means something totally different. The integers are countable but infinite for example. The reals are also infinite but they are not countable. Seeing as there is an iterative process being used to generate these orbits the set of points in that orbit would appear countable to me. Funny thing. I meant to say, it generates a countably infinite set  so, oops. That's an embarrassing mistake to make. But! Thinking about whether the set is countable vs. uncountable... it might actually be uncountable. We're not actually using an iterative process, in the sense of applying a single function over and over. We're generating points from sequences of the form LRRLLRLRRLRLL... etc. In other words, each point in the image is in correspondence with some binary decimal. But, how dense is the set? My sense is now, having actually thought about it, that it likely is uncountable! Or rather, that the fully unbandaged set of pieces is of uncountably cardinality. Which would mean that we could not unbandage it even with a (countably) infinite number of sequential unbandaging steps.


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 08, 2013 11:07 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

bhearn wrote: Thinking about whether the set is countable vs. uncountable... it might actually be uncountable. We're not actually using an iterative process, in the sense of applying a single function over and over. We're generating points from sequences of the form LRRLLRLRRLRLL... etc. In other words, each point in the image is in correspondence with some binary decimal. But, how dense is the set? My sense is now, having actually thought about it, that it likely is uncountable! Or rather, that the fully unbandaged set of pieces is of uncountably cardinality. Which would mean that we could not unbandage it even with a (countably) infinite number of sequential unbandaging steps. The binary decimals are still countable. Though I do agree this set could be uncountable... I just don't think this argument proves that it is. If this does actually produce areas that are just a sea of jumble dust without islands then IF you could prove ANY two points in that sea are in the same orbit then yes I would think the set would HAVE to be uncountable. The reals in a line segment are uncountable so surely the points in a region of positive area are also uncountable. Would that be the only way to prove it? I highly doubt it but its the one that came to mind and I wouldn't have a clue how to go about proving any two arbitrary points were in the same orbit. Carl
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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 12:26 am 

Joined: Tue Aug 11, 2009 2:44 pm

wwwmwww wrote: The binary decimals are still countable. I should have said binary fractions, not binary decimals, but anyway, they're countable only if you mean the finitelength binary strings. But now maybe we're getting into areas not really relevant to understanding what jumbles and why. Anyway this little digression has inspired me to reopen Smullyan's "Set Theory and the Continuum Problem", which I never finished. Quote: Though I do agree this set could be uncountable... I just don't think this argument proves that it is. If this does actually produce areas that are just a sea of jumble dust without islands then IF you could prove ANY two points in that sea are in the same orbit then yes I would think the set would HAVE to be uncountable. The reals in a line segment are uncountable so surely the points in an area of positive volume are also uncountable. Would that be the only way to prove it? I highly doubt it but its the one that came to mind and I wouldn't have a clue how to go about proving any two arbitrary points were in the same orbit. I agree that my speculation does not amount to proof that the jumble dust is uncountable. However, it could still be uncountable without this continuous area criterion you propose, so my answer is no, that would not be the only way to prove it. Consider the set of all (infinitelength) binary strings, such that every even bit is 0. Cantor's diagonalization still works on this set, but there is no "continuity", if we interpret the strings as binary fractions. Between any two values in the set are other values not in the set. More obviously, the set of irrational reals is uncountable, but between any two irrationals are some rationals. That, perhaps, is analogous to the situation we have here. We could, in fact, prove that a fully unbandaged puzzle has uncountably many cuts / pieces if we could  for example  find two generator sequences A and B (each a sequence of Ls and Rs) such that all strings of A and B, applied to some arc point, produce points in the set. I'm not sure whether this is true, though. However, I will go out on a limb here and conjecture that when we understand the rules governing which move sequences generate points in the set, it will be seen that those rules allow a similar proof of uncountability. But  again, I think this countability question is really a tangent here.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 12:56 am 

Joined: Tue Aug 11, 2009 2:44 pm

wwwmwww wrote: However if N=12 and R just less then 1.39 does NOT jumble (Picture PLEASE!!!) then that makes the title of this thread very interesting. It means there are two puzzles topologically equivalent to Gizmo Gears... one jumbles and the other does not. WOW!!! Hmm, when I make the step size small enough to fully resolve R = 1.38, that also jumbles. However, N = 12, R = 1.37 is fully resolved. Here it is, in its complete, unbandaged glory: 1x: 10x: 100x: 500x:


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 1:05 am 

Joined: Tue Aug 11, 2009 2:44 pm

wwwmwww wrote: I now have a couple questions... which I suspect aren't trivial to answer but I'll ask them anyways.
(1) I believe I see an area of N=12 R=sqrt(2) which contains pieces of measureable area inside the yellow circles. Do the pieces outside these areas (there are more then just the 2 I've circled) contain nonzero area? It appears likely that any piece containing finite area in this region continues to be cut with future iterations and that there may be no islands in this sea of jumble dust.
(2) If the pieces with finite area only appear in areas like the yellow circles, are there a finite number of such pieces? If this area is a perfect circle I could see the edge of this area being made of smaller and smaller pieces and there being infinitely many of them. If so there may be some very interesting fractal like behavior on the shore of this sea of jumble dust.
(3) If there are a finite number of pieces that contain a positive (nonzero area) then how many are there? Would it make for an interesting doctrinaire puzzle if there was an app that just allowed you to play with these pieces which ignored all the jumble dust? Then again even if there are an infinite number of positive area pieces one could always make a doctrinaire puzzle which ignored the ones below a certain size. Great questions! I'll let you know when I can answer any of them.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 1:14 am 

Joined: Tue Aug 11, 2009 2:44 pm

Oh, and just because it's pretty, here's N = 12, R = 1.37, at 5x, colorcoded by which arc angle generates the cuts:


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GuiltyBystander

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 1:58 am 

Joined: Wed May 13, 2009 4:58 pm Location: Vancouver, Washington

I feel like a fish out of water here. Up is down and down is up. I don't have any idea what is happening with this weird infinity stuff so take my comments as those of a layperson. There are no stupid questions here, just an inquisitive idiot. I kind of jumped around as I wrote this so ideas may be fragmented and repeated. Sorry. bhearn wrote: 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). I'm refering to your later one that uses just a single point. Not every point will generate an infinite set. If you start your search with a bunch of different thetas, you're more likely to find one that does make an infinite set I crossed that out because I'm not sure about it anymore. Maybe all points are generatable from a single point. Though I would still recommend using several points to start with. If you start with a point that's on one of those pieces of finite nonzero area, it will have a finite orbit. Now my concern is that if our tests are on a jumbling puzzle and a search result says it has a finite orbit, does this mean that numerical precision has failed us and through computational errors the point has drifted inward away from the cuts and onto a piece? Your posts about listing a point as a history of {R,L} choices has me worried that you're saying that every point on the curve can be generated from a single point. I can't see a way to definitively prove this one way or another. Does your version of the program that handles arcs find any segments that definitely have a short orbit while the other parts have very large/infinite orbits? bhearn wrote: 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. Thought more about it and it's kind of stupid to think that no point would generate an infinite set. The length of all the cuts on a jumbling puzzle will infinite. The starting cuts are finite length. The starting cut also has a uncountably infinite number of points on it so that might throw a wrench into your argument saying how many cuts there. I need to reread this whole thread in the morning a few times. bhearn wrote: 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? I'm excited too. I haven't spent much time tuning my data structures. I'm currently using Java's TreeSet which seems good enough for a few million points. To account for floating point comparison issues, I round to the nearest 0.000001 and compare those numbers to test if 2 points are the same. I'm sure I need to tweak the rounding precision and test some hashing schemes. I was going to comment some more on some of this infinite theory but then I realized that I don't have a clue what is happening. I started to assume that there was an infinite number of pieces and each piece had a nonzero area. But infinities are weird and I'm seeing how that might not be true. Can we start a list of facts we know for sure? Oskar wrote: I do not understand what I am looking at. I see islands that are not connected by any cuts. Am I missing something? I believe they are islands because he generated his set of points starting with a single point. If he started with more points, it would fill in the gaps bhearn wrote: I haven't added any zoom/pan function to my program yet. Can we get a little more detail around that small circle on the right?bhearn
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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 3:25 am 

Joined: Tue Aug 11, 2009 2:44 pm

GuiltyBystander wrote: Your posts about listing a point as a history of {R,L} choices has me worried that you're saying that every point on the curve can be generated from a single point. No, not at all. But the existence of a single point on the original arc that has an infinite image indicates jumbling, and I want to better understand where that jumbling comes from. Why / how it happens is encoded in those strings of Rs and Ls. Some property of the sequences I listed became enabled at the critical radius. Quote: Does your version of the program that handles arcs find any segments that definitely have a short orbit while the other parts have very large/infinite orbits? Yes, absolutely. That't the norm when you hit the jumbling transition. Quote: I haven't added any zoom/pan function to my program yet. Can we get a little more detail around that small circle on the right?bhearn Well my zoom/pan is rudimentary, hard coded into the program, so it took a little hunting, but yeah, here you go.


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 10:20 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

bhearn wrote: Oh, and just because it's pretty, here's N = 12, R = 1.37, at 5x, colorcoded by which arc angle generates the cuts: You should add a legend. I assume blue is 0 degrees and yellow is 15 degrees? Also if the smallest piece has an area of 1 square centimenter... what is the value of R? How many iterations are needed to generate all the cuts? bhearn wrote: But  again, I think this countability question is really a tangent here. Agreed. But the topic of cardinality I find facinating and I can think of a few more questions... some also off on this tangent and others totally unrelated to Gizmo Gears. Tonight or this weekend I may start another thread for those interested in this topic. Granted I have no formal traning on this subject but I have a good friend that got a math Ph.D years ago that I used to discuss these sort of topics with. Carl
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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 7:39 pm 

Joined: Tue Aug 11, 2009 2:44 pm

Oskar wrote: I am fascinated by the qualification "sort of fractallike pattern" that you use. By definition "classic fractals" are self similar, That is, when you zoom in to a specific area, followed by a blowup by a Feigenbaum constant and a rotation+translation, that you see the same image again. However, "Hearn Fractals" seem to be quasicrystalic. What does that mean? Could you illustrate your finding of the "roughly 2.24", please? That is, take an image, identify a 2.24 times smaller box in the image, and show that that smaller cutout is similar to the original. Is that possible? Or how else should I interpret the 2.24 number? Well, from the Wikipedia page on fractals you linked to, "Authorities disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of selfsimilarity and an unusual relationship with the space a fractal is embedded in. One point agreed on is that fractal patterns are characterized by fractal dimensions," So perhaps a better question here is what is the fractal dimension of these point sets, but I can't answer that at present. When you say "you zoom in to a specific area, followed by a blowup by a Feigenbaum constant and a rotation+translation", I think that only describes a particular kind of fractal. It's easy to produce a particular fractal with any selfsimilarity scaling constant you like. E.g., take a unit square centered on (0, 0), and redraw scaled by .25, centered on the points (.5, .5), (.5, .5), and (.5, .5); repeat. Now whether there is some structure hidden in that fractal that reflects a Feigenbaum constant, I can't say, but it's not obvious from what I can find on the web or in my math texts that Feigenbaum constants must be manifested in all fractals. Really, I don't feel very qualified to state opinions here, because about the extent of my knowledge of fractals is that I spent some time writing fast Mandelbrotset generators, Back in The Day. (Meaning, the day when the Mandelbrot set was discovered. Yeah, I'm old. And computers were slowwwww then.) The "roughly 2.24" comes from the distances from centers of rings of 7 smaller rings to the smaller rings' centers, in the following image. I took the ratio of what looked like those distances at two adjacent scaling steps.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 8:45 pm 

Joined: Tue Aug 11, 2009 2:44 pm

Bram wrote: 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? Working on it. Here is a Google spreadsheet with the values I'm filling in as I find them.


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 09, 2013 11:02 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

bhearn wrote: Working on it. Here is a Google spreadsheet with the values I'm filling in as I find them. Not in the spreadsheet yet but above you appear to pin the critical R for N=12 down to something between R=1.37 and 1.38. Just looking at N=12 for a bit its clear that values like: R=1/cos(15 degrees) R=1/cos(30 degrees) R=1/cos(45 degrees) are interesting as that is when new interactions are introduced basically at the initial iteration state. At these levels, I expect big jumps in piece count and its where my intuition would have told me to expect the transition from finite to infinite to occur. My intuition though has failed me here and I'm not surprised as I'm far from what I'd consider an expert on this subject. Though it does make me beg the question... what makes some value in this particular range so special? Is there some critical interaction that is just starting to be introduced at some buried iteration state? Yes... I don't expect a simple answer to that question but I certainly find it puzzling. Carl P.S. I spun off my tangential questions to here.
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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sat Aug 10, 2013 5:08 pm 

Joined: Tue Aug 11, 2009 2:44 pm

Bram wrote: It is of course possible to generalize the question to the radiuses of the two circles being different and the fraction of a complete rotation they can do being different. Coming back to this... actually, we can generalize the question further to consider circle puzzles with any number of overlapping disks. This turns out to be a very interesting thing to do, because once the problem can be parameterized like this, you can ask questions of complexity and decidability. Jaap has already shown that bandaged circle puzzles are PSPACEcomplete. That is, if I give you a set of N disks and specify how they overlap and are allowed to turn, and where the cuts are, and a starting position and goal position, how can we determine whether the puzzle is solvable? Many questions like this are computationally intractable; this was the topic of my Ph.D. thesis. In this case, the problem is PSPACEcomplete. Without getting into exactly what that means, these problems are (almost certainly) even harder than the NPcomplete problems such as Traveling Salesman, Boolean Satisfiability, etc. There's no efficient algorithm to solve them. The really interesting thing to me here is that, instead of "can this bandaged circle puzzle be solved", if we ask "can this bandaged circle puzzle be unbandaged", we have a problem that is at least a candidate for being undecidable. What does that mean? If it's undecidable, it means that there is no algorithm, even in principle, that can answer the question correctly, for any given problem instance. Undecidable problems are really, really cool! But before we could consider trying to show undecidability for the general case, first it is necessary to be able to tell when the simple cases jumble! And so far, we are still figuring that out. Edit â€“ I should mention that Scott (VeryWetPaint) and I were discussing whether decidability could be an issue for whether circle puzzles jumble, at IPP. It didn't seem to me at the time like the concept would be relevant here. But that was for two disks. For N, yes, it's a possibility. Well, actually, hmm, it might be relevant for two disks as well, but only if a problem instance contains more information than just turning angle and radius. Maybe a set of initial cuts, or a more complicated set of valid turning angles. But it would be easier to prove (but still not easy, probably!) in the Ndisk case.


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Bram

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sat Aug 10, 2013 10:40 pm 

Joined: Sat Mar 22, 2003 9:11 am Location: Marin, CA

bhearn wrote: Bram wrote: 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? Working on it. Here is a Google spreadsheet with the values I'm filling in as I find them. It's notable how large the values for 10 and, especially, 5 are. I'd be interested in seeing what a notquitejumbling depth 5 looks like. Also, are you completely sure that 3, 4, and 6 never jumble at any depth?


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sat Aug 10, 2013 10:48 pm 

Joined: Tue Aug 11, 2009 2:44 pm

Bram wrote: It's notable how large the values for 10 and, especially, 5 are. I'd be interested in seeing what a notquitejumbling depth 5 looks like. Also, are you completely sure that 3, 4, and 6 never jumble at any depth? Pretty sure. I ran R up to a few hundred for some of those values. Working on the notquitejumbling N = 5. There's a ton of cuts when R is close to critical; this might take a while. I'm running R = 2.14. But you can already see R = 2 back on the first page. The most notable thing to me is that the even numbers tend to need a larger radius than the overall trend would suggest.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sat Aug 10, 2013 11:07 pm 

Joined: Tue Aug 11, 2009 2:44 pm


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Aug 11, 2013 10:42 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

bhearn wrote: Here's N = 5, R = 2.14, at 1x, 10x, and 100x. Dinner time now... Math certainly makes some of the prettiest art. It boggles my mind just trying to think about how many pieces might be in this puzzle... or how many moves it would take just to thoroughly scramble the puzzle. And I'd venture a guess even the best solvers here wouldn't know how to approach solving something like this. If you need an algorithm for each piece type... the number of piece types is huge beyond imagination and I'd guess the algorithms would need to be huge as well. Back to the art side... I love the way something so apparently simple (two over lapping circles) can produce something with so much order and complexity. Carl
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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Aug 18, 2013 9:49 pm 

Joined: Tue Aug 11, 2009 2:44 pm

So... I have made a lot of progress here. First, Bram and Eric Vergo both commented on my table of results, noting that N = 2, 3, 4, and 6 don't seem to jumble, but all other N do. Eric noted (on Facebook) that "all values for N that don't produce jumbling are numbers with rotational symmetry that produce tessellations in 2d". Indeed, this is key to understanding what is going on. First, consider what happens when R is very large. To unbandage means we need to propagate images of the cut edges under all possible move sequences. With R large, we can only unbandage when the image formed by all the cuts is closed under rotation about two distinct points (the disk centers). But the Crystallographic restriction theorem implies that this is possible only for N = 2, 3, 4, or 6. For other N, we instead can have quasicrystals. So why is this not a problem for all R? Because when R is small, effectively a large class of move sequences is disallowed. I remarked earlier that when propagating cut images around, we can ignore moves that move cuts outside the disk intersection. Those moves are legal, but serve to put a cut where it can't be moved by one of the disks, until it is moved back. So we might as well just assume it's moved back, and omit the sequence of moves that takes it out of the intersection and moves it back. This means that the cut image now doesn't have to be closed under arbitrary Nfold rotations about two points, only under certain restricted sequences of those rotations. When R is large enough to induce Nfold rotational symmetry about the two centers, we will necessarily jumble unless N = 2, 3, 4, or 6. The only issue now is studying exactly what happens at the transition to jumbling, and why it occurs at the particular R values it does. Here, I haven't made much progress. Instead, I've spent a lot of time improving my program  now I can easily navigate around and change all the parameters without recompiling. Also I can play with color mappings and make lots of pretty pictures. Really, it's way too fun to just sit around and surf through the spaces. If there's interest I'll share the program (Mac only!), but it needs a little more polish first. So what about those quasicrystals? I said I thought I saw them hiding in there. Indeed, they are there. This is clearest when N = 5. Here's a sample screen shot (with full parameters, for others who want to look  GuiltyBystander, how is your program coming along?). There is a clear 5fold lattice structure. Now there is no such thing as a regular 5fold lattice, but this does look like a quasicrystal lattice. And indeed, it is possible to place Penrose tiles along the "grid lines": Again, thanks to VeryWetPaint for first suggesting that jumbling might have something to do with Penrosetile geometry. It's been clear to me for quite a while that this quasicrystal structure is there, but it wasn't until today that I actually sat down and laid tiles on it. Yep, it works!
Last edited by bhearn on Mon Aug 19, 2013 12:43 pm, edited 1 time in total.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Aug 18, 2013 9:57 pm 

Joined: Tue Aug 11, 2009 2:44 pm

Here's the kind of thing that happens when N = 6  you get a regular hexagonal grid structure. This is R = 35.


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atompuz

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Wed Aug 21, 2013 6:30 pm 

Joined: Sun Aug 11, 2013 2:44 pm

Gizmo Gears, Farmland Gears, Magical Gears, and Battle Gears were all designed with the same cut depth so that fewer parts are necessary, ie. only the pieces were different, the frameworks are the same. Possibly several other designs are possible with this same cut depth. Gizmo was designed to look like it has both triangular and square symmetry translating to dodecagonal symmetry when doing full unbandaging attempts.
Bob Hearn has produced some amazing results. One could think of expanding circles as waves with a single crest and very thin. Of course the waves can not rotate like the twisty puzzles. Yet the electron orbitals supposedly circulate about and are waves, but any real connection is a question with no answer at present. These results seem to connect twisty puzzles to physics and crystallography at the least. Many thanks to the forum for this cool stuff.
Thus here is a thought experiment. Write a computer program using quantum theory to simulate photons going thru a triple slit and see what the interference pattern looks like. You could off set the slits by having the middle one located closer to the source. For instance locate the slits like poles on three adjacent vertices of a flat hexagon. In addition the photographic place would be slanted a steep angle to the horizontal so that the interference pattern appears two dimensional. Would the pattern be hexagonal? Now try it with the slits on the adjacent vertices of a pentagon. Would the pattern be Penrose Tile like? Try the slits at an angle to each other. Many other experiments possible. Try having the computer record 3D interference patterns. Perhaps similar experiments could be done with real apparatus to cross check the computer results.


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Wed Aug 21, 2013 7:21 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

bhearn wrote: When R is large enough to induce Nfold rotational symmetry about the two centers, we will necessarily jumble unless N = 2, 3, 4, or 6. The only issue now is studying exactly what happens at the transition to jumbling, and why it occurs at the particular R values it does. Here, I haven't made much progress. Oh I wouldn't say "only issue". Along the lines Bram has mentioned we could introduce N1 and N2 and allow the two circles to have different N values. We could also do the same with R and introduce R1 and R2 and allow the circles to be of different sizes. So say if N1=3 I might guess that N2=2, 3, and 6 may still not result in jumbling but that N2=4 may. I'm just guessing that as the least common multiple of N1 and N2 still falls in the set of (2, 3, 4, and 6) but the least common multiple of 3 and 4 is 12. And I have no idea what allowing R to differ between the circles may cause. bhearn wrote: Here's the kind of thing that happens when N = 6  you get a regular hexagonal grid structure. This is R = 35. Looking at the red lines which appear to form the perfect hexagons.... as these aren't strait lines but parts of arcs are each of these really two arcs, one concave and one convex? Or are the hexagons formed of alternating concave and convex segments? Carl
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Andreas Nortmann

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 22, 2013 11:36 am 

Joined: Mon Aug 02, 2004 7:03 am Location: Koblenz, Germany

atompuz wrote: Bob Hearn has produced some amazing results. One could think of expanding circles as waves with a single crest and very thin. Of course the waves can not rotate like the twisty puzzles. Yet the electron orbitals supposedly circulate about and are waves, but any real connection is a question with no answer at present. These results seem to connect twisty puzzles to physics and crystallography at the least. Many thanks to the forum for this cool stuff. I assume it is you Douglas? Beyond that I can't do more than second your praise. Amazing, Bob! I am happy that this problem is in good hands.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 22, 2013 12:18 pm 

Joined: Tue Aug 11, 2009 2:44 pm

I am still assimilating atompuz's interesting comments. And yes, Carl, soon I will get around to trying independent R and N for the two disks. Good question about which sets jumble; I'd guess that your guess is right. In the meantime I just wanted to mention that right at the jumbling transition, if you look at the image of a single point, there is actually a clear fractal structure. Here's N = 7, R = 1.6236, theta = 0.226675, at three scales: Also it occurred to me that there's a way to plot an image here more akin to the way the Mandelbrot set is drawn. What we have above is the image of a single point on the cut arc. Instead we could plot, for each pixel in the image, not the number of iterations until it "escapes", as for the Mandelbrot set, but the number of distinct images points (in the intersection) generated by starting with that point and making all moves. Or similarly, the number of iterations of "apply all four single moves to this point set" before the point set stabilizes, again starting with an arbitrary point. That might look very different from this. Or it might not. All the stuff that's dark blue above would still wind up in the same set, and would still be rendered with the same color. But the currently unrendered area might show some interesting structure as well. Also I've just received this from Amazon; hoping to learn a lot!


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 22, 2013 4:57 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

wwwmwww wrote: bhearn wrote: Here's the kind of thing that happens when N = 6  you get a regular hexagonal grid structure. This is R = 35. Looking at the red lines which appear to form the perfect hexagons.... as these aren't strait lines but parts of arcs are each of these really two arcs, one concave and one convex? Or are the hexagons formed of alternating concave and convex segments? Ok. I've studied this image a bit more and I think I can answer my own question. Notice there are 2 orange segments coming out of the ends of each red segment. This tells me there are many many very thin parts on can't resolve at this scale. A few new questions: (1) Can you produce an image of the entire puzzle? I'm sort of curious how the hexagons terminate at the puzzles surface. Hmmm... that may require too images. One of the entire puzzle and one at this same scale except taken at the outer edge of the puzzle. (2) Any ideas on how one might be able to count the number of pieces present in some of these finite puzzles? (3) Assuming we could count the number of pieces present in the puzzle. I'd expect constant piece count over a small range of R (assuming N=N1=N2 and R=R1=R2). So for the puzzles which jumble... are their a finite number of puzzles which contain a finite number of pieces? Or I could see that as one approaches some critical R that the range with a given piece count could get smaller and smaller and their could actually be an infinite number of puzzles with a finite number of pieces. Follow my logic? I hope that question makes sense. (4) If there are a finite number of puzzles which produce a finite number of pieces for certain values of N (allowing R to very) which produce jumbling, I'd be curious how many pieces were in the puzzle with the largest value of R to produce a finite piece count. That and the piece counts as R changes could produce some very interesting numer series. (5) Another way to ask question 3 is to ask if these series are finite or infinite. They are likely interesting in either case. Thanks, Carl P.S. I agree with Andreas. I've very happy with the hands this problem has gotten into.
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atompuz

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sat Aug 24, 2013 1:01 pm 

Joined: Sun Aug 11, 2013 2:44 pm

Yes, Andreas, Doug Engel here. Martin Gardner would have liked this stuff. I am new to the forum so hope I am not too crude, or rub the wrong way. My Circle Puzzler's Manual talks about puzzles with an infinite number of pieces but did not anticipate the jumbling idea. Also CPM mentions non periodic circle puzzles, but that is a another idea. I wonder if they would make any sense and lead to more jumbling and non perodic patterns with Bob's program. Thanks to Andreas for doing something with my question, asking if Gizmo gears had been checked for jumbling, and thanks to Oskar for the link. Andreas clued me that Battle Gears had been unbandaged. This got me interested so I registered and now doing more posting. Curiously some of Bob's patterns have a straight line or polygon look even though generated with circles. Almost like a sawtooth Fourier series in two dimensions if such a thing were possible. Perhaps some of these patterns might be accompanied in one corner by a small circle with a number inside showing the magnification such as 10x, 100x etc. That way you get an idea of the size of the generating circles that made the pattern. One wonders if there would be flat surface looking areas if this were done with 3 dimensional jumbling puzzles. I don't want to get too far off subject but think these results deserve more study. For instance the Mandelbrot set looks cool but one does not immediately see natural connections in the patterns it produces. The patterns Bob has produced look immediately natural and might have some real world and certainly have simple as well as more complex abstract mathematical connections. Could this method generate more kinds of non periodic tile systems? If so could it generate an unlimited number of different kinds? Or is the Penrose Pentagonal system the only one that can be made in so simple a manner, only two tiles, whereas a heptagonal system, for instance might require 10000+ kinds of tiles or an unlimited number. As puzzles go circle puzzles are not as cool as Rubiklike 3D twisty puzzles, they just don't have the charisma. You can not have a more perfect puzzle than Rubik's Cube(tm), and its offspring. Also the circles move along arcs while 3D puzzles move on flat planes offering less propensity for locking up. So the two circle unbandaged examples do finally give these circle puzzles some separate interest in their own right and represent cool symbols(Bob's examples) for the simplest two circle interactions even though having too many pieces to exist as real or virtual puzzles for a sane mind to attempt. I have worked with them a long time and like this stuff.


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atompuz

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 29, 2013 2:21 pm 

Joined: Sun Aug 11, 2013 2:44 pm

Some more thoughts here. The Mandelbrot set actually has some natural analogies, a bit like climbing vines, whorls, thistles, very organic in fact. Here is an image by Bob Hearn earlier in this forum. I have added the equal sided polygons showing the strange non periodic like structure. Note how the whole pattern seems both wave like and particle like. Very interesting. . I created this image by copying dodecagons. This gave me straight edges to make the experiment easy to do. . Would be interesting if Bob were to make a web site to allow experimentation with these patterns, or write a book with images. When the intersection radius increases the number of pieces in a circle puzzle can suddenly decrease. This happens because where previously only two circles intersected at a point, now three or more circles may intersect at a point. In physics, although there is no known connection I know of, this would be a point of greater stability or lower energy. How would this affect chaos of whirling liquids or gases, etc.? It would be cool to write a program that found these points of multisection. An interesting and perhaps easy question for puzzles that are based on a periodic grid is what is the greatest number of circles that can intersect in one point for a given radius range. What about non periodic? Perhaps some simple formula can be found to calculate these radii. Does anyone know of a formula for the number of pieces in a periodic circle puzzle circle as R increase, or an approximate formula like the prime number theorem? My conjecture is that it would be based on log R and might require pi and e in it.


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JasonSmith

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Thu Aug 29, 2013 11:23 pm 

Joined: Sat Apr 21, 2007 11:21 pm Location: Marin, CA

atompuz wrote: Yes, Andreas, Doug Engel here. Martin Gardner would have liked this stuff. I am new to the forum so hope I am not too crude, or rub the wrong way. My Circle Puzzler's Manual talks about puzzles with an infinite number of pieces but did not anticipate the jumbling idea. Also CPM mentions non periodic circle puzzles, but that is a another idea. I wonder if they would make any sense and lead to more jumbling and non perodic patterns with Bob's program. Thanks to Andreas for doing something with my question, asking if Gizmo gears had been checked for jumbling, and thanks to Oskar for the link. Andreas clued me that Battle Gears had been unbandaged. This got me interested so I registered and now doing more posting. That image reminds me of graphene: Thinking of this as a pattern (the cuts) repeatedly overlaying after some rotations, I feel these patterns must be related in some way to Moire patterns. For example, this mathematica demo feels related. Also, Moire patterns display fractallike similarity at different scales. I need to find an example of that. EDIT: Here's a weak example. I've seen stronger examples that don't use rectilinear grids: EDIT: Ah, this is better. A graphene Moire pattern shows the graphene structure at an apparent larger scale. Thinking over exactly why this happens is really fun.
_________________ Jason Smith posted here as 'io' through 2012. Visit Jason Smith's PuzzleForge on Shapeways! Jason Smith's Puzzles  YouTube Channel.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 30, 2013 12:34 am 

Joined: Tue Aug 11, 2009 2:44 pm

Wow, so many interesting comments here to be pursued and explored. I'm sorry that I have not been keeping up. Work has been keeping me busy, and it's only going to be worse for the next couple of weeks I'm afraid. I have been sneaking in time here and there to continue work on the program. This is what the control panel looks like now: Not everything there is hooked up  in particular I don't have distinct parameters for the two disks yet. Jason's comments about MoirÃ© patterns seem likely on the mark to me. So it may be that we have nothing new there with the connection to quasicrystals. A quick search turned up these interesting links: Complex quasicrystals created using new nanofabrication technique (MoirÃ© nanolithography) Hofstadterâ€™s butterfly and the fractal quantum Hall effect in moirÃ© superlatticesBut I still think the really fascinating thing here is the transition from nonjumbling to jumbling at a critical radius. Right at the transition we seem to have a pure fractal, that verges into a quasicrystal as R increases. This seems very much like a phase transition in physics. atompuz wrote: Yes, Andreas, Doug Engel here. Martin Gardner would have liked this stuff. Welcome, Doug! Great to have you here. And I still need to digest your thoughtful comments. But yes, Martin Gardner would have liked this. I am tentatively planning a talk "From Twisty Puzzles to Quasicrystals" at the next Gathering for Gardner, in March. In the meantime I have been figuring out who to contact about this that would help better place it in the context of known mathematics. I'm torn, because I really want to learn more and understand it better myself (still not finished with the book on Quasicrystals, and I think I need some remedial reading on group theory and representation theory); I want to be an active part of the discovery process here. But really I don't have the time right now, and it may be time to hand this off to the experts.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 30, 2013 12:57 am 

Joined: Tue Aug 11, 2009 2:44 pm

wwwmwww wrote: bhearn wrote: Here's the kind of thing that happens when N = 6  you get a regular hexagonal grid structure. This is R = 35. A few new questions: (1) Can you produce an image of the entire puzzle? I'm sort of curious how the hexagons terminate at the puzzles surface. Hmmm... that may require too images. One of the entire puzzle and one at this same scale except taken at the outer edge of the puzzle. Here's an image of the upper right quadrant. I've forgotten the color settings I used for the previous image. For future reference, this is color scale = 2, color phase = .2, color by theta. Here I'm showing only the cuts in the overlap (draw rotations = false): And here's the edge (with draw rotations = true):


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 30, 2013 1:10 am 

Joined: Tue Aug 11, 2009 2:44 pm

wwwmwww wrote: (2) Any ideas on how one might be able to count the number of pieces present in some of these finite puzzles? I would love to be able to do that. Could bruteforce it now, maybe, but it would be really slow once it gets at all complicated. Next step would be to generate exact (arc) cuts; now I just sample theta in points or line segments. One improvement I have made is adaptive step sizing for the segments, which normally speeds it up quite a bit: if the image of a segment never crosses the disk intersection boundary, it's fully resolved, and you don't have to go any finer to get accurate results. But the breaks, which partition the original cut arc into tiny segments, each of which has its own orbit, are not resolved any finer than a min. theta step cutoff currently. I have some code which tries to split them exactly when they have an image that crosses the intersection boundary, but it doesn't work. I have to go back to exact arc math for that, I'm afraid; one more thing on my list! Given an exact set of all the cuts, one per piece edge, the thing to do is use Euler's formula V  E + F = 2. Then we just have to accurately count edges and intersection points. Normally, when tracking points/segments, I reflect them to always positive x & y: by symmetry, we only need to compute one quadrant of the image. Also, I ignore cuts that leave the disk intersection, because they are not relevant for what goes on in the disk intersection, and you get an image of the complete puzzle just by stamping rotations (and reflections). But this either won't work or will have to get more sophisticated if we want a total count of edges and intersection points.


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JasonSmith

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 30, 2013 3:02 am 

Joined: Sat Apr 21, 2007 11:21 pm Location: Marin, CA

The crossover with quasicrystals is amazing, and I'm not sure how VWP had the intuition that it would be the case. But the image overlaying the penrose tiles is fantastic!
I think you're absolutely right about the transition boundary from jumbling to unjumbling being a potentially VERY interesting area to explore, and even possibly a unique exploration.
_________________ Jason Smith posted here as 'io' through 2012. Visit Jason Smith's PuzzleForge on Shapeways! Jason Smith's Puzzles  YouTube Channel.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 30, 2013 3:36 am 

Joined: Tue Aug 11, 2009 2:44 pm

JasonSmith wrote: The crossover with quasicrystals is amazing, and I'm not sure how VWP had the intuition that it would be the case. But the image overlaying the penrose tiles is fantastic! Thanks! I was pretty thrilled to see that it worked. OK, one more image then off to bed. This is a zoom on the N = 7 fractal above. Those small circles you see in the complete puzzle? Look closely  they are, yes, gears. Really!


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kastellorizo

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 30, 2013 7:16 am 

Joined: Fri Nov 04, 2005 12:31 am Location: Greece, Australia, Thailand, India, Singapore.

JasonSmith wrote: I think you're absolutely right about the transition boundary from jumbling to unjumbling being a potentially VERY interesting area to explore, and even possibly a unique exploration. I also completely agree with everyone on this, and I am very certain that any past research did not (and how could it?) utilise this type of pattern constructions. In the worst case, it would a theory converging to the results of another theory. I am watching this thread as it unfolds and I am astonished!!! Simply amazing. And Doug, it is absolutely brilliant to have you here! As I told you in person last February, you are in my humble opinion, one of the best puzzle designers of all time! Pantazis
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Oskar

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 30, 2013 2:40 pm 

Joined: Mon Nov 30, 2009 1:03 pm


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Fri Aug 30, 2013 5:43 pm 

Joined: Tue Aug 11, 2009 2:44 pm

bhearn wrote: OK, one more image then off to bed. This is a zoom on the N = 7 fractal above. Those small circles you see in the complete puzzle? Look closely  they are, yes, gears. The gears are a lot clearer at R = 1.641 (theta, at first jumbling angle, = 0.99):


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Oskar

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sat Aug 31, 2013 1:48 am 

Joined: Mon Nov 30, 2009 1:03 pm

bhearn wrote: OK, one more image then off to bed. This is a zoom on the N = 7 fractal above. Those small circles you see in the complete puzzle? Look closely  they are, yes, gears. Bob, I still do not understand what I am looking at. Pardon my ignorance. Some of your images show choppedup islands with arcs, surrounded by solid unchopped white. How do these islands get chopped up? Now you are showing an image with solid white gears. However, if these gears are solid, then how does the volume between the teeth get chopped up? Oskar
_________________ Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club .


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sat Aug 31, 2013 1:06 pm 

Joined: Tue Aug 11, 2009 2:44 pm

Oskar wrote: I still do not understand what I am looking at. The short answer is that you're looking at the image of a single point, not of part of an unbandaged puzzle. Here is the long answer: Let's start from literally unbandaging circle puzzles and show how we get to fractals with gears in them, step by step. I apologize if this is too much gory, irrelevant detail, but it does help explain how my program works. Let's say N = 7, R = sqrt(2), so we start with this, before any unbandaging: Where we only have two cuts. Pretty fully bandaged. Fully unbandaged, we have this: Now, how do we get that image? Effectively by making all possible move sequences, reapplying the two original cuts, as an unbandaging step, after each move. But actually, we avoid the complicated geometrical calculations involved in chopping up pieces and rotating around all the pieces we have so far. Instead, we simply propagate images of the original cuts around the puzzle, again following all possible move sequences. This gives the same result, because in the unbandaged image above, every cut must be an image under some set of moves from one of the original two cuts. Make sense? How do we know we've explored all possible move sequences? We have to keep track of which cut images we already have. We make sure that for each, we have explored where it goes under all four moves (left disk, right disk) x (clockwise, counterclockwise). When all such moves lead to cuts we already know about, we're done. Next, rather than propagate around images of entire arcs, we simplify the task by chopping them into small line segments. So the task might begin like this, with the image of a single piece of an original cut: When we do this for all the little pieces of arc, accumulating all the cut images, again, we have the complete unbandaged image above. We do still have the following problem: say we propagate a line segment around to where it crosses one of the disks. The next move would split it in half. How do we handle that? By reducing our step size and trying again. We have a minimum segment length (actually, minimum angle (theta) step) that we don't go below, so we only accurately resolve the image to a certain level of detail. (This is something I would like to improve, to get exact results for the nonjumbling cases.) When we have a minimumlength cut that still winds up crossing a disk edge after some move, we simply throw it away, so we have little tiny gaps here and there. Next, we observe that everything outside the disk intersection is irrelevant to what's really going on. All we care about is getting this: because the rest of the puzzle is just rotationally stamped copies of it. So, we don't have to track cuts that leave the disk intersection. We don't have to worry about, wait, what if they get moved all the way around, and come back in from the other side? Because making some number of e.g. clockwise moves (leaving the intersection) is the same as making some other number of counterclockwise moves (staying within the intersection). Then, by symmetry, we really only need to care about one quadrant of the intersection. So what we do is only image half of one of the original cuts, broken into little segments, and after every move, if the cut leaves the upperright quadrant, we reflect it back in. So we might start with this, applying the two possible moves (that don't leave the intersection!), left clockwise and right counterclockwise, to a segment (new cuts shown in light blue): After reflection to the upperright quadrant, we have this: So, the complete image of the original red cut segment there (shown above) is just this image, reflected and rotationally stamped. Do this for the entire upperright half cut, add them up, and we're done. (Further symmetry observations show we don't even need to image this entire half cut, but I'll skip the details.) OK. Now... what about jumbling? Let's start with this line of reasoning: if we only have a finite number of images of the original cuts, we can only generate a finite number of pieces (no jumbling). Also, if we only have a finite number of images of each small segment, we only have a finite number of images of the original cuts. Reversing this, if a puzzle jumbles, it must be the case that some small cut segment has an infinite image. In fact, some point from an original cut must have an infinite image. We can't actually detect this, because we have finite memory and time, so instead we set an upper limit on the number of cut images. So technically, this kind of approach can never prove jumbling, but it can illustrate what happens at different parameter values, allowing us to deduce that jumbling is occurring. Because we detect jumbling based on the behavior of a particular tiny segment, or point, I have tended to show images of just that point. So those are not images of unbandaged puzzles, and we are leaving twistypuzzle land proper. But they do communicate more information about the mathematical structure we are observing than the complete unbandaged image would, I think. In particular, two patterns emerge. One, at high R, the image of a single point can produce something with the symmetries of a quasicrystal. The fact that I could overlay Penrose tiles on an image of a point from N = 5 is highly striking. It's almost like... looking at the decimal digits of pi, and finding a message from God there (as was part of the plot of Sagan's book "Contact"). I haven't talked much here about what Penrose tiles are, but they are mathematically very interesting objects, with the property that they only tile the plane aperiodically. There is a lot of research behind them, and a strong connection to quasicrystals. There are Penrose tiles hiding in twisty puzzles! This is awesome!Two, at critical R, for all N except 2, 3, 4, and 6 (which we now understand why), jumbling occurs, and the image of some single point from an original cut produces a clear fractal pattern. It was that criticalR fractal for N = 7, again, starting with a particular point (theta = 0.226675), that had the gears hiding in it. My program is set to stop when a single cut segment hits the maximum cutimage threshold, so I haven't actually generated any images yet of a complete, unbandaged, jumbling puzzle. But just so we can see what one looks like, I've changed that setting. Here is N = 7, R = 1.6236 (critical value), fully unbandaged (limited, of course, to a finite number of cut images): Zooming in on the "gears", we don't see much unless we take some additional visualization steps; the jumble dust is too dense. Here I've colored every cut segment based on which part of the original cut arc it came from. Zooming in further, here are two of the "teeth": Again, not quite as clear as in the original image of just a point, but at least you can see that there are proper cuts and pieces in there. I hope everything is understandable now!


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Oskar

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Sep 01, 2013 2:38 am 

Joined: Mon Nov 30, 2009 1:03 pm

bhearn wrote: I hope everything is understandable now! Bob, Many thanks for the detailed description. Now I better understand your analysis. That does not mean that I understand why everything looks the way that it looks. I notice that you only analyze (part of) the intersection between the two circles, as the rest of the circles follows from there. If that is the case, there is no reason to limit yourself to rotation angles of 360/n degrees with integer n. The picture below shows an example of two intersecting circles, where each circle can turn just one step up followed by one step down. Attachment:
Unbandaging.jpg [ 49.62 KiB  Viewed 3996 times ]
Consider yourself Brammed Oskar
_________________ Oskar's home page, YouTube, Shapeways Shop, Puzzlemaster, and fan club .


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Sep 01, 2013 8:22 am 

Joined: Tue Aug 11, 2009 2:44 pm

Oskar wrote: I notice that you only analyze (part of) the intersection between the two circles, as the rest of the circles follows from there. If that is the case, there is no reason to limit yourself to rotation angles of 360/n degrees with integer n. The picture below shows an example of two intersecting circles, where each circle can turn just one step up followed by one step down. Consider yourself Brammed Actually, the justification for ignoring everything outside the intersection is based on two properties that no longer hold here: (1) that we are exploring all move sequences, and (2) some number of clockwise moves is equal to some other number of counterclockwise moves. I thought that the technique would probably work anyway, but simple changes to my code are not quite enough. I will try to get an image of this for you, but I am slammed now, and it might be a while.


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Sep 01, 2013 6:20 pm 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Oskar wrote: kastellorizo wrote: Doug ... ... you are in my humble opinion, one of the best puzzle designers of all time! +1 Doug, I must confess I'm not as familiar with your work as I should be and I hope to correct that soon. I just wanted to say welcome to TwistyPuzzles and its very nice to see such a well respected designer join the community. Oskar wrote: there is no reason to limit yourself to rotation angles of 360/n degrees with integer n. The picture below shows an example of two intersecting circles, where each circle can turn just one step up followed by one step down. Interesting problem... Though not much of a puzzle. Like the 1x2x2, it seems at each move you have 2 options, either undo the last move or proceed forward with what must be the Devil's Algorithm. But I assume in this case that algorithm may be infinite. I must confess I never found planar circle puzzles that interesting before. I had no idea they were hiding so much. After the Rubik's Cube it took us over 2 decades before we had our first jumbling puzzle. I have no idea how old the first circle puzzle is but it appears this jumbling concept has been hiding in the dark for a very long time. Seeing the interesting fractal behavior and quasicrystal nature jumbling can produce it temps me to ask what happens in 3D. If instead or arcs what if we were talking about the cut planes of a helicopter cube. Do we get quasicrystals there too? Viewing this stuff in 3D I can imagine must be complicated and I don't want to sidetrack any of the time Bob has to play with these circle puzzles... but it does make the mind wonder. Carl
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atompuz

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Tue Sep 10, 2013 6:38 pm 

Joined: Sun Aug 11, 2013 2:44 pm

Quote: Doug ... ... you are in my humble opinion, one of the best puzzle designers of all time! Pentazis and Oskar +1, thanks for the compliment. Coming from imminently and equally if not more capable fellow designers this very humbling. wwwmwww you now know most of the stuff about circle puzzles. They can be pretty difficult to solve but Jaap's site has several with solutions. Bob's images are quite remarkable. I notice that some of the images show cut segments that stop short inside a piece. Is this just an anomaly of the program? Or does the cut continue but in a different color? The "gear" explanation is quite revealing. Bob this is great work. These twocircle puzzles could be compared(and often have been) have to rotating just two faces of a 3D twisty puzzle. I wonder if any of the 3D puzzles would jumble with this restriction. For instance rotating just two faces of a dodecahedron does not jumble even though it is pentagonal. In that sense the circle puzzles seem to do something that 3D puzzles don't do in some cases. There is much interesting mathematics hiding in these flesh and blood 2D and 3D twisty puzzles that jumble. One would like to know much more about them. Bob has blazed an interesting trail to tantalize us. Great work!!! I saw fractals in the last set of images Bob produced. Complex numbers involve abstract rotations and the Mandelbrot set comes about that way. It is possible that complex numbers could also produce some of these circle puzzle patterns but I have no idea how to even try that.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Oct 13, 2013 12:28 am 

Joined: Tue Aug 11, 2009 2:44 pm

So, finally I have finished the code to generalize this problem to independent N1, N2 and R1, R2, as Bram suggested: Bram wrote: It is of course possible to generalize the question to the radiuses of the two circles being different and the fraction of a complete rotation they can do being different. And Carl seems to be right here: wwwmwww wrote: Along the lines Bram has mentioned we could introduce N1 and N2 and allow the two circles to have different N values. We could also do the same with R and introduce R1 and R2 and allow the circles to be of different sizes. So say if N1=3 I might guess that N2=2, 3, and 6 may still not result in jumbling but that N2=4 may. I'm just guessing that as the least common multiple of N1 and N2 still falls in the set of (2, 3, 4, and 6) but the least common multiple of 3 and 4 is 12. And I have no idea what allowing R to differ between the circles may cause. There is just so much cool stuff here I hardly know where to begin. Yes, it's the LCM of N1 and N2 that we care about. So when N1 = 3 and N1 = 4 we do get jumbling. In the original problem, for a given N, there is some critical R at which jumbling occurs. Right at this transition, the image of some point on the cut arc generates a fractal. This means we have a discrete family of fractals generated, one for every N > 1 except for 2, 3, 4, 6. These are, as far as I know, new fractals to the world of mathematics. I've shown some of them above. But with R1, R2 independent, we have a continuous family of fractals. Why? Because in general, for any N1, N2 (discrete), and R1 (continuous), there is a critical R2 which generates a fractal. (Here we are still assuming, without loss of generality, that the circles are centered at (1, 0) and (1, 0).) So how exactly should we investigate where criticality (jumbling) occurs? Well first, we can vary N1, N2, but keep R1 = R2. Then, this is the fractal we get at the jumbling transition for N1 = 3, N2 = 4, and R1 = R2 = 3.465: And this is what the entire unbandaged (jumbling!) puzzle looks like: For N1 = 3, N2 = 5, this is the fractal we get, at R1 = R2 = 2.41146: Zooming in: Again: Again: (That's 50,000,000 cuts, about as high as I can go.) And the complete puzzle: Zooming in: Again (here the dense green pentagon is where the fractal part is, seen in zoom above): Continued in next post.
Last edited by bhearn on Sun Oct 13, 2013 1:16 am, edited 1 time in total.


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bhearn

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Oct 13, 2013 12:50 am 

Joined: Tue Aug 11, 2009 2:44 pm

So  that's one way to investigate the behavior for N1, N2 without having to worry about which R1 and R2 to look at: set R1 = R2. But there's another way. It turns out that there is a minimum R1 for which, below this value, there is no jumbling even if R2 = âˆž. So we can look at this minimum R1, then set R2 to the minimum value for which there is jumbling. Here is this critical fractal for N1 = 3, N2 = 5, R1 = 2.07 (the minimum jumbling value for any R2), R2 = 3.285 (the minimum jumbling value for R1 = 2.07): And the full puzzle: But wait! After playing with this a while, I realized something very cool. Now we can investigate jumbling for two disks, with any N, by finding the critical R. And for two disks with differing N, N1 and N2, we can look at R1 = R2, or R1 = minimum jumbling value. But what if we wanted to investigate puzzles with more than two disks? I don't want to write that code. The program is complicated enough already, believe me. But guess what... I don't have to. Suppose R2 is large. Then what do we have? We have a puzzle With N2 disks, each with N = N1, R = R1! Like this, for N1 = 5, N2 = 4, R1 = 1.805 (minimum jumbling value), R2 = 5 (or anything large). This is a twodisk puzzle, with the stated parameters, but it's exactly the same as a FOUR disk puzzle, with N = 5, R = 1.805, disk centers at (1, 0), (1, 2), (3, 0), (1, 2): Isn't that cool?!!! I will leave you with this pretty image from N1 = N2 = 5, R1 = 1.8142 (slightly subcritical), R2 = 3.


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wwwmwww

Post subject: Re: Are the Gizmo Gears jumbling? Posted: Sun Oct 13, 2013 1:27 am 

Joined: Thu Dec 02, 2004 12:09 pm Location: Missouri

Very very nice!!! Certainly wall paper worthy 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 POVRay 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.
Again great work... I really enjoyed the conversations we had at IPP on the topic, Carl
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Last edited by wwwmwww on Sun Oct 13, 2013 5:55 pm, edited 2 times in total.


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