The shapes of the pattern will have negligible effect on the rate it deteriorates. It will still have an effect because people will walk differently (even if it's very slight).
[deleted] ยท 14 points ยท Posted at 07:06:53 on March 2, 2015 ยท (Permalink)
How and what do you mean by this? You mean for some finite iteration you can tile the plane with copies of it? If so, this isn't really a demonstration of that.
The same transformations being applied to the square can be applied to the whole planeโso if the original plane were tiled with squares, the transformed plane would end up being tiled by twindragons.
I think the idea is to imagine the plane being tiled by squares in a grid pattern. As you do these transformations to the entire plane, every square will become a twindragon simultaneously. I guess you would need to check that each transformation doesn't affect the fact that it is being tiled, but that doesn't seem too hard.
It doesn't immediately follow from the linked animation, but it's not a huge generalization.
[deleted] ยท -61 points ยท Posted at 09:32:58 on March 2, 2015 ยท (Permalink)
Yeah, I got that now. Just confused on why a technically incorrect comment was getting upvoted, so I asked :p
Say you have a plane cut into square tiles (like a coordinate system, 2d of course). If you were to apply this to every square, I think the guy says you would have infinite twin dragons. Idk if this would work, but hey, cool idea... actually I think this would work and it would be a nice tesselation
[deleted] ยท -1 points ยท Posted at 20:02:35 on March 2, 2015 ยท (Permalink)
I understand the explanation.. but the gif doesn't show this and hence doesn't demonstrate it. So therefore, apparently, for pointing this out, I get downvoted :)
Can you elaborate on why the gif fails to illustrate that it tiles the plane? It seems to me that the gif is sufficient to conclude that it does tile the plane. Sorry you're getting downvoted.
[deleted] ยท 1 points ยท Posted at 22:38:00 on March 2, 2015 ยท (Permalink)*
Well, firstly, it isn't sufficient. It certainly suggests it's possible, however, but strictly speaking, a pretty gif is by no means conclusive. Secondly, the conclusion drawn requires extra steps to be drawn and hence the gif is not a demonstration of what the comment says it does. Please correct me if I'm missing something because I'm genuinely not understanding how this is technically a demonstration.
Edit: I don't really care about the downvoting. People on this sub seem to downvote pretty arbitrarily.
While this is far from a rigorous proof, one can imagine a large number of adjacent squares (all identical to the one in the gif) covering a plane, and each being simultaneously manipulated in the manner shown, with each iteration leaving no spaces or overlaps in the pattern. I am by no means saying that the gif is absolute proof of this, or that you are wrong. After all, just because something is intuitive doesn't mean it's demonstrably true. I would, however, like to know what specific information not available in the gif you would require in order to conclude that it does tile the plane.
In other words, if the gif doesn't demonstrate it (which it very well may not) what else is required?
[deleted] ยท 1 points ยท Posted at 23:18:44 on March 2, 2015 ยท (Permalink)
Well, perhaps tiling the plane viewed in the gif with squares and then doing the process might be what I would call a demonstration. If you have to use intuition and fill in steps, it's not really a demonstration is it? Like, by definition, a demonstration requires things to be demonstrated, not left to the viewer.
Anyways, the gif doesn't even attempt to convey the idea that the plan can be tiled. The viewer necessarily needs to make a leap to conclude a tiling is possible.
Edit: notice how every one had to describe what would tiling the plane would look like because it wasn't even present in the gif.
I think it's not perfectly obvious that the shape will remain connected under an infinite number of applications of this transformation.
It should be able to be nailed down somehow since the splittings are getting smaller as factors of 2, which means that the smallest squares are always twice as big as the splitting grid, so they always stay connected.
But that's not a well-worded explanation. I think that if someone asked me this question, I would grant that the answer is not obvious.
Nonetheless, even if the shape could somehow split, the results would still tile the plane. One could experiment with this by increasing the distance that the strips slide and seeing what you get out.
japed ยท 1 points ยท Posted at 00:50:58 on March 3, 2015 ยท (Permalink)
I think part of the issue is that you are interpreting the comment as saying the gif demonstrates that the curve tiles the plane, but the "it" in the comment probably refers to the "way to generate the dragon curve". In other words, this construction of the twindragon demonstrates that it is a shape that can tile the plane.
The point is not that this gif makes it immediately obvious to everyone (as always, it depends on what the viewer can already take for granted), but that unlike some other constructions, this one is very simply linked to the tiling property.
[deleted] ยท 1 points ยท Posted at 05:13:27 on March 3, 2015 ยท (Permalink)
As far as I can tell, the method given says nothing about dealing with the curve in the limit and hence cannot be used to say anything about the limit curve tiling the plane. So, I still don't see how anyone can claim the demonstration can be made.
japed ยท 1 points ยท Posted at 11:41:46 on March 3, 2015 ยท (Permalink)
Given that you want object to anything that requires extra steps, I'm surprised you say that there is a method given at all. I would say that the gif illustrates a construction of the twindragon as a shape, rather than a curve. The curve properties are not necessary for a tiling. That the construction produces a shape that tiles the plain follows reasonably simply from the nature of the construction, and is probably more obvious than, say, the fact that the shape is the twindragon.
However obvious you think any of it was, the original point was that this is effectively a construction of a plane tiling which is the twindragon, not that this gif gives all the details of the tiling or even the construction itself.
Even though the way this question is posed may infuriate some of you, it is really not helpful to downvote this. What follows bellow is by far the most substential explanation to this post and will be burried below threshold very soon.
This particular animation, though, illustrates a twindragon curve, which can be thought of as two dragon curves placed next to each other.
An interesting feature of the twindragon curve is that you can make tiles out of it: If you imagine tiling a floor with the square tile at the start of the animation, each step of the animation would still result in tiles that fit together: http://ecademy.agnesscott.edu/~lriddle/ifs/heighway/twindragon.htm
They (the dragon curve and the Julia sets) can both be found by using iterated function systems. However, in the Julia sets, the iterated functions have to be related by both being inverses of the same transformation. For example, (and this is in the wiki article for the Julia set), a family of julia sets defined by the function
[;z_n \rightarrow z_{n+1} = f(z_n) = z_n^2 + c' ;]
Has the inverse mappings:
[;z_n^{(1)} = +\sqrt{z_{n+1} - c} ;]
and
[;z_n^{(2)} = - \sqrt{z_{n+1} - c} ;]
(Technical aside for the complex analyiticians who are on their toes: Yes, you can put the branch cut for the square root in multiple places, so there are different sets of inverses that can be done here. I'm ignoring that, taking the usual branch cut, and just writing +/- for the different inverses.)
Inverting both of these functions gives you the same thing; the function we started with.
The iterated function mappings for the dragon can be inverted to get something like:
[;f^{(1)}(z) = \frac{2 z}{1+i} ;]
and
[;f^{(2)}(z) = \frac{2(z+1)}{1-i} ;]
Since [; f^{(1)};] is not the same as [; f^{(2)};], I believe that this means that the dragon curve is distinct from any Julia set. They can nonetheless be related to each other in that they are both encodable as Iterated Function Systems.
If anyone has something to add or corrections to this post, it would be lovely if you would chime in.
AbouBenAdhem ยท 154 points ยท Posted at 06:12:14 on March 2, 2015 ยท (Permalink)
And it demonstrates that it tiles the plane at the same time.
wintermute93 ยท 64 points ยท Posted at 15:03:19 on March 2, 2015 ยท (Permalink)
Photoshop confirms.
sinsecticide ยท 27 points ยท Posted at 03:14:11 on March 3, 2015 ยท (Permalink)
I do enjoy a good proof by photoshop
FinitelyGenerated ยท 10 points ยท Posted at 05:35:05 on March 3, 2015 ยท (Permalink)
Has someone built an induction plugin for it yet?
del-squared ยท 43 points ยท Posted at 08:15:14 on March 2, 2015 ยท (Permalink)
Now I want my floor to be tiled with these
BobFloss ยท 15 points ยท Posted at 21:59:17 on March 2, 2015 ยท (Permalink)
It would deteriorate too quickly
isthisagoodusername ยท 7 points ยท Posted at 03:10:42 on March 3, 2015 ยท (Permalink)
Or at the very least need an awful lot of grouting
[deleted] ยท 2 points ยท Posted at 03:55:00 on March 3, 2015 ยท (Permalink)
What's the optimal tile shape for it to last the longest? Squares, triangles?
BobFloss ยท 1 points ยท Posted at 04:44:58 on March 3, 2015 ยท (Permalink)
The shapes of the pattern will have negligible effect on the rate it deteriorates. It will still have an effect because people will walk differently (even if it's very slight).
[deleted] ยท 14 points ยท Posted at 07:06:53 on March 2, 2015 ยท (Permalink)
How and what do you mean by this? You mean for some finite iteration you can tile the plane with copies of it? If so, this isn't really a demonstration of that.
AbouBenAdhem ยท 134 points ยท Posted at 07:12:54 on March 2, 2015 ยท (Permalink)
The same transformations being applied to the square can be applied to the whole planeโso if the original plane were tiled with squares, the transformed plane would end up being tiled by twindragons.
joelthelion ยท 6 points ยท Posted at 19:30:21 on March 2, 2015 ยท (Permalink)
Amazing demonstration, thank you.
[deleted] ยท -21 points ยท Posted at 07:17:21 on March 2, 2015 ยท (Permalink)
Not quite sure how this technically demonstrates that, but who cares, that's pretty damn awesome. Thanks for bringing that thought to my attention :)
christes ยท 29 points ยท Posted at 09:14:18 on March 2, 2015 ยท (Permalink)
I think the idea is to imagine the plane being tiled by squares in a grid pattern. As you do these transformations to the entire plane, every square will become a twindragon simultaneously. I guess you would need to check that each transformation doesn't affect the fact that it is being tiled, but that doesn't seem too hard.
It doesn't immediately follow from the linked animation, but it's not a huge generalization.
[deleted] ยท -61 points ยท Posted at 09:32:58 on March 2, 2015 ยท (Permalink)
Yeah, I got that now. Just confused on why a technically incorrect comment was getting upvoted, so I asked :p
DrugsAreBad4U ยท 3 points ยท Posted at 14:47:05 on March 2, 2015 ยท (Permalink)
Say you have a plane cut into square tiles (like a coordinate system, 2d of course). If you were to apply this to every square, I think the guy says you would have infinite twin dragons. Idk if this would work, but hey, cool idea... actually I think this would work and it would be a nice tesselation
[deleted] ยท -1 points ยท Posted at 20:02:35 on March 2, 2015 ยท (Permalink)
I understand the explanation.. but the gif doesn't show this and hence doesn't demonstrate it. So therefore, apparently, for pointing this out, I get downvoted :)
vendetta2115 ยท 2 points ยท Posted at 21:05:27 on March 2, 2015 ยท (Permalink)
Can you elaborate on why the gif fails to illustrate that it tiles the plane? It seems to me that the gif is sufficient to conclude that it does tile the plane. Sorry you're getting downvoted.
[deleted] ยท 1 points ยท Posted at 22:38:00 on March 2, 2015 ยท (Permalink)*
Well, firstly, it isn't sufficient. It certainly suggests it's possible, however, but strictly speaking, a pretty gif is by no means conclusive. Secondly, the conclusion drawn requires extra steps to be drawn and hence the gif is not a demonstration of what the comment says it does. Please correct me if I'm missing something because I'm genuinely not understanding how this is technically a demonstration.
Edit: I don't really care about the downvoting. People on this sub seem to downvote pretty arbitrarily.
vendetta2115 ยท 2 points ยท Posted at 22:58:08 on March 2, 2015 ยท (Permalink)
While this is far from a rigorous proof, one can imagine a large number of adjacent squares (all identical to the one in the gif) covering a plane, and each being simultaneously manipulated in the manner shown, with each iteration leaving no spaces or overlaps in the pattern. I am by no means saying that the gif is absolute proof of this, or that you are wrong. After all, just because something is intuitive doesn't mean it's demonstrably true. I would, however, like to know what specific information not available in the gif you would require in order to conclude that it does tile the plane.
In other words, if the gif doesn't demonstrate it (which it very well may not) what else is required?
[deleted] ยท 1 points ยท Posted at 23:18:44 on March 2, 2015 ยท (Permalink)
Well, perhaps tiling the plane viewed in the gif with squares and then doing the process might be what I would call a demonstration. If you have to use intuition and fill in steps, it's not really a demonstration is it? Like, by definition, a demonstration requires things to be demonstrated, not left to the viewer.
Anyways, the gif doesn't even attempt to convey the idea that the plan can be tiled. The viewer necessarily needs to make a leap to conclude a tiling is possible.
Edit: notice how every one had to describe what would tiling the plane would look like because it wasn't even present in the gif.
ViridianHominid ยท 1 points ยท Posted at 01:59:48 on March 3, 2015 ยท (Permalink)
I think it's not perfectly obvious that the shape will remain connected under an infinite number of applications of this transformation.
It should be able to be nailed down somehow since the splittings are getting smaller as factors of 2, which means that the smallest squares are always twice as big as the splitting grid, so they always stay connected.
But that's not a well-worded explanation. I think that if someone asked me this question, I would grant that the answer is not obvious.
Nonetheless, even if the shape could somehow split, the results would still tile the plane. One could experiment with this by increasing the distance that the strips slide and seeing what you get out.
japed ยท 1 points ยท Posted at 00:50:58 on March 3, 2015 ยท (Permalink)
I think part of the issue is that you are interpreting the comment as saying the gif demonstrates that the curve tiles the plane, but the "it" in the comment probably refers to the "way to generate the dragon curve". In other words, this construction of the twindragon demonstrates that it is a shape that can tile the plane.
The point is not that this gif makes it immediately obvious to everyone (as always, it depends on what the viewer can already take for granted), but that unlike some other constructions, this one is very simply linked to the tiling property.
[deleted] ยท 1 points ยท Posted at 05:13:27 on March 3, 2015 ยท (Permalink)
As far as I can tell, the method given says nothing about dealing with the curve in the limit and hence cannot be used to say anything about the limit curve tiling the plane. So, I still don't see how anyone can claim the demonstration can be made.
japed ยท 1 points ยท Posted at 11:41:46 on March 3, 2015 ยท (Permalink)
Given that you want object to anything that requires extra steps, I'm surprised you say that there is a method given at all. I would say that the gif illustrates a construction of the twindragon as a shape, rather than a curve. The curve properties are not necessary for a tiling. That the construction produces a shape that tiles the plain follows reasonably simply from the nature of the construction, and is probably more obvious than, say, the fact that the shape is the twindragon.
However obvious you think any of it was, the original point was that this is effectively a construction of a plane tiling which is the twindragon, not that this gif gives all the details of the tiling or even the construction itself.
RotateMe ยท 5 points ยท Posted at 16:31:26 on March 2, 2015 ยท (Permalink)
Even though the way this question is posed may infuriate some of you, it is really not helpful to downvote this. What follows bellow is by far the most substential explanation to this post and will be burried below threshold very soon.
whirligig231 ยท 70 points ยท Posted at 08:41:45 on March 2, 2015 ยท (Permalink)
This is more properly a twindragon curve. The original dragon curve is asymmetric.
๐๏ธ fatn00b ยท 10 points ยท Posted at 17:07:34 on March 2, 2015 ยท (Permalink)
Oops. Thanks for pointing that out!
eat-KFC-all-day ยท 29 points ยท Posted at 05:39:07 on March 2, 2015 ยท (Permalink)
Can someone explain to me what I just saw?
nerd65536 ยท 59 points ยท Posted at 06:38:32 on March 2, 2015 ยท (Permalink)*
A Dragon curve is the shape you would get if you fold a long strip of paper in half repeadedly and unfold it enough to let the creases sit at 90ยฐ.
This particular animation, though, illustrates a twindragon curve, which can be thought of as two dragon curves placed next to each other.
An interesting feature of the twindragon curve is that you can make tiles out of it: If you imagine tiling a floor with the square tile at the start of the animation, each step of the animation would still result in tiles that fit together: http://ecademy.agnesscott.edu/~lriddle/ifs/heighway/twindragon.htm
(Source of the animation: http://www.eschertile.com/kids/animate.htm )
ChalkboardCowboy ยท 5 points ยท Posted at 19:14:47 on March 2, 2015 ยท (Permalink)
This construction makes it obvious why the twindragon tiles the plane. Nice!
eat-KFC-all-day ยท 5 points ยท Posted at 20:49:34 on March 2, 2015 ยท (Permalink)
Thanks a lot. You explained that clearly and well.
AveragePoster ยท 55 points ยท Posted at 06:17:12 on March 2, 2015 ยท (Permalink)
Mmm baby is that y = |sin(x)| + 5exp(-x100)cos(x) from -3 to 3 or are you just happy to see me?
apocalypsedg ยท 31 points ยท Posted at 10:46:31 on March 2, 2015 ยท (Permalink)
hahaha
http://www.wolframalpha.com/input/?i=y+%3D+%7Csin%28x%29%7C+%2B+5*exp%28-x%5E100%29*cos%28x%29++from+-3+to+3
TakeOffYourMask ยท 16 points ยท Posted at 15:14:55 on March 2, 2015 ยท (Permalink)
Wasn't expecting math penis.
But of course, I should have.
๐๏ธ fatn00b ยท 11 points ยท Posted at 17:05:41 on March 2, 2015 ยท (Permalink)
Great, now I have to tag it NSFW.
SleepyHarry ยท 5 points ยท Posted at 17:47:57 on March 2, 2015 ยท (Permalink)
for this:
y = |sin(x)| + 5*exp(-x100)*cos(x)
do this:
Trekky0623 ยท 3 points ยท Posted at 23:30:48 on March 2, 2015 ยท (Permalink)
Better yet
[; y = |sin(x)| + 5*exp(-x^{100})\*cos(x) ;]SleepyHarry ยท 1 points ยท Posted at 02:21:57 on March 3, 2015 ยท (Permalink)
TEST POST PLS IGNORE
[; y = |sin(x)| + 5exp(-x{100})\cos(x) ;]
edit: Nope, I don't get it. What is it meant to do?
Trekky0623 ยท 2 points ยท Posted at 02:40:02 on March 3, 2015 ยท (Permalink)
If you use the LaTeX browser plugin linked in the sidebar, it displays this.
SleepyHarry ยท 1 points ยท Posted at 04:19:23 on March 3, 2015 ยท (Permalink)
Just got the plugin. Thanks for pointing this out!
Fiskerr ยท 21 points ยท Posted at 06:01:37 on March 2, 2015 ยท (Permalink)
Goddamn this turns me on.
justbeane ยท 26 points ยท Posted at 12:34:38 on March 2, 2015 ยท (Permalink)
I might have a subreddit for you... /r/gonwild
[deleted] ยท 7 points ยท Posted at 12:57:07 on March 2, 2015 ยท (Permalink)
โกโกโก
agumonkey ยท 6 points ยท Posted at 16:19:32 on March 2, 2015 ยท (Permalink)
Too bad /r/polygonwild is already taken as a parody.
[deleted] ยท 7 points ยท Posted at 09:29:05 on March 2, 2015 ยท (Permalink)
[deleted]
๐๏ธ fatn00b ยท 9 points ยท Posted at 17:01:59 on March 2, 2015 ยท (Permalink)
No idea. So I guess that's left as an exercise for the reader.
ViridianHominid ยท 2 points ยท Posted at 02:21:02 on March 3, 2015 ยท (Permalink)*
They (the dragon curve and the Julia sets) can both be found by using iterated function systems. However, in the Julia sets, the iterated functions have to be related by both being inverses of the same transformation. For example, (and this is in the wiki article for the Julia set), a family of julia sets defined by the function
[;z_n \rightarrow z_{n+1} = f(z_n) = z_n^2 + c' ;]Has the inverse mappings:
[;z_n^{(1)} = +\sqrt{z_{n+1} - c} ;]and[;z_n^{(2)} = - \sqrt{z_{n+1} - c} ;](Technical aside for the complex analyiticians who are on their toes: Yes, you can put the branch cut for the square root in multiple places, so there are different sets of inverses that can be done here. I'm ignoring that, taking the usual branch cut, and just writing +/- for the different inverses.)
Inverting both of these functions gives you the same thing; the function we started with.
The iterated function mappings for the dragon can be inverted to get something like:
[;f^{(1)}(z) = \frac{2 z}{1+i} ;]and[;f^{(2)}(z) = \frac{2(z+1)}{1-i} ;]Since
[; f^{(1)};]is not the same as[; f^{(2)};], I believe that this means that the dragon curve is distinct from any Julia set. They can nonetheless be related to each other in that they are both encodable as Iterated Function Systems.If anyone has something to add or corrections to this post, it would be lovely if you would chime in.
graycrawford ยท 5 points ยท Posted at 06:26:10 on March 2, 2015 ยท (Permalink)
Does anybody have an animation of this method with more iterations?
Kebble ยท 3 points ยท Posted at 02:28:45 on March 3, 2015 ยท (Permalink)
This actually sounds like a fun programming experience (at least to me)
Let me get back to you with any findings within a couple hours
graycrawford ยท 1 points ยท Posted at 04:50:00 on March 3, 2015 ยท (Permalink)
Sweet, please do.
thejester541 ยท -5 points ยท Posted at 06:27:29 on March 2, 2015 ยท (Permalink)
Anyone??
Would like to see this, but I think it may be in danger of destroying itself...
DrugsAreBad4U ยท 3 points ยท Posted at 14:43:11 on March 2, 2015 ยท (Permalink)
Holy shit I had no idea you could generate them that way. Now there's like 20 ways to do it.
HappyGlucklichJr ยท 3 points ยท Posted at 15:45:47 on March 2, 2015 ยท (Permalink)
Is the boundary a fractal?
๐๏ธ fatn00b ยท 6 points ยท Posted at 17:00:35 on March 2, 2015 ยท (Permalink)*
It's my personal favorite fractal. The dragon curve
Edit: It's actually a twindragon curve.
oogaboogacaveman ยท 2 points ยท Posted at 19:26:26 on March 2, 2015 ยท (Permalink)
Am I the only one who keeps seeing this title on their homepage and thinking it has something to do with the new Magic: the Gathering set?
[deleted] ยท 4 points ยท Posted at 06:52:13 on March 2, 2015 ยท (Permalink)
What is this witchcraft.
Floxxomer ยท 0 points ยท Posted at 04:27:56 on March 3, 2015 ยท (Permalink)
started watching, was sure that it was going to be a trogdor joke, now i'm very disappointed