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Dragon Curve - Numberphile
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- Опубліковано 15 сер 2024
- Beautiful Dragon Curves, Fractals and Jurassic Park. Featuring Rob Eastaway.
More links & stuff in full description below ↓↓↓
Rob's website: www.robeastaway...
Thanks to Matthew Ward and Faraz Barzideh who helped Brady out with some curves!
The book Jurassic Park is by the late Michael Crichton.
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The book is also really really recommendable as a good read. Very realistic with lots of detailed (but a little fictionalized) descriptions of the science. Differs from the movie too, some characters die in the book that did not in the film etc.
Dude they use a secret tunnel behind a waterfall. How cool is that?
The book is more or less a critique of the reckless use of science, and the belief that humans can predict and control everything. The movie scenes of the group having dinner, and Ellies chat with Hammond over the ice cream, pretty much sums up half of the book.
If you want an adventure story, see the film. If you want a lecture, read the book.
The whole point of Malcolm iterations serving as chapters was to visualize to the reader that something which appears predictable quickly becomes unpredictable and ultimately takes its own form. The park goes from human control to chaos to animal control.
But it is used in the book Jurassic Park, which is about dinosaurs, not dragons!
Go figure!
Fun fact: When the Chinese found incomplete dinosaur bones in ancient times, they thought it was the bones of a dragon. So, dragons and dinosaurs do have a correlation
Dinosaurs are just irl dragons
In Japanese dinosaurs are called 恐竜 "scary dragon"!
years ago this was one of my first encounters with this fascinating shape
Dragon Curve Enthusiast it’s kinda one of my favorites that I always come back to
Wow you’re _really_ into dragon curves
@@connorcriss I'm not a mathematician, but I love fractals. And this is one of my favorites.
I plan to make myself earrings in this shape by bending some wire. I just don't know yet how I'm going to do this to get regular segment lengths and nice 90 degree angles...
Dragon Curve Enthusiast here’s a hint for if you do do that:
Take a zero:
0
Add a 0
00
Take every number before the 0 and turn it into a one if it’s a zero, and a zero if it’s a one, reverse that sequence, and add it to the end
0|0|1
Repeat this step
001|0|011
0010011|0|0011011
001001100011011|0|001001110011011
Now for every 0, do a left turn, and for every 1, do a right turn, and you will get the dragon curve. Pretty neat!
Dragon Curve Enthusiast
There’s a pattern you can use for that:
Start with a zero, and every step you copy the sequence and reverse it, then replace every 0 with a 1 and vice versa, add a zero to the end of the original sequence, and add the new sequence, to get the following pattern:
0
001
0010011
001001100011011
0010011000110110001001110011011
And so on.
Then, for every 0 bend the wire 90 degrees to the left, and for every 1 bend it 90 degrees to the right. You will get a dragon curve.
"I like the book because of the pretty pictures"
"We've all seen the film, but very few people have read the book."
I must be about the only person who's read the book and not seen the film, then.
+Andrew Ray You probably are...
I have done both. The book was better than the film. Wasn't a huge fan of The Sphere though.
I have read the book as well. I have read a few Michael Crichton books!
TIL there was a book.
@@Jivvi there's two, actually
I first learned about the dragon curve from Martin Gardner. You can construct the pattern without folding paper using a simple iterative process: Take the current sequence, switch all the peaks and valleys, then reverse the order - then add this "upside-down and backwards" sequence to the beginning of the current sequence, and put a Valley in between them. This becomes the next iteration.
Starting with nothing, nothing upside-down and backwards is still nothing, put the two nothings together with a V in between, and you have:
V
V upside-down is P, and backwards is still P, so put P and V together with a new V in between:
PVV
Upside-down is VPP, backwards is then PPV. And this to the PVV with a V in between:
PPVVPVV
Upside-down is VVPPVPP, then backwards to PPVPPVV, so the next one is
PPVPPVVVPPVVPVV
Then:
PPVPPVVPPPVVPVVVPPVPPVVVPPVVPVV
Then:
PPVPPVVPPPVVPVVPPPVPPVVVPPVVPVVVPPVPPVVPPPVVPVVVPPVPPVVVPPVVPVV
etc.
Note that there will never be four P's or V's in a row - if there were, the path would cross itself.
What I got (while watching the video) is (i do not know wether it is right or not):
you take the first Iteration:
V
now you take the alternatig pattern PVPVP... (n+1 elements long were n represents the number of elements in the previous pattern) and merge them, so that you take the first of the alternating pattern, then the first of the previous pattern, then the second...) and you will get the same result.
***** I never thought of it that way before, but I think you're absolutely right!
+PhilBagels I learned about it from Vihart
+PhilBagels I remember you! :D
i had a similar approach with 0 and 1 instead of P and V. heres part of my java code i wrote to make the strings:
order = new String[lenght][2];
order[0][1] = "1"; //[x][1] is the "real" string
order[0][0] = "0"; //[x][0] is needed to build strings of higher complexity
public void makeString(int x) //only call with a for loop, singel calls might cause errors (x-1 might not be defined)
{
if(x>0)
{
order[x][0] = new String(order[x-1][1] + "0" + order[x-1][0]);
order[x][1] = new String(order[x-1][1] + "1" + order[x-1][0]);
}
}
You can see if you look at the "end" of one line segment, that the lines appear to rotate through 45 degrees each time the "paper is folded".
You can also see why the "sections" get smaller by the square root of 2. If each line segment has to rotate 45 degrees, it has to fit a straight line into a diagonal segment, which is equal to root 2 times the length of the "edge". So its better to think of the sections getting bigger by that amount as you come in towards the middle.
This was my introduction to the idea of fractals, way back in the day. Jurassic Park was one of the first "adult" books I read growing up.
I’m about 8 years late to this party, but to further go into how it’s used in the book, Ian Malcolm (played by a Jeff Goldblum in the movies) essentially used fractals and chaos theory to explain that a small issue is indicative of a much bigger issue. One tiny technical failure, like a small fractal pattern, can be used to predict a catastrophic failure, like that small pattern repeating itself and iterating on a large scale.
In Jurassic Park, of course, it’s that one small technical oversight by the people running the park shows that, inevitably, everything will come crashing down. And as we all know, that’s exactly what ends up happening. Interestingly, the book begins with a very small dinosaur: the Compy (roughly chicken-sized) having escaped its confines. Then much later, the larger dinosaurs also escape. Just like the iterations of the fractal curve. A small instance indicative of a bigger pattern.
Pretend you're me, and you're in math class...
Vi Hart
I love you
snakes are not stainless steel
I much prefer to pretend I'm in my infinite pink steel dragon dungeon
YES!
Great video!
Ayyy RedKB
Woah
I wonder...would it be possible to have a Dragon Curve that iterates in 3 dimensions?
Asdrubal Vect oooh
:o
So you should have a 4 dimensional hypersheet of paper
@@hamsterdam1942 A sheet of paper in the 4th dimension would just be 3-d.
yueshijoorya our 2D shapes on paper are made out of 1 dimensional lines which means in the fourth dimension the 3D shapes on 4D paper are made up of 2D lines
Oh, the dragon fractal! I love this one; I really think that it deserves more publicity, since it is, as the person in the video says, a beautifully complex design created from a simple set of rules (i.e. folding a sheet of paper).
Its appearance fee is massive!
That is absolutely beautiful...
Thank you so much for explaining that! It's always been one of my favorite books and I had never understood the illustrations
"The whole world knows about fractals..."
You reckon this guy spends much time outside of his mathematics department?
Didn't the whole world hear the Frozen song "Let it go"?
Yup, never heard of these til i started reading the book lol
Should probably have said "anyone with an education"
Wow I thought fractals were common knoledge. I guess that's what watching so many Numberphile videos does to you.
The reason the dragon curve is used in Jurassic Park is to represent Chaos Theory, imagine if you will that each fold is an impossible event. At the first iteration only a few impossible events (such as dinosaurs being created) have taken place to lead were you have gotten, while if you look at the sixth you will see that many impossible events have taken place to get to there. It is quite interesting if I do say so myself!
I'm so glad they did a video over this!!! Jurassic park was one of my favorite books as a kid and this fractal was my favorite fractal.
The dragon curve is also a space-filling curve. When iterated an infinite number of times those blocks in the center are completely filled. In other words, The curve actually touches every single point in a two-dimensional area on the inside of the curve. The length of the curve is also infinite. Fractals are awesome.
1. Saw the movie 2. Read the sequel 3. Read the first book 4. Saw the sequel. There's a fractal for ya.
Watched this video and instantly went and made a dragon curve tesselation. It's AWESOME.
great video, i used to make dragon curves when bored at school, but i like them more when junctions are not crosses, but when the edges are rounded so that the curve never touches itself or crosses itself. All generations have this property that the curve never touches or crosses itself, and drawing it out or generating it with that in mind, with rounded edges, REALLY makes it look awesome.
also, with some practise, due to shape recognition, humans can freely draw them in high iterations.
Was reading a book just because of this pictures. Never watched a film.
It's wonderful that this pictures where recreated in Russian translated version.
0:22 "just happens to be brown"
me from 2016 : yea OK.
ikr
And it's 3 blues.... And one Brown
life, uh, finds a way.
I love how they included Jurassic Park, which is how I learnt about this
I love that book, read when I was in sixth grade. One of my favorite books of all time.
SO COOL GUYS!! SO INTRESTING!!
And Ian Malcolm is a chaotician
Oh, man, I have loved the fractals in the Jurassic Park novel for years. I had no idea they had anything to do with folding paper. That is totally awesome.
In the Jurassic Park video game, the background of the menu was an animation of a dragon fractal growing. It was hypnotizing.
Not sure if this has been mentioned before but one can predict the pattern of peaks and valleys.
First it's easy to picture what the next iteration will look like (where here 'iteration' means the next fold, not the next picture in the book!) by looking at the current iteration, taking a copy of it, rotating it 90 degrees, then sticking the ends together. For example, after the first iteration, we have an L shape. If you rotate the L so you have something like _I then stick the ends together (with the rotated version on top) you'll get
_I
L
which is the "saucepan", second iteration.
Now, this shows that in the (n+1)th iteration, we get a duplicate of the nth iteration (but rotated), then the nth iteration. So, if you take the pattern for the nth iteration and rotate it, you actually get the BACKWARDS pattern, and you SWITCH all the valleys and peaks. So, if we have PVV, then the rotated pattern will be PPV.
To conclude, take the pattern for the nth iteration. Write it backwards and switch the letters. Then add in a V (because to get from the nth to the (n+1)th iteration you have to fold it inwards) then write down the pattern for the nth.
For example, the 2nd iteration has pattern PVV. Therefore the 3rd iteration will have pattern PPV, then V, then PVV, so we obtain PPVVPVV. For the fourth iteration:
Write the 3rd backwards: VVPVVPP
Switch letters: PPVPPVV
Add a V: PPVPPVVV
Concatenate with the original 3rd: PPVPPVVVPPVVPVV.
Repeat!
Sorry the explanation is poor. I thought it would be easier to explain than it actually was!
EDIT:: I remembered this TOTALLY incorrectly. The way I found it was quite insane but that wasn't my point. the point I was making was that I had come across this while playing with turtle graphics and named the source code file "dragon_pattern". That is what I found interesting, it is an actual known pattern named "Dragon", which is what I called it as well...What blows my mind is that I came across this pattern on my own while playing with turtle graphics by using an iterative function that begins with am axiom of left and right and iterates through the items appending to a list the opposite of each item, then iterates through the resultant list in the same manner then after x amount of iterations (not very many needed as it grows exponentially) I used the resultant list by mapping the left and rights to a turtle graphic that moves forward n pixels and makes a 90 degree turn in the desired direction.
Anyways!
What blows my mind. Is that I named it the dragon pattern, for obvious reasons, not knowing that it was an actual known pattern WITH THE SAME NAME!!!
I can't imagine how great you must have felt when you realized this was turning into something meaningful and with structure.
I think what you are describing in your comment is the Thue Morse sequence. What I understood was you started with
L R
and appended to it R L (opposite of these elements)
But that doesn't make the dragon curve (I've tried). The pattern I used to generate next iteration from a given one was inserting alternate L and R after each element in the list. (LR changes to L L R R). You could also do this starting with an R (LR changes to L R R L).
+Prateek Saini let me review the code. I know I have it somewhere on an old USB stick. This was years ago so I may be a little cloudy on the order in which i appended the items as I did come across a few meaningful patterns while fiddling about with turtle graphics.
This is going to be an exercise, going through piles of thumb drives to search for it.
+Miles Williams No need for that. I read some other comments and it turns out you also have to add an L before appending the interchanged Ls and Rs and you also have to iterate from the last element to the first. So, L becomes L L R which becomes LLR L LRR which becomes LLRLLRR L LLRRLRR and so on.
I'm sure I did it differently. I am curious now exactly how I achieved the same result so I will have to find the code and post what I find. The is no way I can let it pass without looking into it now, my curiosity has gotten the better of me.
Ohhhh it does look like Roshar.
So excited for Oathbringer!!
Anyone else has never seen Jurassic Park?
+KasabianFan44 -_- bruh...
Search for "Jurassic Park T Rex breakout" here on UA-cam and you will want to watch that movie.
Also keep in mind that that scene was recorded in 1993.
No.
me
I wouldn't exactly wear that as a badge of honour, the film is a genuine masterpiece.
Oh so that's what they are. Cool.
Yes please! I have been fascinated by fractals for a long time, especially the mathematical formulas. It would be really great if you would make videos explaining the formulas of famous fractals.
Numberphile in the day and vihart nights.....heaven!
lol Read the book friend. Its my favorite. I read it every couple of years.. :)
1. Now I know why I always get problems when re-folding some map!
2. The emerging patterns at the end very much remind me of some Julia-Sets or the "seahorse tails" of the Mandelbrot set. As you mention fractals - can you please explain what is the mathematical connection between folding a paper and these sets?
Fractals are a never-ending recursive pattern. And folding a piece of paper like that is the same thing (well, except that paper having a thickness and a limited surface area means it's not really endless). Imagine for a second if after folding the paper like this some 7 or 8 times, instead of completely unfolding it and then putting them back into 90' angles, if he instead undid each fold and "locked" it at 90', and then undid the next fold, over and over you'd see that the results would be the same as looking at it after each additional fold (ie undoing the first fold would just give you a valley, undoing the second fold, keeping everything at 90' would give you the peak-valley-valley, undoing the next would be the same as when he looked at 3 folds: peak-peak-valley-valley-peak-valley-valley, etc). In this way you can see how unfolding the paper is essentailly a fractal pattern, undoing each fold turns it into itself and its' reflection, which then unfolds into that pattern and its reflection, etc.
So you can think of folding a piece of paper as a real-world analogy for what this fractal is doing.
phiefer3 very well explained my friend :)
Thanks.
I found the dinosaur's contribution very enlightening. I'd like to see more videos featuring this dinosaur!
At first I think this is boring, but I was wrong in the end! Stunningly beautiful.
I read the book but I haven't seen the film.......... Am I normal??
Andrew Sutton As usual, the book was better.
PhilBagels
Not in this case.
For iteration N, duplicate iteration N-1, rotate it -90 degrees and attach it's end point to N-1's end point.
Simple stuff.
Now find the non-sequential formula.
When I read this book at age 10 I didn't really understand what these pages meant, only that the shape was getting more and more complex. I concluded at the time that it meant that something was changing, whether it be the dinosaurs themselves, the park going to into chaos, or something else. This video is very fascinating.
I always knew there had to be a pattern to regular paper folding, but never knew how to start to express it, or even how to look it up online. Cheers for showing that there is one!
Why is this stuff in Jurassic park, anyways?
That's what I wonder 😂😂😂😂
dinosaurs are not dragons
Nianyi Wang and nobody is saying they are
Yes, they are, *stumps feet on ground and throws a tantrum*
Awesome video, I love fractals, especially the mandelbrot set, but the little known ones are also very cool.
Oh my god, I used to do this in elementary school with the bit of paper left over from tearing your paper out of a perforated notebook. I'd fold it and then make it go into right angles. This kind of makes me happy that there's actually something behind it.
I read that 20 years ago when I was 6. I know this because it was the first book i checked out with my first personal library card. First adult novel, in fact. It's an amazing feeling to have the dragon curve finally explained to me! 20 years! You know what that's like? I'm so excited right now. I have to show somebody. You guys are THE shit!
Funny how he says "everyone knows about fractals but nobody knows about the dragon curve", and now the dragon curve is one of the most known fractals.
This is why I love Michael Crichton!
With the advent of the internet, my mind ceased to get blown as often as it used to be. Good job.
we're happy you found us too!
This was actually one of the first fractals i was introduced to. A way to cheat your way to more intereations with paper is to use multiple pieces pre folded and stick one end to the other.
dragon curve + paint = feel like an artist
THIS kind of thing is why I love Numberphile.
I am stunned! I have seen that! I saw it in a computer program (forgotten its name) in which you could watch beautiful pictures of different patterned curves, and I was amazed at how you could zoom in at one spot infinitely far and it kept producing the same patterns. I never knew where this pattern came from or what the mathematics behind it were. Thank you for brightening my world!
These are seriously some of the most well done videos on UA-cam
That last part took me by surprise... Much more than the dragon curve!
Yes, please, one or several videos about fractals in general would be awesome.
Wow. This video has to be my favorite numberphile.
The course covered (function) sequences and limits in ℝ, (uniform) continuity and derivatives in ℝ, (power) series in ℂ, (sequences, limits, functions and topology in) metric spaces and (improper) riemann integrals. Self-similarity was briefly touched in the chapter about completeness of metric spaces.
It was the main culling course of the first year.
The dragon curve is surely the most beautiful thing you can draw in LOGO.
This is my favorite fractal, surprised not many people have heard of it.
I love the dragon curve! When I read Jurassic Park about 20 years ago (I was 11), we were beginning to learn in school about fractals, and we all folded parts of a dragon curve out of paper and put them together. It was awesome. :)
this is pretty much a small demonstration of complexity in our universe. if you can create something so complex with simple iterations of the same shape, you can imagine how complex everything in the universe is and can be given time and energy.
Jurassic Park has to be one of my favorite book because of Ian Malcolm's character. I'm very glad you made this video. Plus fractals are cool.
Each iterations preserves blocks from earlier iterations.
This is sooo cool! I was higher than normal the other night and saw amazing fractals.
I watched the video [light at 100,000,000,000,000 times a second] that was amazing.
This is one of the coolest Numberphile vids I've seen
THAT. WAS. AWESOME!!! I had read the book a few years back and had always wondered what exactly those were. Awesome episode!!!
Jurassic Park, amazing book!
Amazing! Best Numberphile video in a while.
I think that this a lovely explanation of fractals. I did wonder at the time when I read the book.
The sequence of iterations goes that if you had the sequence pvv (p for peak and v for valley) you just take the reverse sequence (vvp), make every peak a valley and vi versa, add a valley in the end and then add the original sequence.
I've constructed an iterated function system for this in an exercise for "Analysis", but never saw this construction or even its name. Nice.
That's really interesting! So often there is order in what seems completely random and chaotic. It makes you think about the basic substance of reality which brings another movie to mind: the Matrix. I love it when videos give those small insights into things that lead to bigger questions to think about.
WOW that is crazy. I love this sort of thing.
That reminds me Greatly of The Mandelbrot set.
Its amazing
love listening to people like this talk. so intriguing and intellectual. very satisfying to listen to.
This is the sort of stuff I subscribed to numberfile for. Thankk you
All your movies are really great, but this one is particularly cool!
At 6:14 you see 4 dragon curves nested together to tile a surface. But when you nest them like that, they don't tile a surface *perfectly*. They never meet along an edge, but always at a vertex, so there's always a blank square between the colored squares of two adjacent dragon curves. A squiggly line of diagonally-connected blank squares separates each of the 4 dragon curves. In order to make the image shown at 6:14 you need to add in extra squares that aren't part of the original design.
First start with a cylinder with the radius larger than the height of the cylinder. Then start to shrink the radius. The volume of a napkin ring around the cylinder will always be the same, and the volume of one of the other two caps goes from larger than the napkin ring to smaller than it. Therefore, there is somewhere between where the volumes are equal.
I love your enthusiasm!
More chaos stuff like this please, it's really interesting.
This strikes me as having a better potential for creating music from math than many things that have been tried.
I have actually never seen Jurassic Park. I'm saving it for when originality is COMPLETELY and UTTERLY absent in movies.
The whole world knows about fractals - overstatement of the century :P
incredible book
I love the voiceis of the people that appears on Numberphile.
Each new iteration is the previous iteration rotated 180° (also known as mirroring the order and making valleys become peaks and vice verse), plus a valley in the middle, plus the original form of said previous iteration.
There are also video games based on Jurassic Park. In one of them there are computer screens on which you can see all kinds of fractal patterns in the background. I think the dragon curve is one of them.
There is also another pattern. I'm not sure if anyone has thought of this yet, but here it is: First, the number of peaks+valleys is 2x+1 where x=the # of peaks and valleys in the most previous sequence. We can divide the new, larger sequence int three parts: two are as long as the previous (so each is a little less than half the entire sequence. Separating these is a single fold. This middle single fold will always be a valley because the first fold was looked at as a valley.
MIND = BLOWN
Some other Forms of these Fractals have verry important uses in the Computer Sience. I like this one too, since it was one of our practices to Code this and some similar Fractals into a simple Painter.
I've seen this before, and it actually came up in a class once (or at least a variant of it). I took a class on wavelets last year, and at one point the professor was talking about how they had tiled the plane, and it wound up being the Twin Dragon Curve. It was neat to see that these fractals can still wind up in serious mathematics.