How to Code the Mandelbrot Set 2

The Mandelbrot Set is one of the most incredible things I've ever witnessed in life. This is part 2 of a 2 part series where I share it with Huiqi and you. I give a step by step tutorial for how to create a Mandelbrot Set visualization yourself, using nothing by HTML and JavaScript. Enjoy! If you'd like, the source code produced in the episode can be found at https://github.com/airportyh/thumbprint-of-god

Transcript

The following transcript was automatically generated by an algorithm.

00:01 : i tried all kinds of
00:03 : prints like i went to fedex
00:06 : and i used their
00:08 : higher quality printers
00:10 : some photo printing as well
00:13 : the larger paper
00:15 : oh i don't know if you can see the
00:17 : quality from here
00:19 : the print quality is really good that's
00:22 : what i was going for so so like i i'm
00:24 : making a website that lets people print
00:27 : out posters like these so that part has
00:30 : been figured out i think maybe you
00:32 : cannot have like a store that sells like
00:35 : t-shirts and
00:38 : so yeah last time we ended up with this
00:41 : uh where
00:42 : what you're seeing is you start with the
00:45 : point on your cursor
00:47 : we're canceling out the
00:49 : sort of vertical axis
00:51 : we're ignoring that for the moment so
00:53 : it's always plotted at the middle
00:54 : vertically but
00:56 : we start at the point that your pointer
00:59 : is pointing to
01:01 : and then
01:02 : use that as the c
01:04 : in this equation f of c
01:07 : of z
01:10 : is z squared plus c
01:13 : we start with
01:15 : z
01:15 : is equal to zero
01:18 : we get the calculation and then we take
01:20 : the result and feed it back into z and
01:23 : do it again and we do it iteratively and
01:26 : the idea is we want to figure out if it
01:29 : goes to infinity or not there's a
01:30 : theorem that tells us if at any point in
01:33 : time the output goes outside of this two
01:36 : by two circle then we know for a fact
01:40 : that it will go into infinity so anyway
01:42 : that's what's going on so this is each
01:44 : iteration this is iteration 0 this is
01:48 : iteration 1
01:50 : iteration 2 etc
01:53 : we put this label element here let's
01:56 : make use of it let me uh
02:01 : play block so it'll be placed nicer
02:06 : and i'll give you the background
02:08 : color
02:10 : of light yellow
02:12 : and then we will display
02:18 : t e c r
02:24 : and the ci
02:29 : on the label
02:31 : there it is it's kind of too long but
02:33 : i'm not gonna bother fixing it last time
02:35 : you asked me is it true that um if it's
02:38 : less than one then it'll go to zero
02:41 : right and i sort of wanna answer why
02:44 : that's not the case if you take c equals
02:47 : to zero point five for example
02:50 : so c is 0.5 0 squared is 0. initially we
02:55 : get
02:56 : just c which is 0.5 and we plug it back
02:59 : in
03:01 : this time we're going to do
03:03 : 0.5 times 0.5 for z squared which is
03:07 : going to be 0.25
03:10 : and that part gets smaller the problem
03:13 : is we have to add add it back to the
03:16 : original c oh okay i see that's going to
03:19 : make it bigger so
03:20 : so just by being smaller than one
03:24 : doesn't make it go to zero
03:27 : needs to be a really tiny number okay
03:31 : um for for it not to go to infinity one
03:34 : thing i want to do before we go to 2d
03:36 : really quick is because the mandelbrot
03:38 : sets definition is the set of complex
03:41 : number c for which if you do this
03:44 : iteration of this equation it will not
03:47 : eventually go to infinity the c is my
03:50 : pointer basically c is where my pointer
03:53 : is what we could do to visualize the
03:55 : mandelbrot set is to color the point
03:58 : black if it is in the set it doesn't go
04:00 : to infinity and to not color it if it
04:03 : does go to infinity so leave it be white
04:06 : and that's how you get the shape of the
04:08 : mandelbrot set
04:10 : maybe i can just do that for the real
04:13 : numbers to show you what that might look
04:15 : like so instead of generating these
04:17 : color bubbles i'm just going to say
04:19 : after going through these 100 iterations
04:23 : is the number
04:24 : greater than 2 because if at any point
04:28 : in time you go outside of this circle
04:31 : then it will for sure go to infinity if
04:33 : it is greater than two then we're gonna
04:35 : plot the point
04:38 : uh otherwise we will not plot the point
04:41 : so i move that code that plots the point
04:43 : in inside this if statement oh so one
04:46 : question so after you look through it
04:49 : for a hundred times then you plot the
04:52 : point in the original
04:55 : yeah
04:56 : yeah in the original place but we we
04:58 : don't plot the point that's like going
05:00 : towards the infinity we plot the point
05:02 : from the original place wait a minute
05:04 : let me see if this works
05:09 : oh i should say the reverse if it's
05:11 : still inside two i want it to be black
05:14 : it didn't go to infinity at least for
05:16 : the number of times we've tried if i
05:18 : just don't clear the canvas every time
05:22 : then i leave a trail
05:25 : for the places on the real number line
05:29 : that are inside the mandelbrot set
05:31 : starting negative 2 which is all the way
05:33 : at the
05:34 : left side of the canvas to this point
05:37 : here which is about
05:39 : 0.264 more than that it goes to infinity
05:43 : so basically in order to pin this black
05:46 : line you have to go through
05:49 : uh the whole
05:50 : horizontal boundary in your canvas in
05:53 : order to figure out which one goes to
05:56 : infinity which one doesn't exactly now
05:59 : i'm going to revert back and now we're
06:01 : going to do two dimensions now we're
06:04 : going to do imaginary dimension as well
06:06 : how do we do that um i want to give you
06:08 : an example of how you would do it by
06:11 : hand let's say the point 1 1 1 1 means c
06:16 : the constant c is equal to 1
06:19 : plus
06:20 : i i is just 1 times i if we iterated
06:23 : this equation which is
06:26 : that so f of
06:28 : 1 plus i
06:31 : starting at 0.
06:34 : is going to be zero squared which is
06:36 : still zero
06:38 : like the zeroth one you always ignore
06:40 : the z squared because it's going to be
06:42 : zero
06:43 : so that the first one is just going to
06:45 : be one
06:46 : plus
06:46 : i
06:48 : and then we feed that back into here
06:58 : so this is going to be 1 plus i
07:01 : squared
07:02 : plus 1 plus i
07:05 : right and how do you calculate that well
07:08 : you eat this part
07:12 : we do the
07:13 : we distribute
07:15 : the
07:16 : multiplication if you pair this one with
07:19 : this one
07:20 : then you pair this one with the i which
07:23 : is plus i and then you pair this i with
07:26 : this one
07:28 : which is another i
07:30 : and then the two eyes get paired
07:32 : together
07:34 : which is i squared
07:36 : what's the value of i squared oh it's
07:39 : negative one yeah exactly it's negative
07:42 : one
07:43 : because i is the square root of negative
07:46 : one
07:47 : so we end up with two
07:49 : i one
07:51 : plus two i
07:52 : minus one oh the ones cancel out
07:56 : so we just get 2i at the end
08:00 : and then this whole thing is 2i and
08:02 : we're going to add 1 plus i
08:05 : it's going to be uh
08:07 : 1
08:09 : plus 3i
08:11 : that's the answer
08:12 : for for this one and 1 plus 3i if we
08:16 : were to plot it on the canvas is
08:18 : basically the dot
08:21 : 1 3. we started with 1 1 and then we get
08:26 : and we ended up with one three it's all
08:29 : the way up there somewhere
08:31 : uh
08:33 : where this manual is
08:35 : that kind of took a while so i won't
08:37 : keep going but but if you kept going i
08:40 : if you did it one more time you will end
08:42 : up at negative seven seven which is kind
08:44 : of crazy because that's all the way over
08:47 : there so it started here and it went up
08:50 : there and then
08:52 : flew all the way back there
08:54 : that's kind of great but let's get the
08:56 : general equation of this thing let's say
08:59 : z was a plus b we need to be able to
09:03 : perform
09:04 : z squared how do we do z squared well z
09:07 : squared is going to be a plus b
09:11 : times i
09:13 : squared
09:15 : which is going to be
09:20 : we're going to
09:24 : break those terms out and then we're
09:26 : gonna do the distribution thing a gets
09:29 : paired with a and then a gets paired
09:32 : with
09:33 : bi
09:34 : and then this gets paired with a
09:38 : and then the two bis get paired with
09:41 : each other so it's gonna be b squared
09:44 : and then i squared
09:46 : these two guys become two times a
09:50 : b i
09:52 : and then
09:53 : this guy just becomes negative
09:56 : b squared so the end result is this
10:01 : and if you take a look at what the real
10:04 : part is and what the imaginary part is
10:07 : um
10:08 : the real part is these two combined
10:11 : right and that's the imaginary part
10:14 : so
10:15 : we call it maybe the
10:17 : new a
10:18 : a prime or something a prime is going to
10:21 : be
10:22 : a squared minus b squared
10:25 : and then the b prime the new the new
10:28 : sort of imaginary number
10:30 : is going to be 2 times a
10:33 : b and then we don't write the i because
10:36 : we just want the b part so so actually
10:38 : that's the equation of of of just
10:40 : squaring it
10:42 : and then and then we still want to add
10:44 : the c's real part
10:47 : and then add c's i part to complete this
10:50 : equation oh
10:52 : where the c part i kind of get lost oh
10:55 : okay so so before adding the c we're
10:58 : just doing z squared
11:00 : oh okay
11:01 : assuming z was a plus b i
11:05 : oh okay so this is the answer of the
11:08 : sort of the z squared
11:10 : right but then because the equation
11:12 : wants us to add c
11:14 : so we tag on the c part the real real
11:17 : part of the c goes into a prime which is
11:21 : the real part of the original number
11:24 : and then we add the imaginary part of
11:26 : the c as well
11:30 : so these two equations are what we're
11:32 : going to use
11:34 : in our iteration now
11:37 : and i'll show you that so now before we
11:39 : were just ignoring the imaginary part
11:41 : we're going to bring it back
11:43 : and before we were just using the
11:46 : half of the height now we're actually
11:48 : going to use the coordinate of the y
11:51 : from the mouse and that's going to go in
11:53 : here so c should have a correct value
11:57 : based on this screen to world
11:59 : function and here now we're going to
12:02 : have two equations
12:04 : we're going to say this one is like the
12:07 : a prime
12:08 : and this one is like the b prime
12:10 : actually
12:12 : we need to be careful
12:13 : and have a separate one
12:17 : have a separate one because we don't
12:19 : want to override the original one until
12:21 : both of the calculations are done i have
12:24 : made that mistake many times so you have
12:26 : to cache them before
12:28 : overwriting
12:29 : the current variable
12:31 : [Music]
12:32 : so this should be
12:35 : a squared minus b squared which would be
12:38 : same as
12:39 : zr
12:40 : squared
12:42 : minus zi squared
12:46 : so
12:47 : and then plus the cr
12:49 : that's what i had over here and then for
12:52 : the the b prime which in the code is zi
12:57 : is 2 times
12:59 : a times b
13:01 : and then plus c i which is just 2 times
13:04 : dr times zi
13:07 : plus ci okay
13:09 : see if that works
13:13 : oh no that did not work oh i know why i
13:16 : didn't swap into zi
13:18 : to be plotted we were ignoring it before
13:22 : and now we get this
13:24 : hey
13:25 : oh let me see this swirl
13:28 : yeah the swirls are pretty amazing uh if
13:31 : i stick to the
13:33 : real number line like i just put my
13:36 : pointer on the center you still get what
13:38 : you saw before
13:40 : this is this is kind of
13:43 : uh we can still if i align it correctly
13:46 : it'll still all line up as long as my
13:50 : mouse is exactly on the
13:53 : center
13:54 : but the moment you move a little bit
13:56 : from the center
14:00 : then crazy stuff can happen
14:02 : um but also
14:05 : depend on where you
14:07 : move your mouse
14:09 : interesting
14:10 : patterns
14:12 : can form like sometimes you have this it
14:15 : goes back to the same five spots
14:17 : phenomenon sometimes two spots sometimes
14:20 : it's four spots
14:22 : but then in between oh look at that star
14:26 : um this is like it's at one point it's
14:29 : going back to those same spots but
14:33 : sort of around that point
14:35 : you can sort of see it's like converging
14:38 : to those spots and then it's like
14:40 : drawing these streaks or spirals around
14:43 : those spots
14:46 : this this one looks like there's three
14:48 : spots
14:50 : and then there's three spirals coming
14:52 : out of three spots and that i mean just
14:55 : looking at this
14:57 : is fascinating like i could play with
14:59 : this for a long time here
15:01 : oh i look at that one
15:05 : some of the times you can see when it
15:07 : goes like this you know it went to
15:10 : infinity when it goes crazy you know
15:12 : it's going to infinity
15:14 : but when there's this sort of pattern
15:17 : it's probably staying in
15:19 : it it's like sort of bouncing back and
15:21 : forth within this restricted area and
15:24 : can't get out it feels like and and
15:27 : that's those are the points where it's
15:29 : within the mandelbrot set it's not going
15:31 : to infinity here we're inside the set
15:35 : and then
15:37 : and we're still inside the set we're
15:39 : still inside the set
15:41 : you can sort of feel out
15:43 : hey in this
15:44 : spot it's in the set and then here boom
15:47 : it went off it's not in the set anymore
15:49 : and then it's like oh okay you can kind
15:52 : of feel out which areas are in the set
15:55 : and which areas are not in the set but
15:57 : you could also just plot a dot there
16:00 : black
16:01 : if it's in the set so let's do that
16:02 : again
16:06 : for
16:06 : the for the complex numbers now this
16:09 : time
16:10 : i actually have to calculate the
16:12 : distance from the center because there's
16:14 : two coordinates now
16:17 : um to calculate the distance from the
16:19 : center we're gonna add their squares and
16:21 : then do the square root of that
16:24 : number
16:25 : so the distance from the center is going
16:27 : to be
16:29 : zr times zr plus zi times zi
16:33 : so squaring both of them and then take
16:36 : the square root of that that gives us
16:38 : the distance to the center
16:40 : and i'm going to say if the distance to
16:41 : the center is less than two
16:45 : i'm gonna paint it black
16:47 : okay
16:48 : i'm not going to clear the canvas every
16:51 : time so we can kind of see the residue
16:55 : and i can
16:57 : i can finger paint
17:02 : the great reveal of the mandelbrot set
17:07 : it looks a little bit like a beetle yeah
17:10 : right a bug of some sort but yeah
17:13 : obviously i mean this is a too tedious
17:15 : to do so what you might want to do is
17:18 : just
17:19 : extract this code out
17:21 : and just say uh
17:23 : plot inset maybe
17:34 : move it out and then
17:36 : double check it still works and then we
17:38 : can
17:39 : loop through all pixels on the canvas
18:01 : and let it do it automatically
18:07 : there we go
18:08 : okay
18:09 : um
18:10 : that still looks like a kids painting
18:13 : when we plot the dot
18:15 : we can use a smaller radius because we
18:18 : were using five as the radius now let's
18:21 : just use uh one as the radius
18:24 : and maybe we'll get a finer picture
18:27 : that's what the mandelbrot set looks
18:29 : like why those are there are dots like
18:32 : separated from the main image oh yeah
18:35 : these little dots um there's actually
18:37 : points here that are still inside the
18:39 : set even though it's separated from the
18:42 : main body it might be connected actually
18:45 : i'm not totally sure there's there are
18:48 : actually these
18:49 : antennas these very thin antennas that
18:52 : that come out from this bulb here
18:55 : and then connect with these little dots
18:57 : and those dots if you zoom into it it
18:59 : also looks like the speed holding
19:03 : smaller versions of the mandelbrot set
19:06 : now let's talk about coloring though so
19:07 : you asked last time what do the colors
19:10 : represent um there's multiple ways to
19:12 : color the mandelbrot set but basically
19:15 : it has to do with how many times did you
19:17 : have to iterate before it went outside
19:21 : that circle of that two radius circle
19:23 : for example
19:25 : if i only did one iteration
19:27 : instead of 100
19:29 : and i rerun this algorithm
19:32 : we would just get a black circle
19:35 : okay if i did two we get this little
19:37 : oval and then
19:40 : three it looks like a pair
19:42 : um
19:43 : this one looks like a stingray
19:45 : um
19:47 : this looks even more like a stingray so
19:49 : so the more iterations you do the more
19:52 : it resembles the actual mandelbrot set
19:54 : one simple way to color it that also
19:57 : gives really nice looking results
20:00 : is simply color each one of these case
20:04 : with a different color
20:08 : i can
20:09 : move this distance check
20:11 : inside of the loop
20:13 : and say if you already went outside side
20:16 : of the circle i can just come out of the
20:19 : loop right away first of all
20:22 : and then i'm gonna color you with this
20:24 : current k
20:27 : so it each each layer will be colored a
20:29 : different color i'm not even gonna use
20:32 : this i'm just going to calculate a color
20:36 : using a color from my color palette
20:39 : based on this k
20:41 : and then i'm going to
20:43 : just uh
20:45 : set the color to that color and then i'm
20:48 : going to plot the point
20:51 : the original xy not not the points that
20:54 : are bouncing around
20:56 : um
20:57 : one radius
21:00 : so if it if it sort of went outside the
21:03 : circle we're gonna plot that color and
21:05 : be done and skip this part but it if it
21:08 : didn't
21:09 : go after all of these iterations it
21:11 : still didn't go outside the circle it'll
21:14 : come down here and color the point black
21:17 : at that point
21:18 : um i'm not gonna even bother checking
21:21 : anymore here because
21:23 : if if it was outside the circle it would
21:26 : have
21:26 : returned at this point let's see if that
21:29 : works yeah
21:31 : so
21:32 : yeah so now i have only done five
21:35 : iterations but uh we pump that up to 100
21:38 : iterations
21:41 : then we get that
21:43 : okay
21:45 : yep so so that's a very simple coloring
21:48 : method that also looks really good
21:50 : nice the last thing that i wanted to
21:53 : show you is how can you zoom in how can
21:56 : you pick like a coordinate and then zoom
21:59 : in
22:01 : i am going to
22:02 : i'm gonna bring back this code here
22:12 : so i brought back the code that will
22:15 : show you the coordinate um of the mouse
22:18 : but in the world coordinate for example
22:21 : i know zero zero is in the belly of the
22:24 : beetle somewhere here but this allows us
22:27 : to read out
22:29 : the coordinates so we can introduce a
22:32 : origin we can say i want the canvas to
22:35 : be centered on a certain point so
22:39 : if we had that we would be able to shift
22:42 : the image up and down or left and right
22:45 : right
22:46 : right let's do that so i'm going to say
22:48 : there's going to be a origin
22:50 : in the real direction
22:52 : and the origin
22:54 : in the eye direction let's say go into
22:57 : this antenna
22:59 : looking thing right
23:00 : i'm going to zoom in there let's see
23:03 : we already have a scale factor which
23:05 : will control the zoom in by the way like
23:08 : if i do that
23:09 : we zoomed in so we already can control
23:12 : this scale factor which will allow it to
23:14 : zoom in
23:15 : but but having the origin will allow us
23:18 : to shift
23:20 : up and down and zoom into the exact
23:22 : point that we want to zoom in to
23:24 : so i'm going to say hey i want to zoom
23:26 : into this coordinate here which is
23:28 : negative 0.112
23:32 : and 0.928
23:37 : i want to zoom in here how do i make it
23:39 : do that uh well all we have to do is
23:41 : change these uh coordinate conversion
23:44 : functions over here currently our origin
23:47 : is effectively zero zero right
23:50 : like let's say if x and y were 250 250
23:55 : we would end up with zero zero here
23:58 : but i don't want the center to be zero
24:00 : zero i want the center to be
24:02 : my origin numbers so i just add it to
24:05 : here my origin r
24:08 : and my origin
24:11 : i i just add them here
24:15 : and the inverse is true here i want to
24:18 : subtract origin r from here before the
24:22 : calculation and i want to subtract the
24:24 : origin i from here does that make sense
24:26 : yep okay now let's see if we can center
24:30 : it on that antenna
24:34 : yeah
24:35 : we're centered on the antenna now and
24:37 : now if we want to zoom in we just uh
24:40 : make this scale factor
24:43 : smaller and we're going to zoom in to it
24:46 : yep there it is
24:48 : we're gonna zoom in much more
24:51 : one over three thousand
24:54 : and we really get to see the details of
24:56 : the antenna
24:58 : yeah there's the antenna you can see
25:00 : these tiny little mandelbrots these
25:03 : beetle looking things they are
25:06 : everywhere if you zoom into the
25:08 : mandelbrot set so we'll take one of them
25:10 : and try to center on it so these are the
25:13 : separated dots
25:15 : yes exactly these are the separated dots
25:18 : that's yeah
25:19 : and uh they're disconnected
25:22 : from the main body by these little thin
25:25 : antennas so let's try zooming in
25:28 : 0.1276 repeating
25:33 : six a lot of sixes okay
25:36 : and then 0.987
25:40 : and then three repeating
25:42 : [Music]
25:48 : i think it's this little one there's a
25:49 : bigger one over here but i think we
25:51 : centered on this small one but then we
25:53 : can zoom into it even further
25:56 : okay
25:59 : yep that's our little beetle and now we
26:01 : can zoom into it even more
26:04 : if we want
26:06 : yeah
26:08 : that's our little guy
26:09 : so yeah that's how you generate
26:11 : mandelbrot set images and the
26:14 : really amazing thing about this
26:16 : is this image is effectively infinite
26:21 : theoretically you can zoom into it
26:24 : infinitely and there's infinite detail
26:27 : inside of it i'll show you the project
26:30 : that i've been working on which has a
26:32 : better zoomer we can zoom back into the
26:35 : little beetle
26:36 : that we were looking at which was
26:39 : probably this one over here
26:42 : nice
26:43 : of course
26:45 : yeah it's
26:46 : this this interface is like google earth
26:49 : and
26:50 : and it progressively gets more and more
26:52 : detail and at this point like it has a
26:56 : lot of detail but the thing about it is
26:59 : you can continue to zoom in even even
27:01 : when you think you've seen
27:03 : so at the end
27:04 : but no it's not the end you can keep
27:07 : zooming in
27:08 : and then and then he's like
27:10 : oh look there's another even tinier
27:13 : beetle
27:14 : on the inside
27:15 : then you can zoom in more
27:17 : and then and then like no there's still
27:20 : no end you can even zoom in even more
27:23 : there's even like what is what is this
27:26 : little coral looking thing
27:30 : and then and then you can even continue
27:32 : zooming in and find the detail uh
27:35 : theoretically
27:36 : like this image is
27:39 : infinite has infinite resolution
27:42 : as in you can continue zooming in and
27:44 : there's still more details at a finer
27:47 : level
27:48 : but we are limited by the computer's
27:51 : precision and the computer numbers can
27:54 : only have remember so many digits per
27:57 : number
27:58 : right and we're gonna run out of those
28:00 : and at that point things do get
28:03 : pixelated now it is possible still to
28:06 : implement
28:07 : unlimited precision numbers to get
28:11 : around this problem
28:12 : so that you can still calculate
28:15 : the image even at this zoom level but uh
28:18 : that is gonna make everything extremely
28:21 : slow and i i did not want to do that for
28:24 : for this zooming user interface
28:27 : okay but there are people who
28:29 : work on this as their hobby and what
28:31 : they do is
28:33 : they just run a job to have the computer
28:36 : calculate you know the image at a very
28:38 : very deep level
28:40 : the job is just gonna run for a long
28:42 : time
28:46 : it's funny that like this like these
28:48 : shapes are like similar but not exactly
28:51 : the same
28:52 : exactly with many things that you see in
28:55 : life like you're looking for the pattern
28:57 : and you're just want to generalize and
28:59 : you're like yeah it's just the same kind
29:01 : of thing i've seen that before
29:03 : and it is true that there's a lot of
29:06 : repeated patterns right
29:09 : every pattern every pattern is
29:12 : like has a variation
29:15 : on top of the last version of it that
29:18 : you see
29:19 : and there are infinite variations even
29:22 : though they look similar they're a
29:24 : little bit different from the next one
29:26 : in in this mandelbrot set which is just
29:29 : really crazy i i don't even know how it
29:32 : works like i can't really explain any of
29:35 : this stuff
29:37 : you saw the code that we wrote we didn't
29:39 : write that much code 88 lines of code
29:42 : yeah but like i feel like my mind can't
29:45 : like
29:46 : predict like how this will go in terms
29:49 : of like shaping yeah it's completely
29:51 : unpredictable yes like usually you think
29:54 : oh i'm gonna write an algorithm and it's
29:57 : gonna give me a
29:59 : regular
30:00 : shape like a circle or a star or
30:02 : something like that
30:04 : but no you just have no idea the
30:06 : mandelbrot set is one of the foundations
30:09 : of chaos theory and i'm not gonna do it
30:12 : justice i'm sure so summarize chaos
30:14 : theory what chaos theory is
30:17 : a very small number of rules can give
30:20 : rise to
30:22 : chaos that it's very very hard to
30:25 : predict like i haven't seen this pattern
30:27 : before actually this is today is the
30:30 : first time i it looks kind of like
30:32 : broccoli and it's like the meat of a
30:35 : tree