Here’s a more thorough explanation of what’s going on: both the Julia Sets and the Mandelbrot set are defined by the iterative formula z->z^2+c and the points where it stays stable (in other words, find your z and c, start with z, repeatedly square and add z, and see if the result goes to infinity or stays in a stable area. For example, if z is 0 and c is -1, you get 0, -1, 0, -1, 0, -1, etc, which is stable. If they are 0 and 1, however, you get 0, 1, 2, 5, 26, 677, etc, which is not stable.). The Julia sets have z be the point on the plane of the Julia set and c be the constant starting seed; the Mandelbrot has z be 0 and c be the point on the plane. As such, the iterative calculations at any point p of the Mandelbrot set are the same as those of the Julia set with seed p as it’s center (both have z=0 and c=p). Because of how Julia sets work, asking if they contain the point 0 is equivalent to asking if they are connected; due to the strictly self-similar nature of Julia sets, if such a set splits into pieces during iteration, it must continue to split into more and more pieces, leaving the final set as a bunch of disconnected points, known as Fatou dust. If they do contain the point 0, they stay as one solid piece. As such, at any point that the Mandelbrot set contains (which is colored black), the corresponding Julia set is solid (and thus is also black); and point not in the Mandelbrot set has a disconnected Julia set, which is not black.
Very clever and ginous way to visualize the difference between the two sets or rather the two names . Thanks much i would be subscribing and liking your channel .
Eh, it's actually pretty obvious if you know how the Julia Set and Mandelbrot Set are related. There actually isn't just one Mandelbrot Set - just like the Julia Set, it needs a seed value, which is normally 0+0i. Just as a Julia Set with a given seed (say, 3+2i) corresponds to that point on "the" Mandelbrot Set, there is a similar reverse relation. The Mandelbrot Set with seed 0+0i corresponds to the points *at* 0+0i on the Julia Sets. They're complimentary - the seed of one set is a point on the other, and vice versa. So if you take the Julia Sets corresponding to points on the Mandelbrot Set, centered at 0+0i, and zoom in, those squares will all be a solid color, that of 0+0i. And because of the relation above, the point 0+0i on the Julia Sets corresponds to the value at the Julia seed on the Mandelbrot Set with seed 0+0i.
z=0 is known as the critical point, for the Mandelbrot set. It is the solution to 2z=0, the left hand side of which comes from differentiating z squared+c.
i think one interesting implication of thinking in terms of fractal is that on the surface, somethin may seem simple, yet as you become more attentive and delve deeper into the simple looking structure, you find it more complex. its kinda counter intuitive to how we think. usually in nature, we break down matter and expect to find more fundamental, and elementary particles. what if perhaps it actually becomes more complex, like a fractal.
sowing the seeds of ... Julia... and harvest the Mandelbrot ;) nice illustration. Also, it is quite fascinating to see how the Mandelbrot details often mimic the Julia patterns.
Dude, yes. That's awesome. I want more videos on how these things are made and how they relate to each other, because this is way more interesting than watching zooms over and over again. :)
EconomicUplift Close, 4dimensions. The Mandelbrot set is the point(0,0) of every Julia set. The combined function z squared plus c takes 2 complex variables so takes 4dimensions to fully represent all outcomes
For the Mandelbrot set, yes, although it's usually rendered "z^2 +c". You generate your 2D image using the real and imaginary parts of the complex number c as the x and y values (one reason why we use "z" in the definition of the set). To make a Julia set, you pick a single value for "c" (which is why each Julia set can be mapped onto the Mandelbrot set), and then use different complex numbers as your starting value for z. Plotting your result for that second complex number gives you the image.
Because if you take the Julia sets with the function z^2 + c and plot them all, ones with a certain property ( forget the technical name) will "fit" inside the Mandlebrodt set, because it's based on the function z^2 + c as well.
Both fractals are made using the same equation: z' = z^2 + c. The Mandelbrot set sets c (what's known as the parameter plane) to the coordinates in the image, setting the first z = 0. Julia sets set z (the dynamical plane) to the coordinates, setting c to anything you like. A Julia set captures the behavior of iteration at a particular point (or even any function; you can have a Julia set of something other than the Mandelbrot set). So by definition, the center of a Julia set for a given c is equivalent to the Mandelbrot set at the same c, because they start with the same z and c. If it's part of the (filled) Julia set, it's part of the Mandelbrot set, and vice versa. You can imagine an image in two complex dimensions or four real ones encompassing everything, although that's difficult for us to manage.
actually yeah, it kinda does. julia and another bloke who's name escapes me now did a lot of work in the field. they were in europe, and didn't have access to the processing power mandelbrot did, but they did the math that mandelbrot's work is derivative of.
The beginning explanation is so mathematically inaccurate. Its not “one real number and one imaginary number”. A complex number is an imaginary number plus a real number. For example, 3i+2. It’s one number. The complex plane is not meant to represent two numbers, but one complex number.
This is actually insanely obvious and doesn't say anything special at all. The Julia set is just roughly mirroring the Mandelbrot set, so if you put a bunch of Julia sets at small intervals and then blur them it will obviously look like the Mandelbrot set...
Here’s a more thorough explanation of what’s going on: both the Julia Sets and the Mandelbrot set are defined by the iterative formula z->z^2+c and the points where it stays stable (in other words, find your z and c, start with z, repeatedly square and add z, and see if the result goes to infinity or stays in a stable area. For example, if z is 0 and c is -1, you get 0, -1, 0, -1, 0, -1, etc, which is stable. If they are 0 and 1, however, you get 0, 1, 2, 5, 26, 677, etc, which is not stable.).
The Julia sets have z be the point on the plane of the Julia set and c be the constant starting seed; the Mandelbrot has z be 0 and c be the point on the plane. As such, the iterative calculations at any point p of the Mandelbrot set are the same as those of the Julia set with seed p as it’s center (both have z=0 and c=p). Because of how Julia sets work, asking if they contain the point 0 is equivalent to asking if they are connected; due to the strictly self-similar nature of Julia sets, if such a set splits into pieces during iteration, it must continue to split into more and more pieces, leaving the final set as a bunch of disconnected points, known as Fatou dust. If they do contain the point 0, they stay as one solid piece.
As such, at any point that the Mandelbrot set contains (which is colored black), the corresponding Julia set is solid (and thus is also black); and point not in the Mandelbrot set has a disconnected Julia set, which is not black.
Team Fresh you have utterly blown my mind again! Love it!
If anyone is wondering here is the code used for the Julia set in the beginning (-1.25+0i)
Very clever and ginous way to visualize the difference between the two sets or rather the two names . Thanks much i would be subscribing and liking your channel .
very good visualization of the connection between Julia Sets and Mandelbrot
Eh, it's actually pretty obvious if you know how the Julia Set and Mandelbrot Set are related. There actually isn't just one Mandelbrot Set - just like the Julia Set, it needs a seed value, which is normally 0+0i. Just as a Julia Set with a given seed (say, 3+2i) corresponds to that point on "the" Mandelbrot Set, there is a similar reverse relation. The Mandelbrot Set with seed 0+0i corresponds to the points *at* 0+0i on the Julia Sets. They're complimentary - the seed of one set is a point on the other, and vice versa.
So if you take the Julia Sets corresponding to points on the Mandelbrot Set, centered at 0+0i, and zoom in, those squares will all be a solid color, that of 0+0i. And because of the relation above, the point 0+0i on the Julia Sets corresponds to the value at the Julia seed on the Mandelbrot Set with seed 0+0i.
z=0 is known as the critical point, for the Mandelbrot set. It is the solution to 2z=0, the left hand side of which comes from differentiating z squared+c.
ẂHAT??? NO!!! YOU ARE SHITTING ME! WHAT THE FUCK??? HOLY CRAP!
lol
Thank you fractal, very cool!
i think one interesting implication of thinking in terms of fractal is that on the surface, somethin may seem simple, yet as you become more attentive and delve deeper into the simple looking structure, you find it more complex. its kinda counter intuitive to how we think. usually in nature, we break down matter and expect to find more fundamental, and elementary particles. what if perhaps it actually becomes more complex, like a fractal.
i think it is
Hey Fractal, Whats The Song?, I Knew The Name But I Forgot, Can You Help Me?
sowing the seeds of ... Julia... and harvest the Mandelbrot ;) nice illustration. Also, it is quite fascinating to see how the Mandelbrot details often mimic the Julia patterns.
Dude, yes. That's awesome. I want more videos on how these things are made and how they relate to each other, because this is way more interesting than watching zooms over and over again. :)
No isnt real
@DMdarkmatter formula is called "ManyJulia" and it can be found in dmj.ufm (public formulas)
How many Julia sets does it to make a Mandelbrot?
Two, one to call the maid and the other to fix the drinks.
@esgimasd sorry forgot to say! I removed it, because the link at the end of the video is live now :)
1:23 when the point is zero the Julia set is a perfect circle
Awesome! Did you used Ultra fractal? What is the formula which do this?
and now my mind is officially blown....
isnt a mandelbrot a subset of julia set?
How about f(x)=x² +c²-c?
That's cool!
Can you make a video explaining how these things are colored?
0:26 - Actually because of something I learned of the Julia set, there is an infinite amount of Mandelbrot sets!
What's the magnification link?
Looks like a heart
I believe you are correct, that main body, and the largest black section is in the shape of a cardioid, a true mathematical heart.
ScienceBang It's a CARDIOid, even though it looks more like some fruit, or a bum XD
that's exactly why it's called a cardioid
So amazing 😀
Its as if the Julia set is vision of the mandelbrot from the 3rd dimension
EconomicUplift Close, 4dimensions. The Mandelbrot set is the point(0,0) of every Julia set. The combined function z squared plus c takes 2 complex variables so takes 4dimensions to fully represent all outcomes
then, what is a julia set made up of?
Something very complex
@fractalzooms No matter. I've seen the other vid too now haha.
bro why is the link in the description a gambling site
what formula did you use to make the julia sets? i assume it was x^2 + c where c is the complex number you where currently on.
For the Mandelbrot set, yes, although it's usually rendered "z^2 +c". You generate your 2D image using the real and imaginary parts of the complex number c as the x and y values (one reason why we use "z" in the definition of the set). To make a Julia set, you pick a single value for "c" (which is why each Julia set can be mapped onto the Mandelbrot set), and then use different complex numbers as your starting value for z. Plotting your result for that second complex number gives you the image.
Me gustan los fractale
I love how maths works =D
fan-fucking-tastic.
For those into ZF set theory, Both Mandelbrot and Julia sets have the Cardinality of C
well **** me, thats incredible!
why does this work?
Because if you take the Julia sets with the function z^2 + c and plot them all, ones with a certain property ( forget the technical name) will "fit" inside the Mandlebrodt set, because it's based on the function z^2 + c as well.
Ok....
But I think I kinda get it
Both fractals are made using the same equation: z' = z^2 + c. The Mandelbrot set sets c (what's known as the parameter plane) to the coordinates in the image, setting the first z = 0. Julia sets set z (the dynamical plane) to the coordinates, setting c to anything you like. A Julia set captures the behavior of iteration at a particular point (or even any function; you can have a Julia set of something other than the Mandelbrot set). So by definition, the center of a Julia set for a given c is equivalent to the Mandelbrot set at the same c, because they start with the same z and c. If it's part of the (filled) Julia set, it's part of the Mandelbrot set, and vice versa. You can imagine an image in two complex dimensions or four real ones encompassing everything, although that's difficult for us to manage.
@esgimasd Did I remove my comment? I can't remember lol Anyway, thanks for the vids!
wow...very interesting. math is mysterious!
coffeebreakist well we have a 3 pound brain trying to understand an infinite God
the definition of the mandelbrot set is a "map" of "where" the julia sets with a connected area are.
2:42 pixelated mandelbrot
That's a silly question. The answer is obviously infinite.
2:42 mandelbrot pixels
@nhmllr725 it will happen
Pixelated Mandelbrot, eh?
This means that the Mandelbrot set is based on the Julia set.
not really, they just have the same function z = z^2 + c
actually yeah, it kinda does. julia and another bloke who's name escapes me now did a lot of work in the field. they were in europe, and didn't have access to the processing power mandelbrot did, but they did the math that mandelbrot's work is derivative of.
Gaston julian i believe
the voice that came in at 1:34 scared the living hell out of me, I thought someone was calling my name
I hear no voice
the think that goes ooh, ooh ooh, ooh, ooh ooh
The beginning explanation is so mathematically inaccurate. Its not “one real number and one imaginary number”. A complex number is an imaginary number plus a real number. For example, 3i+2. It’s one number. The complex plane is not meant to represent two numbers, but one complex number.
This is actually insanely obvious and doesn't say anything special at all. The Julia set is just roughly mirroring the Mandelbrot set, so if you put a bunch of Julia sets at small intervals and then blur them it will obviously look like the Mandelbrot set...