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The Julia Sets: How it Works, and Why it's Amazing!
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- Опубліковано 15 сер 2024
- A concise description of how the Julia sets are generated, and their similarities to the famous fractal, the Mandelbrot set.
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Some of the music in this video was made by Kevin MacLeod and came from incompetech.com.
The Mandelbrot set looks so nice but imagine having a bad fever and then having a hallucination or nightmare about falling into this.
thats called death bro
buckle up
nah imagine being high asf you’d prolly think you’re dying😂😂
@@joshpollack5936 so it isn't just non existence when you die?
@@naaaaaa7772 a fractal, 3d, mandelbrot or julia set with shepard tone is worse than black peacefull bubble filled with hot gel like substance
@@karolbomba6704 oh, wait do you smoke SALVIA?
Trippy! Thank you for not getting hung up on the math. You explained it in such a way that I didn't have time to panic and instantly tune out. Bravo!
I can feel my brain growing
Gaston Julia appears to be the true founder of the fractal dimension. Mandelbrot used advanced computing to reinterpret Julia's discovery.
"A concise description of how the Julia sets are generated, and their similarities to the famous fractal, the Mandelbrot set."
The madebrot set is a collection of all julia sets, in one set, every julia set is present in the madebrot set , its maths combines all other sets to create it. So its has no similarities to the madebrot set , but exact replicas of it as its contained within it.
Wow. I don’t understand a word of this or the mandlebrot video but you have the best visuals - pretty colors and other people’s are so slow. And I love the end where you back out of it to the start! Enjoyed it thx
I've been studying fractals since the late 80s and yet I'm still amazed by them. That was a lovely series of examples and explanations, thank you.
Thank you so much, I had to do a presentation on Gaston Julia, and the way other sources explained julia sets didn't make sense to me.
This is the most beautiful thing I have ever seen.
Joshua Wagner, yea, I know, I like to call it eye candy. It’s awesome to look at those fractals, I even tried drawing one.
I came to this video by my nerdy self looking on the App Store for science apps. This app had a description and said something about Julia sets, and as my name is Julia, of course I’m interested! So I looked it up online and clicked on this video. Crazy
The colors, the colors...
I almost puked
Lets take you the cleanest and prwetiest forest ever
The colours can be any that you want, you can even cycle them.
That ending music you made was phenomenal
1:29 "A rather simple repeating pattern"
Understatement of the year
Nice video though
Epic!! Thanks for breaking this down, my friend!
Incidentally, how did you completely fail to mention the big connection that the Mandelbrot Set is also defined as where the Julia set of c is connected?
That's not the actual definition of the Mandelbrot set, which is defined solely by the iterative function.
A connected Julia set is also in the Mandelbrot set. Disconnected Julia sets are outside the Mandelbrot set. Julia sets exist independent of the Mandelbrot set. The only relation is due to the form of the function being of the same order.
One does not define the other.
@@t00by00zer That IS the actual definition of the Mandelbrot set in many actual mathematical works on complex dynamics (e.g. Beardon's 'Iteration of Rational Functions'), following the more general principle of partitioning the parameter space of a parametrised set of rational functions into different regions where the Julia sets and complementary Fatou sets for iteration of the corresponding rational functions display essentially the same features. In the special case of the parametrised set of quadratic polynomials Pc(z)=z^2+c (c being the parameter), it is shown that you can define the essentially different behaviours as being of two types: the Julia set is either connected or it is a totally disconnected "Cantor set" (ie, for any c you get one or the other). Then, it is further shown that you get the totally disconnected case if and only if the only finite 'critical point' of
z^2+c, which is 0, is in the 'infinite component' of the Fatou set and that occurs if and only if the iterates 0, Pc(0),Pc(Pc(0)),.. tend to infinity. In the other case, when 0 lies in another component of the Fatou set, these iterates remain bounded.
So you do get the "computational" iterative definition of the Mandelbrot set (the set of parameters c for which the Julia set is connected) and, if you like, you can give that as the definition of the Mandelbrot set, since it is equivalent by the theory outlined, but the more fundamental definition is indeed the one that K1naku5ana3R1ka gave.
Thank you I was confusing myself for over an hour trying to figure this out on my own.
Very cool. And concise.
Fantastic video! I was trying to find a way to generate these sets by messing with a mandelbrot generator i had made, but couldn't find a good explanation and nothing I tried worked. This video explained it perfectly and has it working now!
Error at 0:25
Z2 = (-1.5+.5i)^2 + (-1.5+.5i)
= 0.5 - i
But the video shows the answer to be 2 - 1.5i which is the square without the constant... (-1.5+.5i)^2.
Thank you.
Awesome 🙌 good preparation I’m getting ready for a road trip to search for crop circles in England
It is funny how this is popping up in recommendations along with some Mortal Kombat videos, because you said "Sub-Zero" at 1:02
The Concepts and Elementary Operations of Complex Number
The Magical Complex Number
In the previous sections, we have always represented complex numbers with coordinate forms: (a , b), where a is the real part and b is the imaginary part. This method makes it easier for us to associate complex numbers with the points in a two-dimensional plane, and facilitates understanding the formation of Julia set graph.
In mathematics, a complex number is usually represented with a more professional form, i.e. a+bi. Herein, i is a special mathematical symbol: imaginary unit, i.e. the number 1 in the imaginary number.
The most special property of the imaginary unit i is that its square equals to -1, i.e. i2=−1. For common real numbers, only the square roots of positive numbers make sense. However, with the imaginary unit, negative numbers can also be calculated for square roots. For example, the square root of -1 is equal to the i. After putting the real number and the imaginary number together, a broader concept of the complex number is formed.
The sixteenth-century Italian mathematician Gerolamo Cardano is known as the one who used the concept of complex numbers at the earliest. Many problems that used to be impossible to solve or not so easy to deal with can be solved after introducing the concept of complex numbers into mathematics, although complex numbers look a little strange and are not easy to understand. Not only in mathematics, the complex number also plays an important role in many applications of other specific fields, including physics, chemistry, biology, economics, etc.
Elementary Operations
In the formula of Julia set, z2+c, there are two types of complex operation: Square operation and addition operation.The square operation rule of complex numbers is the same as that of real numbers: z2=z×z, so we only need to know, in effect, the complex addition and multiplication.
For any two complex numbers, a1+b1i and a2+b2i, the sum of them is:
(a1+a2) + (b1+b2)i
That means the respective summation of their real parts and imaginary parts. Moreover, their product is equal to:
(a1a2−b1b2) + (a1b2+a2b1)i
It is very similar to the binomial multiplication rule of the real number, merely joined by the imaginary unit i.
Example
Let's look at an example, and practice the complex operation rules:
(3 + 2i)2 + (1 − 4i)
= (3 + 2i) × (3 + 2i) + (1 − 4i)
= (3×3 − 2×2) + (3×2 + 2×3)i + (1 − 4i)
= (5 + 12i) + (1 − 4i)
= (5 + 1) + (12 − 4)i
= 6 + 8i
Came here because of your nitrogen cycle video... now I'm sucked into the complexity of the universe once again. Thanks for being smart haha.
There is actually a thing that if you place the Julia's set image on the complex plane, respective to the constant, it creates the Mandelbrot set.
this is such a nice explanation!
Why only 4000 sub?
Good video👍👍
Short But Very Nice and Helpful to Understand
What program do you use to generated the Julia sets and the Mandelbrot sets?
Is there any online site where this is possible? XB
You can use python, java, C#, etc. I used C# if you are interested write back and i will publish on github
Back in the day, we used a program called "FractINT". The problem with just doing the math in a simple program is that the normal floating point value has limited precision. Use an extended-precision real number feature, and you can trivially generate this data, slowly.
What's missing in modern visualizations is "palette animation". Plotting the escape value in a range of 0-255 and using 8-bit color, the graphics card can instantly shift the palette entries and create mesmerizing effects.
A *really* big calculator
I am interested FractINT program, How can I have it? could you recommend me?
It's funny no one has mentioned the Xaos open source app for generating fractal images and animations, including the Julia set. I believe a new updated version is coming soon, or may already be out.
fun fact: if you compose all of the julia sets into one image, you can see that it's starting to resemble the Mandelbrot Set.
Best explanation out there
good video, easy to understand!
Wow how nice. Thanks for describing me.
Can there be a program that shows you the Julia Set corresponding to the Mandelbrot Set in the way shown at 2:16?
Mandelbrot set vs Julia set, which one do you prefer?
Great vid...very helpful...thanks!
Can you talk about the burning ship fractal, i went to a fractal generator website and found it and i have no idea what it is or what the formula is but it looks cool
The formula is almost the same as the Mandelbrot, which is z_{n+1} = z_n^2 + c. The Burning Ship fractal uses the absolute value of the components of z after each iteration, before squaring it. This gives the formula for the Burning Ship fractal, which is z_{n+1} = (|Re(z_n)| + i|Im(z_n|)^2 + c, where Re(z) is the real part of z, and Im(z) is the imaginary part of z. See en.wikipedia.org/wiki/Burning_Ship_fractal.
Very nice.
GREAT VIDEO!
I think you made a mistake calculating Z2 at 0:27 min. (Z1)^2=2.25-1.5i-0.25=2-1.5i, so Z2 should be Z2=2-1.5i+(-1.5+0.5i)=0.5-i.
Love the video but i checked the math for the iteration of the point 0.5. Its orbit is not bounded, instead it blows up very slowly.. the calculations for that point are incorrect. Correct me if im wrong.
could you share the music you used at the end 3:30?
0:45 DOUBLEBROT CONFIRMED
this just tripped me the fuck out
Wow.
What is this universe bro🤯🤯🤯🤯
Anyone know how I can make animations that itter between Julia sets and mandlebroths ?
Nice!
thank you!
Who are you and would you be open to the possibility of an email conversation?
there are errors mistakenly overlooked at beginning and middle on showing the basic series calculated
0:36 hold on 0.5 is in the set??
I swear it isn't
z2=0.75
How da hell u have only a thousand subs man
I always thought that the Mandelbrot Set was better as it contained all Julia Sets.
I am so unbelievably high right now and this shit was the BOMB
Does this mean that there are, theoretically, infinite Julia sets?
Does that mean that, in a multiverse of infinite universes, every universe is the shape of a Julia set?
Does that also mean that, since zooming into the Mandelbrot set gives you a Julia set, and the Mandelbrot set could be the multiverse, does that mean that, if used correctly, a spot in the universe could be used as a method of quickly travelling across the multiverse?
well... thats how to gain an instant sub! exellent work!
What if you change the C instead of Zsub0?
Kind of like the butterfly effect 🦋one small change makes a difference
Where are you getting software to do this?
What do you mean by 'zoom in'
OALD=[phrasal verb] (of a camera) to show the object that is being photographed from closer/further away, with the use of a zoom lens; in this case, using digital zoom to show the details of the fractal that are too small when 'zoomed out'
Reality is beautiful
Since he pronounces C and Z the same, some parts of this becomes incomprehensible unless you already kind of know what he's getting at.
Actually, there are some Julia sets that aren't fractals. There is the middle, which is a circle. Outside of the Mandelbrot set is stars, except for the X ones.
what program did you use to zoom into the sets
My own.
Taxi for Holly Krieger!
Could you also call the Julia sets the Gaston sets
No.
@@Foebane72 I dont remember writing this comment
2:17 what did you use???
hey do you remember the video with viruses from 2009? recorded with hyper cam... windows xp... funny song...
0.5 isn't in the Mandelbrot set though. In your iterations, you fail to add on 0.5 again. 1/4 is the greatest real number in the set, where the little inwards bump is. (It's called the cusp)
Z_{1}=0.5
Z_{2}=0.5^2+0.5=0.75
Z_{3}=0.75^2+0.5≈1
Z_{4}=1^2+0.5=1.5
Z_{5}=1.5^2+0.5=2.75
We stop here at Z_{5} because the magnitude 2.75000 > 2 . You have done:
Z_0=0
Z_1=0^2+0.5=0.5
Z_2=0.5^2=0.25
Z_3=0.25/2=0.125
Z_4=0.125/2≈0.063
Z_5≈0.063/2≈0.031
Z_10000=0.00000...25
How many iteration is normally done per point ?
For a clear picture of the whole set, usually several hundred. For deeper zooms, many thousands of iterations are needed. The deepest Mandelbrot zoom videos on UA-cam can use over ten million iterations per point.
@@JimiSol 🤯
Maybe.
A fever dream.
so sick
You're not related to Charles Eiesenstein, are you?
He's his son.
I think you plotted Z0 = 1 + i as Z0 = 1 - i.
What we have *C* een
i keep thinking, why would someone's name be almondbread lol
I saw this from paint.net
Make more videos damnit
Sub-Zero? GET OVER HERE!!!!!
So, is every single Julius set somewhere in the Mandelbrot set
ItsThrowaway yes
good video, even though it doesn't properly explain the color scheme.
The colours can be anything you want.
Hello ..slowly ..a bit ..my brain needs more than 4 mins to understand
Seriously, can native english speakers hear a difference between the letters C and Z?
The alternate pronunciation for Z, 'Zed' is superior for that very reason
I agree!
Yes we can.
Lol yeah , it's easy to tell the difference between the two
@@randomsubject4537 too easy something you pick up in grade school.
You are too smart for your viewers :)
you are too average for these videos
aight i say uh
Julia Set Variants
They're a cute couple
You beautiful bastard.
welcome to the omega ass
With fractals, we can begin to learn about the nature of consciousness
and the next big breakthough in physics is about to come. It will bring us telepathy, telekinesis, remote viewing and more.
CAN YOU MAKE BIRNIG SIHP
Being completely self aware from within the situation one find's one's self is a perception challenge. Here on Planet One, The Most Royal Heliosphere Egg -- Earth True - The Self Fractal Planet. The Identity Fractal contains images of fetus territories. These are basically un-awoken time travel pods, which are designed for residents of Earth during the singularity vortex, which is on presently. However the understanding of general populace is so low, the ambition to become immortal & realize one's self as creator god -- has been lost - because the first fetus incarnation event was not recognized & so went unmet. Earth is the origin of the fractal & humanity doesn't know how to relate to the engine access port -- So I'm developing my own Cyber-Punk Grand Unified System Of Everything. Merry Christmas!
I'm just plain confused. I dont even understand the formula. The only thing I know is complex numbers and Imaginary numbers.
I'm with you here, trying to figure out how to teach myself to do this lol. Anybody know a good resource online for walking someone through these with only Calc 1 knowledge formal knowledge?
you wanna do i tjing things
3:27, Now with that thought, think of the spike in the covid mRNA vaccine, A piece is a map of the whole ,
What kind of drug did you use?
Me:
you sound like timtom
not a clear explanation...you speak too fast...whats the point? if we are still learning...you need to speak ..slow.....
This shit hurts my brain