3Blue1Brown - 2021-10-16
An intro to holomorphic dynamics, the study of iterated complex functions. Video on Newton's fractal: https://youtu.be/-RdOwhmqP5s Special thanks to these supporters: https://3b1b.co/lessons/holomorphic-dynamics#thanks Extra special thanks to Sergey Shemyakov, of Aix-Marseille University, for helpful conversations and for introducing me to this phenomenon. Introduction to Fatou sets and Julia sets, including a discussion of Montel's theorem and its consequences: http://www.math.stonybrook.edu/~scott/Papers/India/Fatou-Julia.pdf Numberphile with Ben Sparks on the Mandelbrot set: https://youtu.be/FFftmWSzgmk Ben explains how he made the Geogebra files on his channel here: https://youtu.be/ICqj7nJbiRI (part 1) https://youtu.be/3BXoNOahFJk (part 2) Excellent article on Acko.net, from the basics of building up complex numbers to Julia sets. https://acko.net/blog/how-to-fold-a-julia-fractal/ Bit of a side note, but if you want an exceedingly beautiful rendering of the quaternion-version of Julia fractals, take a look at this Inigo Quilez video: https://www.youtube.com/watch?v=rQ2bnU4dkso I first saw Fatou's theorem in this article: https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-91/issue-2/On-the-iteration-of-a-rational-function--computer-experiments/cmp/1103940533.pdf Moduli spaces of Newton maps: https://arxiv.org/pdf/1512.05098.pdf On Montel's theorem: https://people.ucsc.edu/~fmonard/Sp17_Math207/lecture11.pdf On Newton's Fractal: https://core.ac.uk/download/pdf/1633889.pdf ------------------ These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ Timestamps: 0:00 - Intro 3:02 - Rational functions 4:15 - The Mandelbrot set 8:12 - Fixed points and stability 12:51 - Cycles 16:25 - Hidden Mandelbrot 21:17 - Fatou sets and Julia sets 26:24 - Final thoughts ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
I recently got my Ph. D. in holomorphic dynamics. We often refer to the "stuff goes everywhere principle" as the "explosion property" of Julia sets. In fact, for the higher dimensional generalization of holomorphic dynamics (known as quasiregular dynamics), this explosion property is used as the definition of a Julia set.
If you’re still able to contact the school where you learned this, perhaps recommend the name “the shotgun property” for this effect, since birdshot shells scatter pellets all over and in a fairly random spread, and from what I’ve seen in the examples from this video, there’s usually one step where the points go from a relatively tight cluster to semi-triangular, fairly random spread.
@Myne33 ....Do you think the school named that? These things aren't officially named, someone discovers something, calls it something in their paper and either it catches on or it doesn't. Someone else might call it something else in their paper, and then that becomes common parlance. Sure, you can call it something different in your paper but if a term is in common use there's little chance a new name will catch on. Clarity in what you're talking about is important
@Yaam Felsenstein So, @myne33, the best way to have your name stick is to write this decade's most important paper on holomorphic dynamics using that name :D
Could you help me out with this: given a neighborhood N of some point in the Julia set as initial values, each point in the plane corresponds to some iteration and initial value pair (k, x0) with x0 in N. It follows each point in Fatou set corresponds to some pair (k', x0'). However, as the recursive algorithm is memoryless, the process (k' + n, x0'), n in 0,1,... must be stable. Unless each point in the plain corresponds to some pair (k, x0) where value at (k - 1, x0) is in the Julia set, I fail to understand how the process can explode.
@MrSuperkalamies Honestly, I can't come up with an intuitive explanation for the explosion property using your idea of individual iterated sequences. The actual proof of this property uses the idea of normal families, and it is remarkably simple once you wrap your head around normal families.
Maybe this idea helps. The Little Picard Theorem says that any map holomorphic on the entire complex plane (plus infinity) omits at most three points, otherwise it is constant. There is a similar theorem in the world of normal families, saying that any family of holomorphic maps on the 2-sphere that omit the same three points is a normal family. This is Montel's Theorem that Grant alluded to in the video.
The Julia set is defined as the set where the iterates of f do not form a normal family. So, if we look at the iterates of f on a neighborhood N of some point z_0 in the Julia set, if the iterates don't cover the whole plane except at most two points (and infinity), then the iterates are normal on N. This is a contradiction since z_0 is in the Julia set.
I highly recommend checking out Milnor's book Dynamics in One Complex Variable. It's a remarkably accessible (in my opinion as a mathematician, so grain of salt for non-mathematicians) introduction to holomorphic dynamics. If you skip partway into Section 3 and read through Section 4, it goes over what I just summarized in detail. From this perspective, the explosion property feels to me very natural and almost obvious, even though the sequence interpretation you give is entirely non-obvious.
When I was taught Newton Raphson many year ago, I was told: "Make sure your initial guess for the root is good, otherwise it doesn't always work." Who knew "doesn't always work" was code for all this incredible beauty? Thank you very much for being our guide.
As a kid of the 80s who iterated Mandelbrot sets on an i386 and would wait patiently for hours to see patterns emerge, I have to draw attention to the computational miracle you’re looking at... Julia sets being near instantly populated with the waive of a mouse!
As someone who has programmed graphics engines before, these graphics are astounding and beautiful to me. Technology these days is amazing, being able to see things that past mathematicians never dreamed of seeing.
I used to program fractals on an 8088 running at 7.44 MHz. I would have to start it before going to bed and see what emerged some time the next day. So I had the same reaction as you.
@Jeffrey L. Whitledge, I too programmed them on my 8088 and would fall apart in excitement at the meager results that emerged eons later. Three fewer pixels and it would have been radio.
Years ago I wrote a Mandelbrot program in C64 assembly. It took 25 hours to compute a 320 × 200 black-and-white image. (And that was with optimisation for the main cardioid, IIRC.)
@Martin Winter, exactly! A handful of pixels, at least a day to run, B&W only. But it was just so damn exciting!!
"Mathematics is like a very good detective novel. At first everything is shrouded in mystery and nothing is clear. But as you dive deeper to understand more, the plot gets crystal clear."
Mathematics is honestly, truly amd genuinely very beautiful and I've fallen in love with this channel.
does it get clear tho?
@sergey there is always a cliffhanger. Like in a good detective novel series.
@sergey It is only as clear as we can make it... we are asymptotically approaching knowing all of mathematics, we will never reach the end, but we can learn as much as we can while we're here...
@Anonymous "... we are asymptotically approaching knowing all of mathematics..." this quote is beautiful! where did you get this?
@Franco Bartolabac Thank you! I didn't get it from anyone... I have been saying that for years...
@3:42 - "I think this distinguishes Julia as one of the greatest mathematicians of all time who had no nose."
Newton: Thank you for adding that critical qualifier at the end of your statement.
Finally, representation for hyper-intelligent Mermaid Man cosplayers
Tycho Brahe was thinking something similar, but for a different word in the sentence.
Gaston Julia is one of the "broken faces" ("gueules cassées") of World War 1
newton is overrated
@QSAnimazione 😂😂 yeah right, I have studied physics for 4 years and he is literally everywhere apart from stuff like quantum mechanics and electricity of course, all of mechanics is based on Newton works, optics is mostly Newton based of course leaving out stuff like YDSE, gave a very important thermodynamics law of cooling, which was the first significant law that described the physical relationship between heat and energy
Yeah, I used to think it was just recreational... then I started doin' it during the week... you know, simple stuff: differentiation, kinematics. Then I got into integration by parts... I started doin' it every night: path integrals, holomorphic functions. Now I'm on diophantine equations and sinking deeper into transfinite analysis. Don't let them tell you it's just recreational.
Fortunately, I can quit any time I want.
its not addictive i swear
Oxygen isn't "addictive" - either. -
But it is essential, -
"dependency"
[e.g. death without it.]
- - -
We find we are dependent upon "purpose":
Math is "sufficient"
I'm on the hard stuff man, Neron models, etale cohomology. There's no turning back once you decide to learn the proof of Fermat's last theorem.
Dang, I really wish I knew this Holomorphic Dynamics sooner. I came from strange attractor of 3D Nonlinear ODEs, going to Poincare Map and Bifurcation Diagram, just realizing that discrete dynamical systems is really amazing. Now, here I am, find out that my acceptance of discrete dynamics led to this beautiful stuff of complex analysis and hyperbolic geometry inside Holomorphic Dynamics. I feel really bad and late to this stuff.😢
@Muhammad Ismail Yunus how do you even get in that stuff tho? algebraic geoemtrey?
@3:44 "I think this distinguishes Julia as easily being one of the greatest mathematicians of all time who had no nose." Actually, his nose was just in a different plane, so he was perfectly capable of detecting complex smells.
If the photographer had waited long enough, he would have seen the nose appearing while the back of the head disappeared
Honestly, why did 3B1B even bother to mention that? That utterly random distinction is unnecessary.
Obviously that would be Tycho Brahe
@@deleetiusproductions3497it's called a joke
I'm at a loss for words. What a fantastically coherent, clear, beautiful and exciting video. And, by the way, I really loved the exercises on this one. Never thought I would get to understand why the Mandelbrot set, of all things, has the shape it does. I felt like I was starting to actually understand the topic, so don't ever feel like you're assigning "too many" exercises, please.
Thank you for your work, Grant.
I quite like this description:
“The Mandelbrot Set is a geography of iterative stability.”
I always wondered about the similarities between control theory and the Mandelbrot set but this description makes it blissfully obvious since control theory is all about feedback loops and iterative processes
@Hoera yes, exactly. This simple description made a certain understanding “click” for me too!
Ben sparks!
@whydontiknowthat …interest in the field.
This an outstanding and brilliant exposition. I am absolutely gobsmacked at the painstaking effort and talent it must have taken to produce a video of this quality on this subject.
Wanted to say something like that but I see you did it already .
Absolutely
I personally will always be easy on 3blue1brown about his deadlines.
That's because..these videos are hard to make, and I mean at every single step.
It's hard to write a nonfiction narrative that's correct, then harder to write a narrative people can learn from and harder still to write a weaved story where listeners can come away feeling like they've seen something beautiful, which is of course what we want to communicate as artists: to convey our personal sense of beauty to someone we don't know.
Right now, I'm making an "explainer" with Manim because it looks incredible when it's done. But rendering and working and trying to make Manim work for me has been both fun and developmental because it's a test in both your fundamental programming, and your ability to articulate your math knowledge to a rigid computer. It's not harder than anything I've ever done. But it takes time, especially when you're caught up with other facets of life. Take it easy on yourself Grant!
have been toying with the idea of making a manim video. any thoughts on getting started?
@Nathan Wycoff same here, math is just too beautiful to not share visually
@Julian H. any thoughts on what your topic would be?
@Nathan Wycoff time series econometrics or non-parametric statistics I think could be an addition
Making 30 minutes worth of Manim video is definitely an insanely large task.
Especially when some of the graphics are not in the library (i.e. how did he do the expanding circle shape in 23:32?)
A second part! It was announced and actually came!
Edit: Thank you for all these wholesome videos. Waiting for them is always worth it, no matter what the topic is!
I like to think all sequel's I once promised will eventually come...it's just that the timeline sits somewhere between 1 week and 10 years.
@3Blue1Brown lol
@3Blue1Brown I can't wait for more awesome videos in the Probabilities of Probabilities series! 🙏🙏🙏
@3Blue1Brown lmao
@3Blue1Brown This has some kind of chaotic behaviour
You're still one of, if not the most, amazing math youtuber out here. Seriously dude, your video are high quality, and you are a very good teacher. Thanks for this two parter, and for the rest of your channel too.
@Jonathan Trevatt lol
And he's not condescending and anti-christianity like mathologer
This channel is an attractor point of math knowledge and beauty.
Fractals such as the Mandelbrot and Julia sets are one of the things that, when I was in high school, convinced me that I would've done math at university. The others were chaos theory, non Euclidean geometries, and Simon Singh's book on Fermat's Last Theorem. Crucially, none of the books that got me interested in mathematics in high school were school books.
As someone doing mathematics at university, god i wish this is what we learned.
@Daniel A Freeman If you choose the courses correctly (assuming your university provides relevant courses), nothing should prevent you learning these things ;). Although Fermat's last theorem needs quite a lot of machinery that no single course would give sufficient knowledge to understand it's proof
@Daniel A Freeman what do you learn instead?
@Daniel A Freeman how far are you into your degree? AFAIK the cool complex stuff starts late
Ok
I think the coolest thing here is that even topics like the Mandelbrot set which seem so abstracted from useful math, have connections to some of the most practical algorithms used in engineering like Newton's method. If this isn't a justification for simply doing math for the sake of doing math, idk what is.
12:26 I did those three exercises, and the Cardioid fell out like magic. I swear, you are the best math teacher on the internet. I never did this kind of stuff when I got my degree, but these kind of recreational applications are by far the most fun and rewarding parts of pure math that I've encountered.
Could you help me with that for a second please? I'm a bit confused. z^2+c=z, if the initial z value is equal to c then the only fixed point is z=c=0, is it not?
If c is constant instead then the fixed points are z=(+/-sqrt(1-4c)+1)/2 and the derivative at those points is +/-sqrt(1-4c)+1
@Random Name the solution to the equation z²+c=z depends on the quadratic formula. But to use to the quadratic formula correctly, it helps to write the equation as z²-z+c=0. Now use the quadratic formula, and think about what a, b, and c should be. Also, note that there will be solutions with a plus and solutions with a minus. You should think, maybe one of those gives attractive fixed points, and one might only give repulsive ones.
@jmcsquared if we just use the quadratic equation, we get points that vary depending on c. I can't wrap my head around it. I mean, after all we are trying to find the attrating points, but what does it mean if they vary with c?
I'm sort of confused, sorry
@Kevin Arturo Urrutia Alvarez go back to 4:15 and watch carefully how the Mandelbrot set is defined. We are iterating the function f(z)=z²+c. Notice that it's a function of z only. That is, c is understood to be a fixed parameter when we do the iterations. However, the Mandelbrot set is defined to be what happens when we iterate the function f(z) specifically for z=0 for different choices of the parameter c.
So, we think of the iteration function as depending only on z, while we think of the fixed points as functions of c. In other words, we first choose a c beforehand, and then f(z) is an iteration map which acts only on z (starting with z=0). But when we ask what kinds of fixed points z we could get, those explicitly depend on which choice of c we make beforehand. So, it helps to think of the fixed points z(c) as functions of c.
@PiercingSight I don't mind at all! I am a teacher, it's sort of my calling card, after all.
I think there are infinitely many fixed points z(c) which depend smoothly on the complex number c. Use the quadratic formula on the fixed point equation f(z)=z²+c=z (make sure to rewrite it in standard form first). Should be a square root function of c somewhere in the answer.
Then try to figure out which branch (plus or minus) to choose in order to make them attractive fixed points (the derivative f'(z) must be less than 1 in magnitude in order for the fixed point to be attractive).
I almost started writing my homework essay, thanks for showing me cool fractals instead
underrated
Every student deserves a teacher like him.. M soo glad that we are living in an era where we can understand and share knowledge so easily.... Ooh and as for all your videos.... They are always a visual treat to watch!
This was my favorite topic in mathematics as a high school student. I didn’t really understand it until much later, but the patterns were so beautiful and captivating I would spend hours torturing my parents home computer with fractint. Thanks for making such a clear explanation and introduction to the subject.
As a kid - it was thinking about what the math I was doing looked like - that got me into enjoying visualization and graphics. I love this topic. Thank you.
This content is fundamentally beautiful and awe inspiring. There is something naturally beautiful about math in general, but videos like this border on a kind of spiritual awakening - for me at least. The hidden relationships between seemingly disparate mathematical functions and the patterns that emerge when taken to great lengths and infinity are something no person throughout the entirety of humanity has experienced until relatively very recently. I feel humbled and lucky that someone like yourself is providing the means and knowledge to see these beautiful aspects of reality.
Yesterday I was literally trying to figure out why we use Stochastic Gradiend Descent instead of Newton-Raphson method in ML, after watching the video, and I ended up deep diving into it and stopped at complex analysis and holomorphic functions. It's so great to see a new video on the same topic today itself 😁
Just wanted to know if you have found the answer to your question about SGD yet? My guess would be that we can't use Newton's method instead of SGD because we're not finding a root of a function, we are finding a minima, and we don't even know if a root would exist. Also you would end up computing the derivative of a high dimensional function if you use Newton's method anyway. But still would like to hear about your research into this matter.
@Kumar Kartikay and maybe with newton's method you can be trapped in those cycles, what would be terrible for an AI, but just a guess too
@Kumar Kartikay I think that's the answer. SGD is simpler because it needs less setup, but otherwise they're quite similar in their approach to solving.
That also applies to the stability too
Newton and quasi-Newton methods are entirely applicable for general minimization and therefore ML. Their main drawback is needing second order information (the Hessian of the function or approximations thereof) which doesn't scale well: for n dimensions, O(n^2) storage and computation are needed. (Reducing this cost is a main feature of quasi-Newton methods.) Another drawback is having to know what even the second derivatives are, or otherwise having to numerically approximate them.
In any case, any proper implementation of an optimization procedure will have checks for instability and (usually) ways to recover so as to avoid the issues mentioned.
He really did the topic he promised this time :o
And gave us homework.
@moonik665 a small price to pay
What I find fascinating about the Mandelbrot set is how well embedded it is in the cross-section of math and art. So often when a mathematical function or concept gets embraced by artists, there's a very consistent inverse relation between how aesthetically interesting it is vs how mathematically interesting it is (ie the principle that cool visuals normally means unexciting math, and exciting math normally means there's nothing much to see). Yet the Mandelbrot set and its associated Julia sets manage to be endlessly interesting in both aesthetics and mathematics.
This might be one of the best channels on the entire YouTube platform, basically the apex of what real, free, and beautifully taught knowledge is. A tear in my eye here.
Just like your video about the Fourrier transform... When I learnt about Newton's method at the university the professor talked about the Mandelbrot set in one of the lectures, we viewed fractional dimensions and the reason why Benoit Mandelbrot came with that shape. Unfortunately, the explanations were so confusing and misleading that it remained misunderstood and mysterious to almost everybody present at that moment. In order to get more into it, I even went as far as writing my own renderer in C back then consulting text books about it, drawing the iteration path, doing the derivative etc... To me it remained mysterious.
Here, you manage to explain the content of these 12 hours of lectures in roughly 1 hour, and now I understand is the link, FINALLY!... you're a mind blowing teacher.
Grant I have to say this. I remember some of these concepts from classes and I was sometimes unable to understand why something worked the way it works. But the way you explain it, suddenly I remember those things and I go "ooooh that's why that thing worked that I once did in class". Without a doubt, you are the best educator I've seen.
I'm just loving what's happening nowadays. For your new initiative on submitting quality math videos, my YouTube recommendation is blessed!
Grant pushing out videos faster than most of us could understand.
Collabs help the channel grow. This channel should
do some with other S-Channels!
Anyway: And theres many Science-Channel who's Fan's dont know each other's channels.
So here comes my plan into account: I drop random comments about 'Hey, want
some recommendations about something? Anything?',
get called a bot sometimes, but who cares,
and sometimes people say 'Thanks, i take a look',
which makes my Day!
Its arguably a hobby and arguably not an Obsession.
XD
You're incredibly talented to be both a great mathematician and a great communicator. It's likely you're single handedly inspiring young people to investigate mathematics. Good work.
Method for solving Exercise 3 at 19:10:
Let r, t, y denote roots of polynomial. Then we can write it as (x-r)(x-t)(x-y). Then multiply it all out and take second derivative.
ABSOLUTELY mind bending, and as a side note, you made it clear to me how the complex plane represents the simultaneous processes of rotation and expansion, in such a simple way that I looked back and realized it was there all along.
As always, WELL DONE.
If we ignore the fact that I couldn't remember my identity and why I clicked on this in the first place, really nice video, I will watch it ten more times after I received my PhD in physics (considering I've just started studying for the undergraduate degree and it would be two weeks from the start of the first class tomorrow). In my eyes, the effort you put in these two recent videos is more worthy than all the money that is on earth right now!
"Gaston Julia is one of the greatest mathematicians of all time who had no nose"
Sad Tycho Brahe noises
Tycho Brahe noses? Oops, I misread. But he was for sure the greatest noseless astronomer ever.
I didn't know Tycho that didn't have a nose. Thanks for the useful fact.
@Guilherme Cavalcanti de Santana He lost it in a sword duel. Replaced it with a brass one.
@Joel Adler I read it on wikipedia, thanks. Living in the past was certainly more interesting than we think.
@Joel Adler I suppose he'll have to have a second sword duel - this time against Julia, in order to determine the greatest noseless mathematician.
This can't be happening! Two great uploads in the span of less than a week!
Well, the upload schedule is stated in percentages after all, so videos with short time between them are bound to happen given enough time.
Right? He's loving these fractals, as am I!
It's obvious that a LOT of work went into this video, and I want to appreciate that. Thanks.
The world is not only stranger than we know, it is stranger than we can ever know.
But that makes it no less beautiful, and no less worthy of striving to understand.
Thank you for helping me understand, and showing me this beauty.
3b1b and SoME1 is the reason why i was inspired to start my own channel. Keep up the great work.
Your animations are so amazing, every time I watch them, I just lost myself in the fascinating patterns. Numbers are really magical. The glory of nature lies in the numbers themselves.
I'm in awe how you use manim to make all these beautiful fractals and even do all the zooming.
Another one in just a few days! Great work, Grant.
Damn this sequel came very fast, really packing heat with these uploads 🔥
I wish I saw your video during my years learning holormophic functions. It helps so much to see the dynamic of the function in full display, which you can't reproduce with a pen and paper. You did an outstanding job, Thank you!
This is one of the best maths videos I have ever seen (if not the best). You didn’t just say “hey here’s the Mandelbrot set, and here’s a pretty fractal!”, you actually went into the intricate detail and even provided exercises for the viewers to do! This is the pinnacle of mathematics education videos. I’ve seen how much better your videos have become in the past 3 years, and I can’t wait to see what’s to come for you.
My favourite part was when you were drawing the two complicated diagrams at 19:40, yet playing extremely calming music 😁
These videos are so humbling. It is impressive what mathematicians of the past have found and created and what the human brain is capable of. I am also impressed with the comprehension, editing and story telling by 3B1B. I always have to watch these videos 3 or 4 times to understand and appreciate it completely, and I have a relatively strong math background.
I think this might be my favourite video of yours by far, it does such a good job at tickling your curiosity and showing just how much there is to explore beyond what is presented. It doesn't feel like an explainer video like most of your other work (which i obviously still love in its own right), it feels so open ended in a sort of new way. I already know how much I like learning about maths and other things, but this way of presenting things felt really special to me in a way I hadn't really seen done to this extent before. Thank you, I hope to see more things like this in the future both from you and others!
Just finished coding a program poorly generating fractals after watching your last video, so glad to see this
That’s awesome! Generating them poorly is the first step to generating them optimally, never stop!
@BobWidlefish thanks man! Ima work on that
I LOVE your channel 3b1b.
Not only is the content ultra fascinating (I learn so much), your graphics are 2nd to none.
I don't think everyone puts enough admiration to your visualisations.
I have no idea how you do it, but the are absolutely INCREDIBLE. So much effort must have gone into creating the capability. In constant awe 🙏🏻
@DrTrefor - 2021-10-16
I enjoy how on twitter you asked recently whether we preferred two 17 minute videos or one 34 minute video. Instead you seem to have given two ~30 min videos:D Best of both worlds:D
@aashsyed1277 - 2021-10-16
Hi there !
@AxxLAfriku - 2021-10-16
WOAH WOAH WOAH!!! Let me get this perfectly straight: You comment something that is completely unrelated to the fact that I have two HAZARDOUSLY HANDSOME girlfriends? Considering that I am the unprettiest YouTuber worldwide, it is really incredible. Yet you did not mention it at all. I am VERY disappointed, dear dr
@h-Films - 2021-10-16
@AxxL new profile picture
@owenweiss9647 - 2021-10-16
He interpolated
@victorscarpes - 2021-10-16
@AxxL if they are so beautiful and you are so ugly, why would you cheat on them?