The Mystery of Hyperbolicity - Numberphile
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- Опубликовано: 8 фев 2025
- Featuring Professor Holly Krieger. See brilliant.org/... for Brilliant and 20% off their premium service & 30-day trial (episode sponsor). More links & stuff in full description below ↓↓↓
Professor Krieger discusses density of hyperbolicity, complex dynamics, iteration, and of course the Mandelbrot Set.
Prof Krieger is Corfield Fellow at the University of Cambridge (Murray Edwards College): www.dpmms.cam....
More videos with Holly: • Holly Krieger on Numbe...
Ben Sparks on the Mandelbrot Set: • What's so special abou...
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More videos with Holly: ruclips.net/p/PLt5AfwLFPxWJ8GCgpFo5_OSyfl7j0nOiu
"I don't do arithmetic in front of people."
I'll have to start using that phrase, it's brilliant!
For real - it's humble, self-assured, and honest. Definitely gonna steal that one.
"I'm a mathematician not a calculator"
I love how she summarizes a difficult problem so succinctly!
When numberphile drops a new holly krieger video ❤
She is back!! Her videos are one of the more memorable ones on this channel for me. Glad she did another one. Hoping for more.🤞
I remember Holly in college (at U of I) and she was exactly like she is in this video: humbly brilliant.
I-L-L
@@Da34BoxINI!!
My favourite Numberphile guest talking about an interesting phenomena around the mandelbrot set - this is like a perfect video :)
7:35 "I'm impressed if anyone remembers". And everyone: "yes, sure omg you're back" 😂
A *new* video with Holly talking about iteration of Zed and the M set.. my day just got substantially better.
"I don't do arithmetic in front of people". I respect that.
This is another one of those things that sound really simple but no one can prove either way, similar to the Collatz conjecture or the twin prime conjecture. I find it fascinating that with all the progress in maths over the last few centuries stuff like this still eludes us.
Bordering spooky
The Collatz conjecture is also a kind of dynamics on integers. So they share some similarities.
"I'll be impressed if anyone remembers (the Mandelbrot Set)." OMG I LOVE THE MANDELBROT SET, HOLLY! Just my inner thoughts coming out.
More mysteries about the Mandelbrot Set. We already know about pi , about Fibbonaci numbers, and now density of hyperbolicity.
Out of all the things to talk about, squaring a number and adding another to it is definitely up there.
Finally came to know some open questions dealing with the Mandelbrot set! Thanks Prof. Krieger, and thanks Brady!
Two other such open questions are the "Mandelbrot Locally Connected" conjecture and a connection to the Catalan numbers.
Well I didn't know before and I still don't know, but now I know nobody else knows. Progress!
Holly is my absolute fav!! So glad to see her back
Professor Krieger's videos are the best. Thank you
Fascinating. Will there be a part 2? I'd love to go deeper in to this topic.
Same. I’d like to go deeper with Holly
Same here!!
Always love seeing more of Holly.
Back in the 80s/90s, the Mandelbrot set was the bases of one of my favorite screensavers for After Dark.
were the flying toasters hyperbolic?
I loved the way it would progressively fill the screen! Watched it for hours.
Wow - up or down till you hit the graph left or right hit the line - love that visual!
This is basically why I love maths. There’s so much proofs and even more to learn. Things like this get my brain juices flowing and why I can’t sleep
I love her enthusiasm! This is top notch!
So happy she's back making videos! :)
Professor Krieger will always have my main cartiod.
"I don't do arithmetic in front of people" is a great libe!
Gooooooood morning Holly! My day just got better.
the minute i see a video with holly i click INSTANTLY
The Holly-Krieger effect as we call it.
I remember an exhibition at the art gallery in Southampton University (where I studied maths) of computer-generated images of portions of the Mandelbrot set. It was beautiful. This would have been in the mid-1980s when such things required expensive computers to make, so a lot of people had never seen it before.
this is the sweetest woman on the entire planet earth. the kind of woman you would want as a parent or teacher when you're a child. the kind of woman you would want to marry when you're an adult and stay together until you're both 200 years old. this isn't hyperbole, I'm sure a few hundred years back poets would write countless books and plays about women like her, and emperors would fight wars over her. her smile is burning my heart
Guess what guv we will be simulating partners orders of magnitudes sweeter, as hard as that is to imagine
@ugiswrong I'm praying every night, I rlly do hope you're right 🙏
Bro, you are coming on a bit strong! 🙏🏻🙏🏻
@@jahnsemtex I really don't think so I'm just being honest
Happy to be reintroduced to the Mandelbrot set in such an intuitive way. Of course I spotted it early on, I watched all your older videos and I'll never forget those.
I didn't spot the Mandelbrot set, but I did arrive at the conclusion that it was connected to the bifurcation diagram very early on. I just didn't remember that those two concepts are _very_ related.
Veritasium's video: "This equation will change how you see the world (the logistic map)" has some excellent perspectives on this concept if anyone wants to check it out.
The Mandelbrot set is my favourite mathematical bug. It has so many weird features. Especially zooming in and in and finding baby Mandelbrots hiding among the hairs.
What an unexpected video and intriguing (bounded and countable?!) result, thanks Professor Holly!
I like how this one and the last -1/12 video revisits on the old hits of this channel and the same professors go much deeper into the same topic.
There are so many talented/intelligent/fun presenters here but Holly Krieger will always be the best one. I know it's not a contest, but if it were, she'd easily win it.
Dr. Grimes too. He appears less frequently but was a must watch since earlier times of the channel
@@nocturnomedieval yes, if I ranked them (which I obviously would never do because that would be immature and unproductive), he would be my second favorite.
May I mention Hannah Fry?
Density of Hyperbolicity, I'll be working that into as many conversations as I can today
Thank you brady and every professor appearing on numberphile for these videos. I started doing a maths degree because of them and will be starting second year next week ❤ 😊
one of the coolest videos I've ever seen
It's nice to know there are things to find out.
I love how this channel makes videos with seemingly the notes of mathematicians.
So nice to see Professor Krieger again, and her midwestern cheer! 😏
It's a bit like the Collatz conjecture, but for real (or complex) numbers.
I hope Dr. Krieger will go back being a frequent guest of the channel.
It's very interesting that such an easily stated problem is still without an answer.
z^2 is a vector operation. While it technically isn't a vector, it's still doing vector stuff. The angle it makes with [1,0] is doubled and the magnitude is squared. Same thing with z^n. That plus 'c' part is a resultant operation. So, 'c' can also be a vector, and you can also square it. 'z' is under iteration, 'c' is not. 'c' is a constant. But it has that vector angle multiplication relationship with the original pixel. Since you know the vector aspect of this, you can now make a Mandelbrot Set based on area, instead of distance squared.
“I’ll be impressed if anyone remembers.”
Professor, you’re dealing with a crowd that watches math videos on RUclips for fun. I’d be more impressed if anyone clicked on this video and didn’t remember. 😂
Super interesting as always. Thank you for your videos!
Love Holly. Always more Holly please!
Soooo.... We need to try to look for singularities in the complex plane, within the bulbs of the Mandelbrot that violate this conjecture?
I see two potential levels to this.
1. Points within a bulb that don't converge.
2. Points within a bulb that have a different orbit period than their neighbors. (They would be hyperbolic, but I think this alone would still be interesting)
I feel like analytical approaches are the only viable option...
-3/4 is exactly at the border of the big blob (the area that have 1 final point) and the smaller blob (2 final points)
so I will say take the average and make it have 1.5 final points :D
Makes sense to me! Maybe they can do something similar to the -1/12 magic to figure it out.
Though I wonder if renormalization would even work on a function like this.
For some reason it seems like it's way harder to find a pattern in these numbers.
In the 1-blob, you have a cycle of 1 step where each step approaches that one point.
In the 2-blob, you have a cycle of 2 steps where each step in the cycle approaches one of 2 different points.
In the 3-blob, you have a cycle of 3 steps where each step in the cycle approaches one of 3 different points.
etc.
Right at the border between the 1-blob and 2-blob (i.e. at -3/4), the "2 different points" are *the same point* (which seems to be -1/2).
Edit:
And right at the border between the 1-blob and 3-blob(s) (i.e. at -1/8 ± i*1/3), the "3 different points" are *the same point* (which seems to be -1/4 ± i*9/20).
You can't have half -an A press- a point!
@@U014Bi think, as non-degree math dude, that this is where hopf fibration dudes dive in to the thread and say "well, äkšjhuli..."
it has 1 final point but converges logarithmically slowly, so it has 1 but takes so long for it ot get there
Two questions occur to me: 1) In the first couple of examples, I would have liked to know what the one or two numbers converged to ARE. 2) I wonder whether you could iterate FROM these numbers and GET BACK TO the original number (zero). Like, instead of square and add, you could take the square root and subtract, etc.
I don't even do arithmetic when I'm alone.
Hyperbolicity? More like "Really interesting; I'd listen endlessly!"
Love seeing the CMS in the background
Another candle of light in the darkness of the Mandelbrot set.
You've got an intersting recursion/iteration there, as the Ben Sparks video about orbits in the different blobs of the Mandelbrot set was visualizing the numbers of the series and how the split up, when you go from one blob to another, and Ben Spark was saying at one point, that this is what Hallo Krieger was showing in an earlier video.
And Holly, I actually do remember the core Meaning of the Mandelbrot set dividing the plane of complex numbers in convergent or divergent, and I also understand the convergent cases can be very different, the first case can even be covered by determinig the point where y=x meets the x^2-1/2 parabola analytically, but I guess only a limited number of such cases exist, especially whenc actually is a complex number. And it's fascinating that even a simpler number like -3/2 is not known to have the hyperbolic feature or not. I haven't tried but I know throwing a program at this you will easily get an answer that you can't decide whether it's due to the precision limits of floating numbers or mathematically true or false.
So does it boil down to finding new mathematically purely analytical methods that can replace the iterative approximation method? Or is it more like proving whether the iterative method works well and which crietria have to be met? Just like you can find counter examples for the Newton's method to finding roots of functions failing?
Who else here is completely in love with Professor Krieger?
Me!!!
Hey Holly, amazing video as always! I am a big fan of the mandelbrot set and love to cumpute rendering videos of it. In the background you got this really cool poster/map hanging at the wall. Is there a chance you can give me hint about where you got it or where you could find one of those? I would love to put it up as well 🙂
I think it might be the Bill Tavis Mandelmap poster.
@@brianrogers9233 Thank you!!
Pulled up my old Mandelbort set generator code after watching this. Now I want to improve its performance see how fast I could make it render.
Holy Holly! ❤😊 Happy to see you again! Come visit the states for a guest lecture here🎉
I wish Professor Krieger had shown the first few steps of iterating -3/2 through this process.
Yeah, from the first several iterations, -3/2 looks to be chaotic, indicating to me that it falls on the boundary of the Mandelbrot set and not in the interior. Maybe the bifurcation diagram for the quadratic map can shed some light on that.
The time I got intersted in fractals was also about the same time kkrieger hit the scene. That's kind of poetic, and I'm properly thrilled that there is still some mathematical mystery around fractals even today. Please visit Holly many times more!
Great discussion!
this problem sounds like it heavily relates to the logistic map bifurcation diagram where there is a period doubling route to chaos as it gets closer to 3.57 and beyond that up to 4 it becomes chaotic with some islands of stability
What limiting behaviors can non-hyperbolic inputs have? Do they all explode to infinity, or do some bounce around forever within a finite region without ever converging to a limit set?
Yes ;)
Holly is wonderful.
Yay, Holly!
From 3:41 onwards it looks to me as it it were still converging to the one intersection point, just a bit slower than before. Why would there be two points?
I thought of it like instead of spiraling in on one point, the shape would begin to look more like a rectangle with corners that intersect the graph at two points
I might just be stupid but they both have 2 points on either side of the line. What makes them different?
Brilliant isn't brilliant. Holly is brilliant!
So cool. I hope to one day find a niche in mathematics interests me enough to work on it.
I created myself a similar conjecture for elliptic billiard (one ball inside ellipse), when you set the reflection law to be, the reflected ray going along the normal at the reflected point : "the ray converges to the 2-periodic orbit, the minor axis....except when you start at vertex of major axis, an unstable starting position". My real mapping function is more complicated than the quadratic you use (z^2 to z^2+c).
Nice stuff ! Thank you.
A fun related fact is Sharkovskii's theorem: for real systems (vs complex like the Mandelbrot set), the possible periods of points can be put in a particular ordering, so that if a system has a point with period m, then it also has a point with period n, for all n which come after m in that ordering. And this is true for any real system at all, using the same ordering!
Sharkovskii's ordering ends with all the powers of 2, so if a system only has finitely many periodic points then their periods must all be powers of 2. And it starts with 3, so if a system has a point of period 3 then it has a point of every possible order.
Way too short. I could listen to Professor K for an hour easily. And Miss Holly, yes I remember the Mandelbrot set and your other videos!
We know the Mandelbot set on the real line ranges from -2 to +1/4. We also know the Mandelbot set is connected (even if by very thin filaments). Doesn't that imply we know that -3/2 is part of the set and will eventually converge on a set of points? What am I missing?
8:49
I think we know all hyperbolic maps are in the Mandelbrot set, but just being in the set doesn’t necessarily mean it’s a hyperbolic map, which if the case with -3/2.
a special example is when z=0, c=-2.
It converge directly to 2. any value of c slightly larger than -2 just give random outcomes, if c is slightly smaller than -2 will spiral to infinity
Density of hyperbolicity.. that is suuuper cool.
Of course we remember Prof Krieger
I was just going to say ... "Very cool, seems reflective of the nature of the cardioid form of the Mandlebrot's non escaping values, that we see in its initial form.". I can't think of the mandelbrot set without imagining myself as the observer, creating the initial cardioid form, out of the circle that is the set when there is no resolution applied to forming it, before iterating. Such a nerd, what else to say! :|
Hey, Holly no public arithmetic; Can we discuss multiplication, perhaps in private? I do apologize, could not resist.
This will be the best video ever!!!!
Now I need to know the reason why those points are called hyperbolic.
Fascinating thank you!
Smart and beautiful as alway Dr Holly
"I don't do arithmetic in front of people"... Ah! Well, that is the mark of a true mathematician.
I need more
The most amazing thing in the video was hearing an American pronounce 'Z' as 'Zed'
lol
Came here to post this. Good call.
What are the exact criteria for establishing whether a value is hyperbolic? Could there be infinitely many hyperbolic values?
Fascinating as I think this branch of mathematics is in its own right, what I'd love to know is whether there are any real world applications of the insights gained from it. Does anybody know?
> I like squaring numbers and seeing what happens with them in the long term.
Hm, okay.
> Let's start with the number z
Hold on...
> And then we subtract 1/2
Mandelbrot sus
> something something convergence
Yeah definitely Mandelbrot
> this is secretly related to the Mandelbrot set
I KNEW IT!!!!
-3/2 at least appears to be in the Mandelbrot set computationally. Is it strictly that we can't prove it doesn't diverge, or could it have an orbit (without a periodic limit cycle) that continues forever without repeating but is still bounded?
Wow, I'm stunned that that's an open problem!
Why is the notion of a finite point attractor called a "hyperbolic set"? Has it anything to do with hyperbolic geometry (say the symmetries of compact hyperbolic Riemannian geometries)? Or is it related to hyperbolic groups? Something else?
Is it only the quadratic transform giving rise to the Mandelbrot fractal set that is hyperbolic in some regions, or is this a general concept?
Had a feeling this was Mandelbrot related as soon as I saw what she was doing. Of course, the number of times I've heard that Jonathan Coulton song (by the name "Mandelbrot Set", naturally) probably helps... XD
Of course it's about the Mandelbrot set!
Professor Krieger ❤
-3/2 is located between the cardoid and the circle (on the x - real axis)?
I proofed the Collatz Conjecture, what's are procedure after ?
I wonder if these kind of iterative functions are related to pseudo-random number algorithms? Linear congruential seems to maybe have similar features in that the output feeds back into the input
Indeed, yes. Derek from Veritasium says in his video on the logistic map that this kind of iterated process was one of the first ways of calculating pseudo-random number sequences.
Ah I want more, can we get a numberphile2 video that goes deeper with Holly?
I didn't get any real idea of what "hyperbolicity" means here, what makes these cobweb plotted iterations "hyperbolic" - can anyone help clarify? Thanks in advance...
read the new article on Quanta magazine
Fascinating! Also, I have that same blue book-keeper-opener on the book shelf. How'd that for hyperbolic???? :)