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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.
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
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 ❤ 😊
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.
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.
@@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.
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.
“I’ll be impressed if anyone remembers.” Professor, you’re dealing with a crowd that watches math videos on UA-cam for fun. I’d be more impressed if anyone clicked on this video and didn’t remember. 😂
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.
-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).
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...
More Holly, more Mandelbrot. I'm really interested in the complex constants producing "stable" cyclic iterations (start at (0+0i) iterate through "n" complex numbers, return to (0+0i) and then start the EXACT cycle over).
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.
9 місяців тому
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.
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 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.
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.
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?
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.
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?
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.
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?
> 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!!!!
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...
We dont know. It is if it eventually converges onto a finite number of points. But it seems we don't know if it does, so we can't know if resides in the Mandlebrot set.
@zornuthank you, this clarification is exactly what I was looking for. as a followup, I wonder if there is anything interesting to say about convergent subsequences within this bounded sequence
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.
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 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).
-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?
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
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?
missed opportunity to visualize how the Mandelbrot set changes as C changes smoothly. It is a fascinating animation, looks like the set is burning. Edit: I was of changing the initial value of Z, not C
The Mandelbrot set is the one describing, for all choices of c, whether the iteration when starting at 0, is bounded. I believe you are thinking of the Julia sets, where the value c in the function being iterated is a parameter , and where a point z_0 is in the Julia set for a value c if the iteration when starting at z_0, stays bounded. And this is the thing where you can change the value c continuously to get a family of images that look wild like that. (... unless you meant like, looking at how the sequence of iterations smoothly changes when starting at 0 and changing c? (or maybe how the set of points it tends to changes? But that seems like, given the open problem the video discusses is open, like it might not be feasible to animate that? Idk. But that probably wouldn’t have the fiery appearance you described, so probably not what you were talking about.)
@@drdca8263 Oh yeah, my bad. I don't know if I was thinking of the Julia set or its family, but I was thinking about gradually changing the initial value of Z each frame, instead of always starting with 0
@@Stdvwr Oh! Ok, I don’t know what that set would be called. I also don’t know what it would look like. I suppose in a sense, Julia sets and the Mandelbrot set are all slices of a larger 4D fractal, Where the Julia sets are slices in one direction, and the Mandelbrot set is one particular slice in the other direction, And, I don’t know what the other slices in that direction, the ones you pointed out, would be called.
@@drdca8263 Someone have wrote about it in a blog, if you are interested the page is called "Mandelbrot set with variations" by B. L. Badger, and it has some animations
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.
I'm no dummy, the last few videos about iteration, the Julian Set and the Mandelbrot Set I can understand upto a point. This one? I didn't get any of it.
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!
The biggest problem with human -invented notions of mathematics is human mathematicians' fundamental misunderstanding of their own invented notions of qualities because all such numbers as 0 , negative numbers, positive numbers, irrational numbers, imaginary or complex numbers never exist anywhere else in the universe other than solely within human mind and its notional operations No human nor AI can ever prove the unprovable notion that negative x negative = positive, and negative x positive= negative because all such human -invented notions are arbitrary. The entire system of human-created notions of mathematics are merely human constructs to describe human subjective understanding of the physical universe, but not and natural intrinsic physical properties of any form of matter at all. That's why there are such irrational notions in mathematics that such non-existent quantities as zero , negative numbers, positive numbers, irrational numbers, complex numbers, infinity which is unknowable unknown can be used in human notional operations . To be able to understand the problem, it requires applying such notions to the real physical world outside human mind's notional operations and imagination. Ex: No negative quantity nor positive quantity can represent any form of matter. It is nonsense to say negative 5 apples, positive 5 humans simply because such qualities do no exist anywhere in the physical universe, while being mere subjective notions in human mind What are the differences among the following expressions? 5 apples, -5apple , + 5 apples Which is a quantity, and can represent a form of matter? The answer is obviously the unsign quantity 5. If the negative sign and the positive sign are removed, they will be the same expression of a quantity of apples. What the symbolic negative sign can actually do is to indicate that the quantity behind it is a decrease in quantity of apples, while what the symbolic positive sign can do is to indicate that the same quantity behind it is an crease in quantity of apples, but it is not that there are such non-existent quantities as negative 5 apples and positive 5 apples as fundamentally misunderstood by all human mathematicians If 5 apples can by multiplied by such non-existent quantities as -5, then you will get an nonsensical expression as: 25 apples x -5 = -25 apples (negative 25 apples) which do not exist in the real physical world Or -5 x 5 apples = negative 5 times of 5 apples is such a irrational expression The number of times of a quantity of a form of matter, an action, an experience can never be negative. Ex: it is nonsense to say that you eat negative 2 times everyday, or you have been to London negative 20 times If negative x negative could actually be positive, then you would get an even more irrational expression: Since + 5 x +5 = + 25 , and -5 x -5 = also +25 , -5 x -5 = +5 x +5 , and thus -5 = +5 which obviously is nonsense, when applied to the real physical world, such as -5 apples = +5 apples, despite the fact that -5 apple represent a decrease, whereas +5 apples stands for an increase in the number of apples, say in a box)for 15 apples, and the number of 15 apples -5 apples decreases to 10 apples, which is opposite to 15 apples + 5 apples increases the number of apples to 20 apples All complex numbers invented by human mathematicians come from their misunderstanding of negative sign and positive sign which do not have any actual value, being used as symbolic labels for humans to distinguish the difference between a decrease and an increase in quantity of a form of matter and the difference between an actual quantity without any number sign, and a decrease or an increase in its quantity value in human mind 's notional operations There is no single point in the entire infinite space of universe which can actually exist in such non-existent being as human -invented notion of zero or nothing or emptiness, even in a volume of vacuum space. The only things which can actually be nothing or zero are the non-existent which do not occupy any space of the universe. A point of vacuum space itself also occupies a spot in the universe, and similarly an extremely small particle, a virus invisible to human naked eyes are also representations of space occupation, and any form of matter which exists in the universe always occupies a space , including light particles. Hence, what human invented notion of 0 can actually represent is the abscence or invisibility of a form of matter in their sight when they cannot see it, and the starting point of all quantities, and thus 0 is not any specific quantity, like infinity. O apple is not any quantity as how zero is always fundamentally misunderstood by human mathematicians, as a number in human -invented notions of number systems, while merely standing for the abscence of apples, and any other forms of matter. Neither zero nor infinity can actually be used in any arithmetic operations as how they are always misused in human manipulations of the flaws of their own invented notion of mathematics, being not any actual specific quantity 0^0 is and any mathematical operations with 0 are nonsensical expressions, and evaluation of a mathematical expression from negative infinity to positive infinity in calculus is even more absurd, while the max quantity which human can understand their own invented notion of quantity is 10^31 which an extremely extremely small quantity, compared to such a boundless quantity as infinity, there can be only one infinity which is the largest value which has no boundary, no beginning nor any end. The only one form of matter in the physical universe which can be infinity is the infinite space of the physical universe. All other forms of matter have to be finite, in order to be in the universe. Human mathematicians ' fundamental misunderstanding of their invented notions of number systems and mathematical operations comes from their own invented coordinate systems , in which they represent such non existent numbers as negative numbers, positive numbers, and zero, in which they ignorantly claims that infinity has a beginning which is the origin at 0, despite the fact that infinity has no boundary or no beginning nor any end, and there is no way for human extremely limited knowledge to locate the location of the Earth, this star system, this galaxy in the infinite space of the physical universe, but ignorant human astrophysicists keep talking as if the infinite universe were a pond in their backyard No form of matter can ever actually resist in any 1-D or 2-D space as how such non-existent space dimensions are always based to express human subjective understanding in mathematics. All forms of matter exis in 3-D, and any space dimensions more than 3-D are representations of human wild imagination and ignorance of the physical universe. All human -invented notions of mathematics are human constructs, and can be applied to only within subjective reality created by human extremely limited knowledge, its subjective understanding, and no where else. All human current destructive development is a living proof of how human species have got from fundamental misunderstanding of their own invented notions of mathematics to misunderstanding of their place on this planet , behaving as if they had created themselves , despite the fact that humans are merely a product of the laws of nature, and have to follow the laws of nature if they don't want to be eliminated or destroyed by the laws of nature
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.
These interations are very easy to understand. If f(x) is the function, then we have.... f(f(f(f(x)))) = f(f(f(x))) Everytime you use f(x), the value changes less and less. The left hand side has one more f(x), but the change is so small that it doesn't matter. If you repeat this infinitely, both sides become the same. If this is true, you can treat the last f(x) on the left side as if it doesn't exist and both sides become the same. That's why the fix point theorem works! This has a lot of fancy proves, but you can only prove the theorem, if you understand what I showed!
6:42 - oh, good! Because I've thought about trying, and... it seemed daunting. Now I can just leave it to Holly and the other mathematicians to puzzle on, and not worry about it. :D (But if I happen to figure something out next time I'm playing with some mandelbrot or related code, I'll let y'all know. :D)
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More videos with Holly: ua-cam.com/play/PLt5AfwLFPxWJ8GCgpFo5_OSyfl7j0nOiu.html
"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 remember Holly in college (at U of I) and she was exactly like she is in this video: humbly brilliant.
I-L-L
@@Da34BoxINI!!
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.🤞
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.
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 don't do arithmetic in front of people". I respect that.
"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.
Well I didn't know before and I still don't know, but now I know nobody else knows. Progress!
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.
Professor Krieger will always have my main cartiod.
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.
the minute i see a video with holly i click INSTANTLY
The Holly-Krieger effect as we call it.
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!!
"I don't do arithmetic in front of people" is a great libe!
Professor Krieger's videos are the best. Thank you
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
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
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.
Holly is my absolute fav!! So glad to see her back
Always love seeing more of Holly.
I love her enthusiasm! This is top notch!
So happy she's back making videos! :)
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 ❤ 😊
Wow - up or down till you hit the graph left or right hit the line - love that visual!
Gooooooood morning Holly! My day just got better.
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.
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?
So nice to see Professor Krieger again, and her midwestern cheer! 😏
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.
“I’ll be impressed if anyone remembers.”
Professor, you’re dealing with a crowd that watches math videos on UA-cam for fun. I’d be more impressed if anyone clicked on this video and didn’t remember. 😂
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.
Density of Hyperbolicity, I'll be working that into as many conversations as I can today
-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
Most charming laugh on Numberphile. 🙂
Holy Holly! ❤😊 Happy to see you again! Come visit the states for a guest lecture here🎉
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...
More Holly, more Mandelbrot. I'm really interested in the complex constants producing "stable" cyclic iterations (start at (0+0i) iterate through "n" complex numbers, return to (0+0i) and then start the EXACT cycle over).
What an unexpected video and intriguing (bounded and countable?!) result, thanks Professor Holly!
JoCo's song about the Mandlebrot Set was actually stating the formula of the Julia set.
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.
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 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.
All my homies love Prof. Krieger 😍
one of the coolest videos I've ever seen
I love how this channel makes videos with seemingly the notes of mathematicians.
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.
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 ;)
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.
I didn't know Amy Adams did math! Great video!
It's a bit like the Collatz conjecture, but for real (or complex) numbers.
It's nice to know there are things to find out.
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.
The second I saw z^2 - a constant Jonathan Coulton's Mandelbrot Set started playing and was waiting for how it relates.
I might just be stupid but they both have 2 points on either side of the line. What makes them different?
Mandelbrot by Holly is a series ! I need to buy colored sharpies for math brain teasers, its so much fun 🤩😂
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?
Love seeing the CMS in the background
> 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!!!!
Love Holly. Always more Holly please!
Super interesting as always. Thank you for your videos!
Who else here is completely in love with Professor Krieger?
Me!!!
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
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.
so is -3/2 in the mandelbrot set? I didn't really understand that part
We dont know. It is if it eventually converges onto a finite number of points. But it seems we don't know if it does, so we can't know if resides in the Mandlebrot set.
@zornuthank you, this clarification is exactly what I was looking for. as a followup, I wonder if there is anything interesting to say about convergent subsequences within this bounded sequence
Hyperbolicity? More like "Really interesting; I'd listen endlessly!"
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.
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!!
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).
So cool. I hope to one day find a niche in mathematics interests me enough to work on 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?
Saw the thumbnail of a new video with Holly Krieger > Immediately clicked
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
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?
I proofed the Collatz Conjecture, what's are procedure after ?
missed opportunity to visualize how the Mandelbrot set changes as C changes smoothly. It is a fascinating animation, looks like the set is burning.
Edit: I was of changing the initial value of Z, not C
The Mandelbrot set is the one describing, for all choices of c, whether the iteration when starting at 0, is bounded.
I believe you are thinking of the Julia sets, where the value c in the function being iterated is a parameter , and where a point z_0 is in the Julia set for a value c if the iteration when starting at z_0, stays bounded. And this is the thing where you can change the value c continuously to get a family of images that look wild like that.
(... unless you meant like, looking at how the sequence of iterations smoothly changes when starting at 0 and changing c? (or maybe how the set of points it tends to changes? But that seems like, given the open problem the video discusses is open, like it might not be feasible to animate that? Idk. But that probably wouldn’t have the fiery appearance you described, so probably not what you were talking about.)
@@drdca8263 Oh yeah, my bad. I don't know if I was thinking of the Julia set or its family, but I was thinking about gradually changing the initial value of Z each frame, instead of always starting with 0
@@Stdvwr Oh! Ok,
I don’t know what that set would be called.
I also don’t know what it would look like.
I suppose in a sense, Julia sets and the Mandelbrot set are all slices of a larger 4D fractal,
Where the Julia sets are slices in one direction, and the Mandelbrot set is one particular slice in the other direction,
And, I don’t know what the other slices in that direction, the ones you pointed out, would be called.
@@drdca8263 Someone have wrote about it in a blog, if you are interested the page is called "Mandelbrot set with variations" by B. L. Badger, and it has some animations
Is -3/2 in the Mandelbrot set?
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.
Isn’t -2/3 represented in the Mandelbrot set?
I'm no dummy, the last few videos about iteration, the Julian Set and the Mandelbrot Set I can understand upto a point. This one? I didn't get any of it.
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!
Quanta magazine just published an article on this.
Do you have any links to papers about x->x^2-3/2 case?
The biggest problem with human -invented notions of mathematics is human mathematicians' fundamental misunderstanding of their own invented notions of qualities because all such numbers as 0 , negative numbers, positive numbers, irrational numbers, imaginary or complex numbers never exist anywhere else in the universe other than solely within human mind and its notional operations
No human nor AI can ever prove the unprovable notion that negative x negative = positive, and negative x positive= negative because all such human -invented notions are arbitrary. The entire system of human-created notions of mathematics are merely human constructs to describe human subjective understanding of the physical universe, but not and natural intrinsic physical properties of any form of matter at all.
That's why there are such irrational notions in mathematics that such non-existent quantities as zero , negative numbers, positive numbers, irrational numbers, complex numbers, infinity which is unknowable unknown can be used in human notional operations . To be able to understand the problem, it requires applying such notions to the real physical world outside human mind's notional operations and imagination.
Ex: No negative quantity nor positive quantity can represent any form of matter.
It is nonsense to say negative 5 apples, positive 5 humans simply because such qualities do no exist anywhere in the physical universe, while being mere subjective notions in human mind
What are the differences among the following expressions?
5 apples, -5apple , + 5 apples
Which is a quantity, and can represent a form of matter?
The answer is obviously the unsign quantity 5. If the negative sign and the positive sign are removed, they will be the same expression of a quantity of apples. What the symbolic negative sign can actually do is to indicate that the quantity behind it is a decrease in quantity of apples, while what the symbolic positive sign can do is to indicate that the same quantity behind it is an crease in quantity of apples, but it is not that there are such non-existent quantities as negative 5 apples and positive 5 apples as fundamentally misunderstood by all human mathematicians
If 5 apples can by multiplied by such non-existent quantities as -5, then you will get an nonsensical expression as:
25 apples x -5 = -25 apples (negative 25 apples) which do not exist in the real physical world
Or -5 x 5 apples = negative 5 times of 5 apples is such a irrational expression
The number of times of a quantity of a form of matter, an action, an experience can never be negative.
Ex: it is nonsense to say that you eat negative 2 times everyday, or you have been to London negative 20 times
If negative x negative could actually be positive, then you would get an even more irrational expression:
Since + 5 x +5 = + 25 , and -5 x -5 = also +25 , -5 x -5 = +5 x +5 , and thus -5 = +5
which obviously is nonsense, when applied to the real physical world, such as
-5 apples = +5 apples, despite the fact that -5 apple represent a decrease, whereas +5 apples stands for an increase in the number of apples, say in a box)for 15 apples, and the number of 15 apples -5 apples decreases to 10 apples, which is opposite to 15 apples + 5 apples increases the number of apples to 20 apples
All complex numbers invented by human mathematicians come from their misunderstanding of negative sign and positive sign which do not have any actual value, being used as symbolic labels for humans to distinguish the difference between a decrease and an increase in quantity of a form of matter and the difference between an actual quantity without any number sign, and a decrease or an increase in its quantity value in human mind 's notional operations
There is no single point in the entire infinite space of universe which can actually exist in such non-existent being as human -invented notion of zero or nothing or emptiness, even in a volume of vacuum space. The only things which can actually be nothing or zero are the non-existent which do not occupy any space of the universe. A point of vacuum space itself also occupies a spot in the universe, and similarly an extremely small particle, a virus invisible to human naked eyes are also representations of space occupation, and any form of matter which exists in the universe always occupies a space , including light particles. Hence, what human invented notion of 0 can actually represent is the abscence or invisibility of a form of matter in their sight when they cannot see it, and the starting point of all quantities, and thus 0 is not any specific quantity, like infinity. O apple is not any quantity as how zero is always fundamentally misunderstood by human mathematicians, as a number in human -invented notions of number systems, while merely standing for the abscence of apples, and any other forms of matter.
Neither zero nor infinity can actually be used in any arithmetic operations as how they are always misused in human manipulations of the flaws of their own invented notion of mathematics, being not any actual specific quantity
0^0 is and any mathematical operations with 0 are nonsensical expressions, and evaluation of a mathematical expression from negative infinity to positive infinity in calculus is even more absurd, while the max quantity which human can understand their own invented notion of quantity is 10^31 which an extremely extremely small quantity, compared to such a boundless quantity as infinity, there can be only one infinity which is the largest value which has no boundary, no beginning nor any end. The only one form of matter in the physical universe which can be infinity is the infinite space of the physical universe. All other forms of matter have to be finite, in order to be in the universe. Human mathematicians ' fundamental misunderstanding of their invented notions of number systems and mathematical operations comes from their own invented coordinate systems , in which they represent such non existent numbers as negative numbers, positive numbers, and zero, in which they ignorantly claims that infinity has a beginning which is the origin at 0, despite the fact that infinity has no boundary or no beginning nor any end, and there is no way for human extremely limited knowledge to locate the location of the Earth, this star system, this galaxy in the infinite space of the physical universe, but ignorant human astrophysicists keep talking as if the infinite universe were a pond in their backyard
No form of matter can ever actually resist in any 1-D or 2-D space as how such non-existent space dimensions are always based to express human subjective understanding in mathematics. All forms of matter exis in 3-D, and any space dimensions more than 3-D are representations of human wild imagination and ignorance of the physical universe.
All human -invented notions of mathematics are human constructs, and can be applied to only within subjective reality created by human extremely limited knowledge, its subjective understanding, and no where else. All human current destructive development is a living proof of how human species have got from fundamental misunderstanding of their own invented notions of mathematics to misunderstanding of their place on this planet , behaving as if they had created themselves , despite the fact that humans are merely a product of the laws of nature, and have to follow the laws of nature if they don't want to be eliminated or destroyed by the laws of nature
What are the exact criteria for establishing whether a value is hyperbolic? Could there be infinitely many hyperbolic values?
Great discussion!
Of course it's about the Mandelbrot set!
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.
Brilliant isn't brilliant. Holly is brilliant!
-3/2 is located between the cardoid and the circle (on the x - real axis)?
These interations are very easy to understand. If f(x) is the function, then we have....
f(f(f(f(x)))) = f(f(f(x)))
Everytime you use f(x), the value changes less and less. The left hand side has one more f(x), but the change is so small that it doesn't matter. If you repeat this infinitely, both sides become the same. If this is true, you can treat the last f(x) on the left side as if it doesn't exist and both sides become the same.
That's why the fix point theorem works! This has a lot of fancy proves, but you can only prove the theorem, if you understand what I showed!
I like to think that mandelbrot and julia set are mathematic visual representations of the edges of infinity. Is this a valid view?
I love when the plot twist is FRACTALS! 😊
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!
Density of hyperbolicity.. that is suuuper cool.
6:42 - oh, good! Because I've thought about trying, and... it seemed daunting. Now I can just leave it to Holly and the other mathematicians to puzzle on, and not worry about it. :D
(But if I happen to figure something out next time I'm playing with some mandelbrot or related code, I'll let y'all know. :D)
Is it possible to find a number for which the process has an infinite number of limit points?