An important distinction: the perimeter is infinite only for the theoretical fractal. Any real world representation will have a finite perimeter limited by the resolution at which the detail can be measured. This is also true for similar situations like measuring the length of coastlines.
I would say that mathematics is Platonic. The real object is the "theoretical" or "ideal" one. Its representations in the "physical" world are just "shadows", always incomplete. So the finite perimeter is a limitation of the physical world, not the "real" one. The same can be said about all geometric objects, and real numbers (let alone complex ones). Maybe even negative numbers. One could argue that large natural numbers are also "ideal", since a googolplex is larger than the number of atoms in the observable universe. Math is a superpower. It allows us to see the realty behind the physical world.
@@anaravena That's an interesting perspective. I see it more the reverse, where math tells us the rules that the world would follow perfectly if it could. It's probably more useful to see it your way in a classroom setting, whereas mine is more suited to science and engineering applications.
My 25 years of experience as engineer taught me that nature follows mathematical rules, that we partially know. The limitation is ours, not of nature. But this is an open debate. My dad, also an engineer, gave me the same answer as you when we discussed this topic some years ago. In this particular case (the Mandelbrot set) we are talking about a geometrical object, so the "reality" is the ideal, not the set of pixels that we can represent on a computer screen. It cannon be correctly represented in a computer, the same way that the length of the diagonal of a 1x1 square cannot be measured in the physical world, since sqrt(2) is irrational. In the physical world, Pythagoras theorem is not correct. It is even worse if you try to measure the area of a disc, let's say of radius 1. This time the value is trascendental. In all these cases what we do in the physical world are just approximations to the reality, which we can only know using the theory we call Mathematics. Theoretical does not mean "fake", on the contrary. There is nothing more practical than a good theory. In engineering we use derivatives and integrals, and we use the "theoretical" functions as much as possible. We only fail back to numerical methods when there is no analytical solution. Numerical derivatives are usually useless. We could (and maybe should) discuss longer, but it seems too large for this margin note.
Great vid! In light of this mathematical object with finite area but infinite perimeter, would you explore a video discussing paradox of Gabriel's horn which has a similar properties only reversed? (Infinite surface area, finite volume)
Excellent video! You are underrated - keep making such quality videos and one day you will become big, only one condition - never stop, unlike most people!
A conjecture is that if you run 0 through f(z) = z² + c an infinite number of times, keeping c as just a variable, you will get an infinite series of terms, where the nth term is c raised to the power n multiplied by the (1 + n)th Catalan number. And the locus of points where the absolute value of this infinite series equals 2 is precisely the boundary of the Mandelbrot set.
Probably naive questions: How is it known that the complexity continues to change, and that the patterns continue to occur no matter how far you zoom in? In other words, how can you tell what's happening as the gaps between values of c become arbitrarily small? On a related note, how are the highly detailed pictures and animations of zooms I've seen actually produced in practice? Is it just larger and larger numbers of calculations having been performed as values of c become closer and closer together? Or are there any other methods employed? Thanks 😀
these are great questions: 1) The rigorous math understanding of how complex the Mandelbrot Set boundary is has to do with something called the Hausdorff dimension, which is a measure of how rough/complex an object is. The Hausdorff dimension of the Mandelbrot Set was proved to be 2 (in 1998), whereas it's topological dimension is 1. This basically means it is as complex as can be. I recommend googling Hausdorff dimension and Fractal dimension if you want to learn more about it. It's quite interesting. 2) The highly detailed pictures and zooms are produced through a mix of larger iterations and also high pixel density. You also need to pick appropriate points to zoom in on. I've played around with this a bit but sometimes end up zooming in to areas where I "lose" the points that are in the set. I plan on making a video related to zooms and how they are produced in the near future, so stay tuned!
An important distinction: the perimeter is infinite only for the theoretical fractal. Any real world representation will have a finite perimeter limited by the resolution at which the detail can be measured. This is also true for similar situations like measuring the length of coastlines.
well said!
I would say that mathematics is Platonic. The real object is the "theoretical" or "ideal" one. Its representations in the "physical" world are just "shadows", always incomplete. So the finite perimeter is a limitation of the physical world, not the "real" one.
The same can be said about all geometric objects, and real numbers (let alone complex ones). Maybe even negative numbers. One could argue that large natural numbers are also "ideal", since a googolplex is larger than the number of atoms in the observable universe.
Math is a superpower. It allows us to see the realty behind the physical world.
@@anaravena That's an interesting perspective. I see it more the reverse, where math tells us the rules that the world would follow perfectly if it could. It's probably more useful to see it your way in a classroom setting, whereas mine is more suited to science and engineering applications.
Because you can only carry out the iterations a finite number of times for each _c_ value.
My 25 years of experience as engineer taught me that nature follows mathematical rules, that we partially know. The limitation is ours, not of nature. But this is an open debate. My dad, also an engineer, gave me the same answer as you when we discussed this topic some years ago.
In this particular case (the Mandelbrot set) we are talking about a geometrical object, so the "reality" is the ideal, not the set of pixels that we can represent on a computer screen. It cannon be correctly represented in a computer, the same way that the length of the diagonal of a 1x1 square cannot be measured in the physical world, since sqrt(2) is irrational. In the physical world, Pythagoras theorem is not correct. It is even worse if you try to measure the area of a disc, let's say of radius 1. This time the value is trascendental.
In all these cases what we do in the physical world are just approximations to the reality, which we can only know using the theory we call Mathematics. Theoretical does not mean "fake", on the contrary. There is nothing more practical than a good theory.
In engineering we use derivatives and integrals, and we use the "theoretical" functions as much as possible. We only fail back to numerical methods when there is no analytical solution. Numerical derivatives are usually useless. We could (and maybe should) discuss longer, but it seems too large for this margin note.
Great vid! In light of this mathematical object with finite area but infinite perimeter, would you explore a video discussing paradox of Gabriel's horn which has a similar properties only reversed? (Infinite surface area, finite volume)
Great suggestion! I'll add it to my list of future planned videos.
Excellent video! You are underrated - keep making such quality videos and one day you will become big, only one condition - never stop, unlike most people!
thank you for the kind comment! I don't plan on ever stopping :)
A conjecture is that if you run 0 through f(z) = z² + c an infinite number of times, keeping c as just a variable, you will get an infinite series of terms, where the nth term is c raised to the power n multiplied by the (1 + n)th Catalan number. And the locus of points where the absolute value of this infinite series equals 2 is precisely the boundary of the Mandelbrot set.
I actually haven't heard this before. Thanks for sharing, seems very interesting!
Probably naive questions:
How is it known that the complexity continues to change, and that the patterns continue to occur no matter how far you zoom in? In other words, how can you tell what's happening as the gaps between values of c become arbitrarily small?
On a related note, how are the highly detailed pictures and animations of zooms I've seen actually produced in practice? Is it just larger and larger numbers of calculations having been performed as values of c become closer and closer together? Or are there any other methods employed?
Thanks 😀
my ans to the 2 question is that computers are probably just strong enough to approximate the set very precisely
these are great questions:
1) The rigorous math understanding of how complex the Mandelbrot Set boundary is has to do with something called the Hausdorff dimension, which is a measure of how rough/complex an object is. The Hausdorff dimension of the Mandelbrot Set was proved to be 2 (in 1998), whereas it's topological dimension is 1. This basically means it is as complex as can be. I recommend googling Hausdorff dimension and Fractal dimension if you want to learn more about it. It's quite interesting.
2) The highly detailed pictures and zooms are produced through a mix of larger iterations and also high pixel density. You also need to pick appropriate points to zoom in on. I've played around with this a bit but sometimes end up zooming in to areas where I "lose" the points that are in the set.
I plan on making a video related to zooms and how they are produced in the near future, so stay tuned!
@@AbideByReason thanks for the reply. I'll check that out 👍 I'm subbed, so I'll look forward to the video on zooms!
Wonderful thank you very much❤
you're welcome!