I love it when people make intuitive explanations for theorems like these. True, it's not a rigorous proof, but it is more useful if you want to actually understand a concept and connect it to what you already know (like the intermediate value theorem in this case).
Wonderful explanation but I think in the tea cup example the fixed point is only valid for a given time. Every time will have some fixed point. The fixed point will appear to be moving, like the eye of the hurricane.
I enjoyed the vector animations, they were very well done. However, I'm worried the assumption you make at 3:45-4:15 is incorrect. In particular, there is no guarantee that we may trace a continuous path of vertical vectors (or, transform continuously between the vertical vectors) in the manner you show, for arbitrary functions satisfying the BFT hypotheses. Indeed, any curve we trace from the 'leftward mapping' to the 'rightward mapping' vectors must contain a point which maps vertically, as the IVT immediately shows. But, there is no guarantee that we can trace a (continuous) curve, comprised of such points, with endpoints mapping in opposite directions; it is *not* in general true that to any point 'x' which maps vertically, a sequence of points that map vertically and that converge to 'x' can be found on subsequent curves connecting the left and right pointing vectors. In other words, there is no guarantee that, e.g., an initial curve of downward mapping points will not suddenly terminate in your construction.
0:39 That's a weird analogy, mixing feels like a pretty discontinuous map. In fact for point particles the probability that that happens is zero but eh, now I'm being overly picky tbh. I think Brouwer did think about this from observing that one point in the coffee (as ripples pass around) is always perfectly still? Something like that. I really liekd the video. Subbed!
You got me lost at the moment you drew that line(orange and blue) on a circle. Why is it that curvy? Is it random? and also what does this line represent?
@@stemcell7200 Fixed points = duality! Photons, light, null rays or the electro-magnetic field = fixed points! Y = X. ct = x where c = 1 implies t = x. Y is equal to X, Y is the same, similar, equivalent or dual to X. Y is dual to X! Photons or light are dual, electro is dual to magnetic! All mathematical equations are dual! f(x) = ct = x, Brouwer's fixed point theorem proves that null rays or light are dual! Duality creates reality! The velocity of light is the same and equal for all observers hence fixed points conform to a principle of objective democracy. "Always two there are" -- Yoda.
@@stemcell7200 Duality implies the following:- What is dual to entropy? Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! "Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy. Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Great video!
What are the prerequisites for this video? I don't understand. “Points that stay the same after some transformation…” how can a point stay the same after being transformed?
Great video, but I don’t understand why a vector that goes after the one pointing up can’t just point down. In 1D I when a line is tilted right the next one can just tilt left to the previous point.
Stirring can be approximated as a continuous transformation (of coffee molecules' positions) from the original volume into itself, at least if it is stirred very gently. Shaking is neither continuous nor volume preserving.
Photons, light, null rays or the electro-magnetic field = fixed points! Y = X. ct = x where c = 1 implies t = x. Y is equal to X, Y is the same, similar, equivalent or dual to X. Y is dual to X! Photons or light are dual, electro is dual to magnetic! All mathematical equations are dual! f(x) = ct = x, Brouwer's fixed point theorem proves that null rays or light are dual! Duality creates reality! The velocity of light is the same and equal for all observers hence fixed points conform to a principle of objective democracy. "Always two there are" -- Yoda.
Because it's a three dimensional fluid with independent molecules that have no reason to stay where they are. The theorem works in three dimensions for sure, just not with something like coffee.
I love it when people make intuitive explanations for theorems like these. True, it's not a rigorous proof, but it is more useful if you want to actually understand a concept and connect it to what you already know (like the intermediate value theorem in this case).
I've always loved this theorem ❤️
Yeah it's an interesting one!
Banach fixed point theorem is better imo
That is a thorougly and very clear explanation. Thank you so much 🙂
Now that's a different cup of tea. Kudos for my fellow country man Bertus Brouwer, and for your video. Love it.
great job!
I think it's a crime not to teach Maths concept like this in the first iteration. Only later can the formulas come.
Wonderful explanation but I think in the tea cup example the fixed point is only valid for a given time. Every time will have some fixed point. The fixed point will appear to be moving, like the eye of the hurricane.
I enjoyed the vector animations, they were very well done. However, I'm worried the assumption you make at 3:45-4:15 is incorrect. In particular, there is no guarantee that we may trace a continuous path of vertical vectors (or, transform continuously between the vertical vectors) in the manner you show, for arbitrary functions satisfying the BFT hypotheses.
Indeed, any curve we trace from the 'leftward mapping' to the 'rightward mapping' vectors must contain a point which maps vertically, as the IVT immediately shows. But, there is no guarantee that we can trace a (continuous) curve, comprised of such points, with endpoints mapping in opposite directions; it is *not* in general true that to any point 'x' which maps vertically, a sequence of points that map vertically and that converge to 'x' can be found on subsequent curves connecting the left and right pointing vectors.
In other words, there is no guarantee that, e.g., an initial curve of downward mapping points will not suddenly terminate in your construction.
This is a great video
Cheers!
you ate with the animation and explanation
Didn't understand it on text. Now i do. Thanks.
Beautiful!
Such a cool explanation, i love it!
Thanks!
great vedio.keep it up
btw,,which software did you used
Thanks! The software is manim, same as in 3blue1brown
this is beautiful . thank you
The fixed point in a circle reminds me of a hurricane eye. I wonder if there is a correlation?
This is a neat theorem
I find in interesting how intuitive it is in 2d (maps) but not in 3d (tea)
0:39 That's a weird analogy, mixing feels like a pretty discontinuous map. In fact for point particles the probability that that happens is zero but eh, now I'm being overly picky tbh. I think Brouwer did think about this from observing that one point in the coffee (as ripples pass around) is always perfectly still? Something like that.
I really liekd the video. Subbed!
You got me lost at the moment you drew that line(orange and blue) on a circle. Why is it that curvy? Is it random? and also what does this line represent?
very well explained!
Great demonstration!
Thanks!
@@stemcell7200 Fixed points = duality!
Photons, light, null rays or the electro-magnetic field = fixed points!
Y = X.
ct = x where c = 1 implies t = x.
Y is equal to X, Y is the same, similar, equivalent or dual to X.
Y is dual to X!
Photons or light are dual, electro is dual to magnetic!
All mathematical equations are dual!
f(x) = ct = x, Brouwer's fixed point theorem proves that null rays or light are dual!
Duality creates reality!
The velocity of light is the same and equal for all observers hence fixed points conform to a principle of objective democracy.
"Always two there are" -- Yoda.
@@stemcell7200 Duality implies the following:-
What is dual to entropy?
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
"Through imagination and reason we turn experience into foresight (prediction)" -- Spinoza describing syntropy.
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Randomness (entropy) is dual to order (syntropy, predictability).
Great video!
What are the prerequisites for this video? I don't understand. “Points that stay the same after some transformation…” how can a point stay the same after being transformed?
f(x)=x is the identity map, yet it is a transformation.
Nice explanation
Great video, but I don’t understand why a vector that goes after the one pointing up can’t just point down. In 1D I when a line is tilted right the next one can just tilt left to the previous point.
They do that but at the fixed point, that's why the surface must be continous so the transition from up to down is gradual
You have nice videos!
How do you make your animations?
They look very good
Stem cell ?
In another video a guy said that shaking the coffee would violate the theorem, is that true? why?
Stirring can be approximated as a continuous transformation (of coffee molecules' positions) from the original volume into itself, at least if it is stirred very gently. Shaking is neither continuous nor volume preserving.
I love topology
I hate it😕
Nice explanation! I am making rn the video about this theorem with manim too, but I am going to present the proof with the help of homology functor
Brilliant
Photons, light, null rays or the electro-magnetic field = fixed points!
Y = X.
ct = x where c = 1 implies t = x.
Y is equal to X, Y is the same, similar, equivalent or dual to X.
Y is dual to X!
Photons or light are dual, electro is dual to magnetic!
All mathematical equations are dual!
f(x) = ct = x, Brouwer's fixed point theorem proves that null rays or light are dual!
Duality creates reality!
The velocity of light is the same and equal for all observers hence fixed points conform to a principle of objective democracy.
"Always two there are" -- Yoda.
What software do you use to make these videos?
This video was done with something called manim
@@stemcell7200 3b1b's library?
Just wow for this .
Thanks
Pretty sure the coffee doesn't work though
how so
Because it's a three dimensional fluid with independent molecules that have no reason to stay where they are. The theorem works in three dimensions for sure, just not with something like coffee.
first
I was first