Great video! I was among those who got the mistaken impression that that bifurcation diagram over the Mandelbrot set was identical to the logistic one; glad you cleared that up.
thanks! that was my primary goal. if you're curious, I've linked another video in the description that talks about a relationship specific to the Mandelbrot Set & the logistic map
What a fantastic take on this relationship! You did a great job of defining each map and explaining the relationship between the two (or lack thereof 😅). I especially enjoyed the comparison table you created at 9:56. Please continue to create these videos!
I've known about bifurcation diagrams for a while, but I never thought to think about what they might look like over a plane of inputs rather than a line. Pretty cool stuff! Thank you for introducing me to stuff that I'd otherwise never think about!
I was on the fence but leaning to "wrong" on the opening question. Side note: In the Sep 1991 edition of Scientific American, there was an exploration of the Lyapunov Exponent under periodic forcing, where r-values (fecundity) were forced between values on a plane during the iterations. The titular exponent for a given point was just an accumulator of the log of the changes, divided by iteration count. But nonlinear behavior forced over an input plane does make for interesting visuals.
visualising the orbits of the IFS (what the escape time fractal accomplishes, among others) lets us see the chaotic regions of the bifurcation diagram as the boundary of the fractal (the 'needle' is where the boundary coincides with the real line).
OMG! First of all: Thank you so much! I was pondering this for a while now (basically since the Veritasium video) and you just answered all my question in one video. 🙏 Secondly: This was by far the *most beautiful* logistic mapping I can imagine. 🤩 Gonna leave some more random enagement to worship the algorithm 😄
Great video, and I really liked the visuals used to explain the iterates for the mandelbrot. One of my favorite ways to render the mandelbrot set is by tracing the individual iterates and incrementing a counter on the pixels they land on, meaning the brightness of pixels corresponds to the intensity of C values that pass through it. It's also sometimes known as "buddhabrot" due to the appearance when rendering the standard z=z^2+c.
ah yes I'm familiar with those, they're very cool! Similar to the 3D bifurcation diagram in the way that you keep track of what the iterates are doing. I haven't tried to make them myself yet, though
@@desden0va At 7:23 you said "All the C values that generated iterates that diverged are not plotted". All you need is to instead only plot those iterates that _do_ diverge, and render a density plot showing where the iterates spend the most time.
Veritasium did not say that the logistic map is hidden in the mandelbrot set, he said that the _bifurcation diagram_ is hidden in the Mandelbrot set, and he also stated how any single-hump function produces a bifurcation diagram.
I know, yes. He never said anything incorrect. But I think people got the impression that the logistic map is in the Mandelbrot set (either from that veritasium video, the numberphile video on the Feigenbaum constant, or somewhere else), because there's numerous comments on my previous video where people say that the logistic map IS the real line of the Mandelbrot set.... and I wanted to set things straight. I wanted to talk more about the relationship between the two, and what kind of functions create bifurcation diagram, but frankly I ran out of time
Rewatching the veritasium video, he does seem to imply that it’s the *same* bifurcation diagram instead of just a scaled version made from a similar generating function.
People put too little thought into plotting complex numbers. Unless these are exponents, the magnitude is much more useful than the real part. You can also use colour to visualise the argument, so you have full information in the plot. These plots work actually really well for recognising patterns.
I'd like to see an animation where you input a complex number using the mouse and can watch the 3D image change. I guess it's hard in real time, but you could do a low-res version or generate the video offline.
wait does this mean every other map has its own 3d and 2d fractal? could u make a vid showing the "quartic 2d fractal" etc? idk how to generate the myself and they probably arent to view anywhere so please make them so we can see more beautiful fractals
@@desden0vaI enjoyed your video. tMS & chaos theory were all the rage when I was at college for engineering. thx for this... I didn't have time for learning much about either back then. ❤
@@john-ic5pzI also went to college for engineering, I just like learning about fractals and chaos for fun.... therefore I can't be 100% sure that I'm right about everything, but I tried!
11:50 Maybe they just don't have their own constants? Not every series has to converge on anything. You say "presumably they do" but none of the information given in this video seems to suggest that.
Is the gauss map just the Gaussian function? I´m pretty sure that it has a single locally quadratic maximum, so i think the feigenbaum constant should apply to that as well. (Great video btw)
I would have thought so too! But from what limited tests I was able to do, it seems to be a different value. Of course, I was brute forcing, but in practice the constant is calculated with number theory techniques that I just don't know. I could certainly be wrong (and thanks!)
@@desden0va I also did some brute forcing with a little bit of python and i´m getting this sequence of the fractions : 7.917429336877362, 4.000552915070775, 4.263276002700649, 4.551692897999981, 4.64052662880076, 4.603252287245357, 5.132284921364028. Im pretty sure the last one and maybe also the second to last one is off because of numerical errors. But this might be converging the the feigenbaum constant. I now understand why you said it´s hard to compute these values numerically. The bifurcations get so close together. Do you still know what numbers you got for the gauss map?
I couldn't find my old work but after trying again, I got 2.38423, 3.32168, 4.13931, 4.47343, 4.6, 5 what value of alpha did you set the map at? I left it at 5 and generated Lyapunov Exponents with beta ranging from -1 to -0.5886 (any bigger and the LE goes positive, indicating chaos, but we wanna stay in the stable zone) so, it seems my initial testing that I described in the video was flawed, and the gauss map seems to have the same feigenbaum constant - good to know! Python is probably better at calculating this sort of thing (though it's also likely that your python skills surpass my mathematica skills, lol)
@@desden0vainteresting, i also used alpha=5 but i started with b at 0.5 end went left to about -0.4, so it makes sense that we have different values. The chaotic region is at around -0.6 to -0.4. maybe ill try going from -1 to -0.6 later.
Great video! I was among those who got the mistaken impression that that bifurcation diagram over the Mandelbrot set was identical to the logistic one; glad you cleared that up.
thanks! that was my primary goal. if you're curious, I've linked another video in the description that talks about a relationship specific to the Mandelbrot Set & the logistic map
ur such a hottie for explaining maths in a digestible manner
What a fantastic take on this relationship! You did a great job of defining each map and explaining the relationship between the two (or lack thereof 😅). I especially enjoyed the comparison table you created at 9:56. Please continue to create these videos!
Thank you very much, I appreciate that! I definitely have more topics I want to make videos of
@@desden0va And again: Those rainbow points going all logistic map… just beautiful! 🤗
I've known about bifurcation diagrams for a while, but I never thought to think about what they might look like over a plane of inputs rather than a line. Pretty cool stuff! Thank you for introducing me to stuff that I'd otherwise never think about!
glad you enjoyed it!
I was on the fence but leaning to "wrong" on the opening question.
Side note: In the Sep 1991 edition of Scientific American, there was an exploration of the Lyapunov Exponent under periodic forcing, where r-values (fecundity) were forced between values on a plane during the iterations. The titular exponent for a given point was just an accumulator of the log of the changes, divided by iteration count. But nonlinear behavior forced over an input plane does make for interesting visuals.
@@casnimotmy previous video is about Lyapunov fractals
This is good work. Very thorough. I hope you win.
visualising the orbits of the IFS (what the escape time fractal accomplishes, among others) lets us see the chaotic regions of the bifurcation diagram as the boundary of the fractal (the 'needle' is where the boundary coincides with the real line).
OMG! First of all: Thank you so much! I was pondering this for a while now (basically since the Veritasium video) and you just answered all my question in one video. 🙏
Secondly: This was by far the *most beautiful* logistic mapping I can imagine. 🤩
Gonna leave some more random enagement to worship the algorithm 😄
Great video, and I really liked the visuals used to explain the iterates for the mandelbrot.
One of my favorite ways to render the mandelbrot set is by tracing the individual iterates and incrementing a counter on the pixels they land on, meaning the brightness of pixels corresponds to the intensity of C values that pass through it. It's also sometimes known as "buddhabrot" due to the appearance when rendering the standard z=z^2+c.
ah yes I'm familiar with those, they're very cool! Similar to the 3D bifurcation diagram in the way that you keep track of what the iterates are doing. I haven't tried to make them myself yet, though
@@desden0va At 7:23 you said "All the C values that generated iterates that diverged are not plotted". All you need is to instead only plot those iterates that _do_ diverge, and render a density plot showing where the iterates spend the most time.
stunning work you're doing amazing
thank you for doing this. I've been wondering about this since the veritasium video
I just left basically the same comment 🤗
Small correction, but at 10:54, the tent map is u*min(x, 1-x), not u*min(x-1)
yep, I've since realized that... luckily just the label is wrong and the bifurcation diagram is correct
@@desden0va _luckily_ 😄
thank you! … great work!! … awesome stuff!!!
really nice work, loved the whole thing :)
Neat! Really nice work.
Thank you, you cleared my exact doubt
This has turned out to be a great format🏆 2 out of 2🏆
YES! someone else finally made a video on this! and also, this is a very good explanation! now i finally understand
And so my week is piqued. Nice.
thx for the new upload bro love ur content its so interesting❤
thank you!!
Veritasium did not say that the logistic map is hidden in the mandelbrot set, he said that the _bifurcation diagram_ is hidden in the Mandelbrot set, and he also stated how any single-hump function produces a bifurcation diagram.
I know, yes. He never said anything incorrect. But I think people got the impression that the logistic map is in the Mandelbrot set (either from that veritasium video, the numberphile video on the Feigenbaum constant, or somewhere else), because there's numerous comments on my previous video where people say that the logistic map IS the real line of the Mandelbrot set.... and I wanted to set things straight. I wanted to talk more about the relationship between the two, and what kind of functions create bifurcation diagram, but frankly I ran out of time
Rewatching the veritasium video, he does seem to imply that it’s the *same* bifurcation diagram instead of just a scaled version made from a similar generating function.
@@Archimedes115 totally! And that's how I and a lot of others understood it.
People put too little thought into plotting complex numbers. Unless these are exponents, the magnitude is much more useful than the real part. You can also use colour to visualise the argument, so you have full information in the plot. These plots work actually really well for recognising patterns.
I'd like to see an animation where you input a complex number using the mouse and can watch the 3D image change. I guess it's hard in real time, but you could do a low-res version or generate the video offline.
Thank you so much for your work, it's truly useful. Tere is any real explanation of this topic on the internet so thanks 😊
wait does this mean every other map has its own 3d and 2d fractal? could u make a vid showing the "quartic 2d fractal" etc?
idk how to generate the myself and they probably arent to view anywhere so please make them so we can see more beautiful fractals
Cutting it close with the deadline I see
This was very nice :D
So the logistic map is hidden twice in a DOUBLE Mandelbrot set? Wow!
Awsome video just like the previous one
Tent map formula at 10:54 is wrong. It should be µ*min(x_n, 1-x_n); min(x_n, x_n-1) is just x_n-1 identically.
aw shoot you're right, good catch. My graph and bifurcation diagram is correct but the label is wrong
@@desden0vaI enjoyed your video. tMS & chaos theory were all the rage when I was at college for engineering. thx for this... I didn't have time for learning much about either back then.
❤
@@john-ic5pzI also went to college for engineering, I just like learning about fractals and chaos for fun.... therefore I can't be 100% sure that I'm right about everything, but I tried!
I had no idea they were related...brain hurts.
Since I read about the
I’m going to use Mandelbrot and bifurcation diagram in Desmos.
11:50
Maybe they just don't have their own constants? Not every series has to converge on anything.
You say "presumably they do" but none of the information given in this video seems to suggest that.
そんな組み合わせもあるのですね
question if the mandelbrot set period doubling ratio is 4 then how is it a projection of a diagram where the ratio is 4.669
where did you hear that the Mandelbrot Set period doubling ratio is 4? I'd like to read about that
@@desden0va i was wrong 💀☠️☠️
LOL no prob
Hi :) which lenguange and interface did you use for the graphics? thanks.
Mathematica and Shadertoy
@@desden0va Thank you! :) Are you providing some online lessons? Thanks.
I wasn't planning on it, but I might someday 🤔
Because the Mandelbrot set is hiding in the logistic map...
Do bifurcation diagram Mandelbrot in desmos
Well done
Is the gauss map just the Gaussian function? I´m pretty sure that it has a single locally quadratic maximum, so i think the feigenbaum constant should apply to that as well. (Great video btw)
I would have thought so too! But from what limited tests I was able to do, it seems to be a different value. Of course, I was brute forcing, but in practice the constant is calculated with number theory techniques that I just don't know. I could certainly be wrong (and thanks!)
@@desden0va I also did some brute forcing with a little bit of python and i´m getting this sequence of the fractions :
7.917429336877362, 4.000552915070775, 4.263276002700649,
4.551692897999981, 4.64052662880076, 4.603252287245357, 5.132284921364028.
Im pretty sure the last one and maybe also the second to last one is off because of numerical errors. But this might be converging the the feigenbaum constant. I now understand why you said it´s hard to compute these values numerically. The bifurcations get so close together. Do you still know what numbers you got for the gauss map?
right, it's hard to get enough precision to calculate the ratio. I probably do still have the data, I'll take a look when I get home later
I couldn't find my old work but after trying again, I got 2.38423, 3.32168, 4.13931, 4.47343, 4.6, 5
what value of alpha did you set the map at? I left it at 5 and generated Lyapunov Exponents with beta ranging from -1 to -0.5886 (any bigger and the LE goes positive, indicating chaos, but we wanna stay in the stable zone)
so, it seems my initial testing that I described in the video was flawed, and the gauss map seems to have the same feigenbaum constant - good to know! Python is probably better at calculating this sort of thing (though it's also likely that your python skills surpass my mathematica skills, lol)
@@desden0vainteresting, i also used alpha=5 but i started with b at 0.5 end went left to about -0.4, so it makes sense that we have different values. The chaotic region is at around -0.6 to -0.4. maybe ill try going from -1 to -0.6 later.
Thanks.
Too bad no one in #SoME3 decided to do a video on Misiurewicz points…
maybe next year! that'd be a good topic
isn't the logistic map is just the Mandelbrot set evaluated over the x-axis with the imaginary part having value 0 ?
no
Click the circle😆
According to Veritasium and Numberphile, yes.
dude just straight up stole that graphic from Veritasium
I wish you success with the channel,your work is really good .
are you really not a mathematician? Then double kudos.
❤
thank you! I'm just a software engineer who likes math
@@desden0va _just_ a software engineer 😄