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Thanks for the video! I have not read the paper yet (dont search libgen to avoid the paywall by the way) but one thing i feel is missing is the explanation on why the solutions do not stay close together which i think is an important property of chaos.

What about when r=1+√8 (= 3.8284..)? According to the bifurcation diagram there is a stable 3-periodic cycle without chaos around that value.

Nice video! One thing I'd like to note is that I was kinda lost a bit starting at 4:06 when both axis had the same intervals a-b-c, implying the interval is the same on x and F(x). Am I just dumb to imply this or is this a possible oversight? If the latter, I think just having capital A-B-C on either axis could be a little helpful to make sure the viewer does not imply the interval lengths being equal.

Thank you for the amazing video!

Thank You

I have been waiting for a long time and i am happy that you have just restarted😃

Ha! I also made a video on population dynamics for SoMEπ! Was a really nice explanation!

my feeling. right now.

Yes, the brightness of the area inside the rainbow is often dramatic compared to the darkness around it. This was a very good explanation for why that is.

So then why does a double rainbow have reversed colors comparatively to the original rainbow. This wouldnt explain this if this were true both rainbows color pattern would be the same. It almost like the truth is they're reflecting of a glass dome like structure above us...hmmmm

Great explanation. Thanks.

Wow wtf

Nice video, but I think the way you explained it is a little misleading in the end. The way you said it makes it seem like the actual rainbow that we see in the sky is made of the light reflected by the droplets at the center of it. But actually, every droplet in the sky creates the circular iris that you showed, and the final rainbow is a composition of light coming from these millions of irises.

This only half explains it though. There isn't one water droplet hanging in mid air. So how is it formed when there are millions of raindrops constantly moving? Ontop of that, why do rainbows only tend to appear after the rain has stopped?

This is the windiest video ive ever watched

I have, once, seen 3 independent rainbows in the sky at the same time... I was 12 years old and it was about 3 days after the local area had recorded about 1½ times its average annual rainfall in 36 hours, filling 2 normally dry lakes nearby... The morning was extremely still, the smoke from the neighbour's farmhouse (4 miles to the north) was going straight up. I was heading to school and was at a point about ½ way between the two lakes, the sun rising over the hills behind me ... The largest and clearest of the rainbows was directly west of me... The other two were _about_ 22° to the north & south of that line and both looked "dirty" as if there was some mud mixed in with the rainbow... It took me a while to realise that the surfaces of the two lakes were probably the light sources for these and *they* were very muddy bodies of water which explained the colour mix.

演算法帶我來了，解說很清楚

Please tell me how you know what I think.

This reminds me of the legendary lecture on rainbows by Walter Lewin (highly recommend it too as a complement of this one! czcams.com/video/iKUSWJWMSk4/video.html ). But your video goes directly to the point and is really well made, good job!

Excellent explanation ❤

woaaaaaaaaaaaaaaahhhhhhhhhhhhhhhhh

There is actualy always two rainbows . One is brighter, the other is less visible

Very nice but why no reference to Lord Kelvin at the university of Glasgow who solved this blooming years ago? He also built a mechanical machine to work out tides anywhere on the planet.

Ga je nog een tweede video over dubbele regenbogen maken?

For a fairly similar reason Starlink Satellites are easier to access around 53 degrees north latitude. (I.e., easier to access in Canada than in the USA!)

I knew about the brighter inside and also correctly understood that is because of how the rays reflect. But this is a much more compete explanation.

Wonderful

Color of rainbow is finite (isnt infinite).

Incredible video

Another effect is that more light hits near the edge of droplets than at the center. This is because there's more area at a larger radius (the area integral formulation of a circle being r*dr*dθ). Your visual integration of the alpha-vs-d graph seems to be missing this effect. Taking it into account would give more weight to light reflected at a higher d, making the rainbow itself appear even brighter than the interior region. You could show this by plotting alpha vs d^2 then doing the visual integral on that.

Intresting, and how would you explaine duble rainbows?

I wondered why the inside is brighter. Great 360 example: czcams.com/users/shortscwQbEo75k70?feature=shared

I've never before a good explanation of why the inside of a rainbow is brighter than the outside. Thanks!

Excellent video and explanation!

How do YOU know what people think?????????

This video shows the second level of explanation. The next level would take into account the degree of internal reflection within a raindrop and polarizing light.

And a double rainbow is light that comes in and has one more reflection in the droplet

This works exactly how i thought they worked. We did the refraction calculations in calculus and it was really interesting to learn

Thank you, rainbows are neat! The arc of the rainbow describes the base of a light cone whos apex is your eye. A person standing next to you sees a different rainbow.

life, the universe, and everything

Челхарош

Ok what about double Rainbows one small and one larger? I have the proof

But what about the color reversal for double and triple rainbows?

A rainbow viewed from above e.g. in an airplane or on a mountain you can see the bright center of the rainbow created by direct reflection in the droplets. In the very middle you can see the shadow of the airplane or your mountain.

I didn’t know rainbows worked. I thought they mostly are unemployed.

You should have considered the graph of the energy of the light returned to the observer, taking into account the division of light at the boundaries of the media (including the effect of total internal reflection from the rear surface of the drop).

Rainbows dont work at all, they're sunbathing. 🤓

You mean there's no pot of gold?? Bugger!

I guess Mats Verneeren found the rainbow connection, I hope Kermit isn’t disappointed