These animations, together with their 'complex analysis', are 'real' gems of mathematical 'imagination'! (all 'puns' intended)
@imaginaryangle
6 ай бұрын
Thank you!
@ILSCDF
6 ай бұрын
Superb, this is what youtube was made for
@imaginaryangle
6 ай бұрын
Thank you! 😊
@TheArtOfBeingANerd
6 ай бұрын
So... if we can visualize 10-dimensional space, we'll have a way to visualize all of these curves.
@TheArtOfBeingANerd
6 ай бұрын
But that's only taking real n into consideration. No idea how many dimensions it would be with complex n.
@imaginaryangle
6 ай бұрын
@@TheArtOfBeingANerd Let's count: +2 for the input, 2 for each of the seeds (+4), 2 more for each of the factors going into addition (+4), so 10 all together, you were right from the start!
@wellscampbell9858
6 ай бұрын
Good heavens, please be careful! You may get him thinking, and we won’t get any new material for years, then he’ll drop a dense 14 hour masterwork that’s best viewed with a volumetric display! :)
@deltalima6703
6 ай бұрын
This is really top tier stuff. I love how everything other than primary solutions are not "left as homework" or swept under a rug somewhere.
@imaginaryangle
6 ай бұрын
Thank you! It's always a challenge to work out representative examples and visualizations of these other solutions, so it makes me happy to see this noticed and appreciated.
@noahtaul
6 ай бұрын
16:10 For your own enjoyment: take S_n + (phi*w)*S_{n-1} + W*S_{n-2}=0, where w and W are the two third roots of unity, and you get a cycle of length 30. This is the longest cycle possible under certain conditions; namely, that the coefficients of the recurrence relation are in biquadratic fields (i.e. that they are combinations of products of elements of two fixed quadratic fields, here being Q(sqrt(-3)) and Q(sqrt(5))). There are a few order 24 ones, namely S_n = (1+sqrt(3))/sqrt(-2)*S_{n-1} + S_{n-2}, and actually 6 others.
@imaginaryangle
6 ай бұрын
Thank you so much! 🥰 Playing with these right now. I'm getting the 30-cycle when using S_n = (phi*e^(2/3pi i))*S_{n-1} + e^(1/3pi i)*S_{n-2} .. so instead of the two third roots of unity, it was any of the two complex ones and its negation.
@amitlanis3104
6 ай бұрын
if you make B=1, S0=0 and S1=1. you can define A=2i sin(pi/N) and get a cycle of length 2N (for any 2
@imaginaryangle
6 ай бұрын
Amazing find!
@jackrubin6303
6 ай бұрын
Amazing. You opened my mind at 67 years old to a realm I never imagined.
@imaginaryangle
6 ай бұрын
I'm very happy to hear that 😊
@wyboo2019
6 ай бұрын
22:21 you can see this sort-of cardioid-shaped curves (if you look at it like 4 hearts each in one of 4 orientations and overlapping). they are not actually cardioids as their cusp meets at a right angle instead of a 0 angle, except ive seen a VERY similar shape before. go watch Morphocular series on wheels and roads, and in his video for rolling wheels on wheels and calculating the ideal shape that rolls around a square, he gets a heart-shaped curve that meets at a right angle like these it would be REALLY really nice if these were the same shape
@imaginaryangle
6 ай бұрын
I'll have a look. There's a little something in an upcoming video about this 😉
@mrbananahead2005
6 ай бұрын
So, one of the things that I noticed and that isn’t really delved into is that at about 12:37 , the shape being formed is very clearly a spiral covering of a taurus or similar object projected onto a 2D plane and “viewed” from inside. I would be very interested in seeing some of these shapes discussed from the viewpoint of projective geometry.
@imaginaryangle
6 ай бұрын
That's an interesting idea I haven't really looked into.
@soupisfornoobs4081
6 ай бұрын
The continuous cycles you showed look a lot like polar graphs! Is there a real connection there, or is that just my imagination?
@imaginaryangle
6 ай бұрын
It's not just your imagination. And there's more coming up about that 😉
@kk__
6 ай бұрын
Do you have an updated desmos file for the graphs? cause I would love to try it out for myself
@imaginaryangle
6 ай бұрын
I didn't get around to doing it for the visuals in this video. I will reply to this message if I get it done so you get notified.
@imaginaryangle
6 ай бұрын
There is now a Desmos graph linked in the description for the combined Lucas and Fibonacci sequence with intersections and multivalued solutions. Enjoy :)
@quasicrystalslog-linmetric3068
6 ай бұрын
in physics, many of these properties occur in quasicrystal diffraction
@quasicrystalslog-linmetric3068
6 ай бұрын
in the diffraction, irrational parts of the indices translate as phase shifts in Euler's formula in the probe. This relates real space of the quasicrystal to the momentum space of the diffraction.
@m9l0m6nmelkior7
6 ай бұрын
23:08 I have a question : we saw that Fn+2 = Fn + 1*Fn+1 and Fn-2 = Fn -1*Fn-1 what if we insert Fn+2i = Fn + i Fn+i and Fn-2i = Fn - Fn-2i ? I mean, we'd have to change the furmula by Isolating Re(n) and Im(n) for the exponents, and there needs to be 2 more numbers to complete a seed, but appart from that, wouldn't that work ? I think though that having a complex function such that for all Z and theta F(z+2exp(i theta)) = F(z) + exp(i theta) F(z + exp(i theta)) isn't possible, though it would be nice…
@imaginaryangle
6 ай бұрын
That sounds very interesting, and a rabbit hole that may go very deep. I haven't tried it and I don't know if I'll be able to give it the time it needs because I'm quite busy on upcoming topics unrelated to this area. I'm honestly also a bit scared to touch it and get nerd sniped for weeks 😅 Maybe something for you to explore and tell us what you find?
@m9l0m6nmelkior7
6 ай бұрын
@@imaginaryangle Yeah indeed that seems like quite a rabbit hole x') I'll try to explore it more, but one thing worth noting is how the matrix representation of the recursion relation Fn+2 = Fn + Fn+1 naturally appears when you plot the thing for gaussian integer values ! I know how captivating this subject can be, don't feel pressured venture too deep in it :')
@soninhodev7851
6 ай бұрын
the fact that when you change the index of the function, the complex versions of the curve change their point of intersection to match, is fascinating! Will that be a future videos topic?
@imaginaryangle
6 ай бұрын
I'm collecting open questions from this video to make a sort of an "after class" that covers them. I haven't fully decided which details I will cover in that one. I'm guessing what you're referring to is these intersections with the direction line behaving so regularly? Or did I get that wrong?
@soninhodev7851
6 ай бұрын
@@imaginaryangle no, at the very end of the video all the "shadows" intersecting the original curve at the same point as you move around the index. (i dont know if they actually intersect, they just look like they do.)
@imaginaryangle
6 ай бұрын
Ah, got it! Yes, it's a true intersection. That point is where the index is zero for both the curve itself and its shadows, and the shadows iterate away from that at some angle (so n is some multiple of a complex unit that is not Real 1). That means that at index zero, the shadows are indistinguishable from the curve itself, because it's just zero times that angled unit.
@madzubmetler
6 ай бұрын
Man, I think you just solved the universe
@imaginaryangle
6 ай бұрын
I appreciate the sentiment 🌌
@madzubmetler
6 ай бұрын
Love your animations, keep on digging that rich soil. I don't know the maths well but the visualizations give me some understanding and inspiration.
@imaginaryangle
6 ай бұрын
Happy to hear that! If you feel like trying to wrap your mind around the math too, Counting in Imaginary, Secret Kinks and The Golden Threeway videos should give you a pretty good shot at understanding the whole stack of what's discussed here - I did my best there to not require anything more than basic arithmetic. Thank you for your support and kind words!
@PermanentExile
6 ай бұрын
Fantastic as always.
@imaginaryangle
6 ай бұрын
Thank you! 😊
@63M1N1
6 ай бұрын
"seeds" is definitely better than "esses", although... xD
@imaginaryangle
6 ай бұрын
YMMV
@GiacomoPerin
6 ай бұрын
I knew fibonacci's sequence for so long and today I understood how much I don't know about it
@imaginaryangle
6 ай бұрын
To be fair, we ventured pretty far from its original definition 🙃 I did my best to keep at least one or two aspects of it around in every extension to get a sense of what relaxing each property does.
@atticuswalker
6 ай бұрын
every 3 turns gives you 45⁰
@hved
6 ай бұрын
pure psychedelics.
@thp4983
6 ай бұрын
You should really add some colors to the spirals and shadows at 12:40 and near the end. In general, visualizations where colorful lines fill out the plane work really well. Visualizations like that is what drew me in to your "Secret Kinks of Elementary Functions" video. I think they would work much better as thumbnails than the ones that are on currently. The current ones invoke a historical perspective, which is neither compelling nor the focus of your videos at all. With all that being said, great video with super interesting content! Always a treat!
@imaginaryangle
6 ай бұрын
Thank you so much for your feedback, this really helps!
@ewthmatth
6 ай бұрын
"the current ones invoke a historical perspective" How so? I think they were meant to look artistic, not old fashioned.
@charlievane
6 ай бұрын
really, going through these 2 videos should be homework assignment for some classes
@sertacatac0
6 ай бұрын
That was a very high quality video of getting deep with fibonacci's sequence. I understood that there always is more to explore, just by using the basics and converting them one to another.
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