The Phiangle!

20:50 Finally, imaginary angles!

I wish I could have seen those last animations on an oscillating 3d graph😭

hi

"Complex Variables" by John W. Dettman (published by Dover) is a great read: the first part covers the geometry/topology of the complex plane from a Mathematician's perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicist's perspective. For practical reasons, a typical Math Methods for Physics course covers the Cauchy-Riemann Conditions, Conformal Mapping, and applications of the Residue Theorem. I've used Smith Charts for years, but learned from Dettman that the "Smith Chart" is an instance of a Möbius Transformation. The Schaum's Outline on "Complex Variables" is a great companion book for more problems/solutions and content.

The real question is Discord serber wen

fun convergence to phi fact: in geometric algebra, you can (kind of) divide vectors (most vectors have multiplicative inverses), and by starting with two vectors and following the Fibonacci recurrence relation with vector addition, these two vectors actually do converge to the golden ratio, i.e. multiplying one by the inverse of the other approaches the golden ratio

"divide the cake into 0 identical pieces" ok I eat the cake, give it a few hours for my digestive system to make infinite cuts and you are left with no cake.

Cool stuff, but something I've come across is partial derivatives and whatnot, and the obvious fact that although mapping from complexes to complexes may appear to require 4D plotting, on the contrary, every x + iy for reals x and y defines a plane, but the conjunction of two planes orthogonal to each other with one of their axes collinear suffices to define an infinite Euclidean 3-space just as the conjunction of two sets A*B forms a plane. Letting C denote the complexes, our Euclidean 3-space formed from C*C would be visualized by mapping all X + iy to f(X+iy), and all x + iY to f(x + iY) where X and Y are held constant as x and y vary over time. Wolfram Alpha just visualizes complex plots by showing the 3D plot of the real and imaginary parts separately since at that point, you have only 3 variables. Alternatively, perhaps it might be shown in 3-space two planes parallel to each other, and the path that each point follows according to the rules of the particular function over the complexes, so for example, f(z) = z would, for some finite rectangular neighborhood of input points, just show a rectangular volume; f(z) = z^2 would probably show for real t, |z|^(t+1) e^(i(t+1) arg(z)) as t increases linearly with time which would show some kind of cylindrical solid when Re(z)^2 + Im(z)^2 is less than or equal to 1 with funny transitional stuff in the neighborhood of 1, but a sort of hyperboloid volume outside that; etc. Sorry, I went on a tangent there (pun not intended). Anyways, is there a definition or analog of the derivative for 3-space for the intuitive idea of the plane tangent to a point on a surface?

0:30 "and along the way, learn how to *extend* the concept of derivatives to complex valued function" That was nice

Your videos are getting better and better. What I like most is your ability to describe familiar concepts (here, the derivative as a ratio of input to output) in unfamiliar ways. It feels like taking the ideas I first learned in high school and slightly tweaking an “abstraction” knob. Fitting, given the beautiful images at the end of this video!

"Even if everything else somehow made sense, we still could not divide vectors." That's where you're wrong kiddo! 😎 It wouldn't make a huge difference though, since the result is effectively a Complex number anyway (assuming the vectors are 2D), which is distinct from a 2D "vector" in this system despite having the same components.

Polar coordinates of a parameterized space are very similar to a Fourier transform.

A truly remarkable video...thank you so much for the education and entertainment!

Nifty AF !

Absolutely beautiful.

I took a complex analysis course a few years ago, so my recollection of it is somewhat limited now. Videos like this really make me stop to think about about questions I don't even really know how to ask, such as questions that involve exploring non-analytical cases with singularities. I love these videos, and they are a massive breath of fresh air to get my mind off my studies and onto something else, if only for a few minutes.

I am just learning the derivative concept, and seeing this with complex numbers is just astounding. Also great explanation, I could understand pretty much everything even though my math level is below this topic, which makes me way more curious. Your content deserves to be at the top

I've just got to say that this is the best explanation of the complex derivative that I have ever seen.

I like your channel, but I really dislike this approach to imaginary numbers, saying that they are just a lateral movement in space. If that's true, we could just as well use 2D vectors. It's the algebra embedded within complex numbers that makes them special, and the deeper intuition is that complex numbers completely classify all translations, rotations, reflections, and scalings of 2D space, just like all real numbers classify translations and scalings of 1D space

Espectacular 💯🙌🏼

I wonder ... well one thing I have wondered about with graphical approaches is: what happens to curves and lines when the axes values are rescaled? It could be either axis on its own or multiple axes on a 1:1 ratio or some other ratios. Would it give further graphical insights into patterns and lead to pattern recognition? Excellent video!

Absolutely fascinating!

1:29 "Group" is a mathematical term reserved for group theory. Please do not use it in this context, unless you are relating complex analysis with group theory.

I tried playing with parametric representation on desmos to visualize space transformation when using complex variables, this really reminded me of that. In a similar vein I wanted to understand and play with raising non-unitary complex numbers (a+bi) to non-unitary complex powers which lead me to finding my favorite number: Gelfond's Constant (-1)^(-i)=e^pi.

2:26 bruhken

ITS beautiful, to See that there ist only one Infinity in complex world, Not a plus and minus Infinity.

in physics, many of these properties occur in quasicrystal diffraction

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.

2:53 two equals signs on the short bar

These animations, together with their 'complex analysis', are 'real' gems of mathematical 'imagination'! (all 'puns' intended)

Wow, subscribed and added to favorites!

ok you sais no fractional powers can be real. but, (phi)^(1/3)-(si)^(1/3) all divided by phi-si is 0.89( approx)

WOW!

🌚🖤💋🧛🏻‍♂️❤️🙌🏻🦇🦂

💕💕💕💕🤸🏻‍♂️🐬🙌🏻

every 3 turns gives you 45⁰

Do you have an updated desmos file for the graphs? cause I would love to try it out for myself

I got lured by the thumbnail,love between Fibonacci and his numbers

really, going through these 2 videos should be homework assignment for some classes

pure psychedelics.

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!

Amazing. You opened my mind at 67 years old to a realm I never imagined.

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.

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…

I'm pretty sure Euclid was not writing The Elements in Ancient Greece. He was a Greek in Alexandria, a Greek city in Egypt built by Alexander the Great.

Can you do a video when you explain where sin cosine tangent come from/are and where the unit cirlce comes from and why it is what it is and how it is important in nature architechture etcccc LOVE THE CHANNEL MATE NICE WORK !

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

Man, I think you just solved the universe

Wish these videos existed without AI art 💀