If you combine the two series and re-order the terms, (which we can do since both are absolutely convergent) we get that phi + 1 = 1 + 1/phi + 1/phi^2 + 1/phi^3 + …, but phi + 1 = phi^2, so phi^2 = 1 + 1/phi + 1/phi^2 + 1/phi^3 + … Amazing video!
These presentations are very bright and easy to remember for a lifetime. One day this channel will undoubtedly become one of the most popular channels in the world. I wish you strength and luck to continue these submissions for a long time. ♥
I assure that you are the next 3B1B for sure. Hope you achieve great heights. By the way love these visual proofs as I am a mathematics student and an enthusiast. Keep going!!!
Less interestingly, but still a fact: We could get the area of the golden gnomon from the sum of the areas of the scaled golden triangles. By Heron's formula, the former is sqrt((5 + sqrt(5))/2)/4. The latter are a bit harder to compute, but should be manageable, although I do not want to figure that out right now. Like I said, it seems to be a less-interesting and messier fact. But we can add it to the repertoire.
Hello! I understand that this question is somewhat unrelated to the video, but I noticed that you use Manim. I was wondering if you have the updated version of the Fourier series animation from 3Blue1Brown. I've been wanting to render that one, but my Manim Cairo version is outdated. By the way, your channel is amazing; the visual proofs really emphasize the tangible beauty of mathematics
Your title is misleading, that's not what a golden series is, and you have to end up with infinitely small decimals. A golden series is a random output of rare sorting of mostly (integer types) and they don't go bigger infinitely neither smaller (Always made using Math and Programming) and they are used in generative arts and Monte Carlo analysis like in economy or terrain simulation, we call them golden because of their usefulness in simulation. What you are talking about falls into the realm of useless mathematics UNLESS YOU CAN SHOW US A USEFUL APPLICATION IN THE REAL WORLD THAT INCLUDES DIVIDING A TRIANGLE INFINITELY. On the other hand what you showed here is used in something called composition, which is used as a subdivision branch of topology but in no way is a golden series, it's just a number that goes smaller every time and that is not what a series is in the first place.
why would we need practical applications for this area of math though? Also I think he's referring to a geometric series, where each successive term is a fixed fraction of the last
@@cheeseburgermonkey7104 Yeah I know what he was referring to, but he is misnaming it just because of the golden ratio. Why do we need an application for such stuff?? Random variables, I'm a computer scientist so everything in pure math can be applied in my screen visually. Real Golden Series are the output of Golden Randomization Engines but not geometry. We call them golden series in software engineering because when we play them as music or display them as pixels they output something beautiful hence the name Golden :)
If you combine the two series and re-order the terms, (which we can do since both are absolutely convergent) we get that
phi + 1 = 1 + 1/phi + 1/phi^2 + 1/phi^3 + …,
but phi + 1 = phi^2, so
phi^2 = 1 + 1/phi + 1/phi^2 + 1/phi^3 + …
Amazing video!
Thanks! Glad you found that bonus series mentioned. Thanks!!
It looks so much like pyramids.
These presentations are very bright and easy to remember for a lifetime. One day this channel will undoubtedly become one of the most popular channels in the world. I wish you strength and luck to continue these submissions for a long time. ♥
Wow, thank you!
I assure that you are the next 3B1B for sure. Hope you achieve great heights. By the way love these visual proofs as I am a mathematics student and an enthusiast. Keep going!!!
Been grateful for his animation tool in this project. Thanks for the comment and glad these help you.
Thank you for pronouncing phi the correct way!
Less interestingly, but still a fact: We could get the area of the golden gnomon from the sum of the areas of the scaled golden triangles. By Heron's formula, the former is sqrt((5 + sqrt(5))/2)/4. The latter are a bit harder to compute, but should be manageable, although I do not want to figure that out right now. Like I said, it seems to be a less-interesting and messier fact. But we can add it to the repertoire.
since the even power sum contains a one, you can replace the one with the odd power sum, rearrange, and you get:
φ = 1/φ+1/ φ²+1/φ³+1/φ⁴+...
Very cool right?
Brilliant, love it! The visuals, and the concise clear narration, really bring the proof to life.
Thanks!
Hello! I understand that this question is somewhat unrelated to the video, but I noticed that you use Manim. I was wondering if you have the updated version of the Fourier series animation from 3Blue1Brown. I've been wanting to render that one, but my Manim Cairo version is outdated. By the way, your channel is amazing; the visual proofs really emphasize the tangible beauty of mathematics
Sorry, I have not played with Fourier series animations. Thank you for the comment!
@@MathVisualProofs thank you for replying :)
Supergolden ratio is tricky as well
Wonderful!
Thanks!
Eventhough you can just find the answer by geometric series formula this is much cooler
😎
awesome!
😀
Wow
how
so nice
Thank you! 🙂
One plus phi. 2.718....
👍
👍
This shows that phi is the solution for x=1/1-x
you actually mean that x = 1/x-1 but yeah that's cool
Your title is misleading, that's not what a golden series is, and you have to end up with infinitely small decimals. A golden series is a random output of rare sorting of mostly (integer types) and they don't go bigger infinitely neither smaller (Always made using Math and Programming) and they are used in generative arts and Monte Carlo analysis like in economy or terrain simulation, we call them golden because of their usefulness in simulation. What you are talking about falls into the realm of useless mathematics UNLESS YOU CAN SHOW US A USEFUL APPLICATION IN THE REAL WORLD THAT INCLUDES DIVIDING A TRIANGLE INFINITELY. On the other hand what you showed here is used in something called composition, which is used as a subdivision branch of topology but in no way is a golden series, it's just a number that goes smaller every time and that is not what a series is in the first place.
Its just cool man, shut up
why would we need practical applications for this area of math though?
Also I think he's referring to a geometric series, where each successive term is a fixed fraction of the last
@@cheeseburgermonkey7104 Yeah I know what he was referring to, but he is misnaming it just because of the golden ratio. Why do we need an application for such stuff?? Random variables, I'm a computer scientist so everything in pure math can be applied in my screen visually. Real Golden Series are the output of Golden Randomization Engines but not geometry. We call them golden series in software engineering because when we play them as music or display them as pixels they output something beautiful hence the name Golden :)
Tell me you hate maths without telling me you hate maths:
Plzz do videos on linear algebra plz
Plzz do videos on algorithms sir 😢