Golden Threeway - Fibonacci, Lucas and the Golden Ratio

Рет қаралды 15,092

Күн бұрын

The Golden Ratio is closely tied to the sequence of Fibonacci Numbers - but this relationship isn't completely described without involving Lucas Numbers. In this video, we explore the many ways they define and connect with each other.
00:00 Intro
00:45 Recurrence Comparison
02:40 Convergence to Φ
05:20 Finding Fibonacci in Φ
07:21 Φ, ψ and the n-th Fibonacci
10:27 Lucas Numbers
13:08 Fibonacci and Lucas off by a Factor
14:43 n+1, n-1
16:27 Linear Combination Using Lucas Numbers
17:35 All Together Now
18:17 Bonus Equality
20:05 Bonus Graph

Пікірлер: 83

@sizur 5 ай бұрын

Start fibs with 0,1... This is one of the best channels! You start with simplest ideas and quickly, step by step get to very interesting aspects!

@imaginaryangle 5 ай бұрын

Thank you! All of this also extends into negative indices: both Fibonacci and Lucas numbers continue on the other side of zero, and you get the same numbers again, only this time, they alternate between negative and positive.

@sizur 5 ай бұрын

@@imaginaryangle Yup, just starting with 0,1 "feels" more natural than with 1,1. I know, fighting established conventions is almost a guaranteed lost cause.

@imaginaryangle 5 ай бұрын

@@sizur To be fair, indexing the Fibonacci Sequence like this is the convention, and starting them as 1, 1 makes sense if you are thinking of counting elements the regular way, that is, starting at the "first" instead of the "zeroth" member. I didn't really do anything rebellious here 😉

@cangrejoxidao 5 ай бұрын

I don't get the efforts to avoid calling the first element the first element when it equals zero. The first on a sequence is just the first one.. whatever its value. Also maybe I'm very used to 0-indexed arrays but starting the sequence with 0,1 since the beginning would've felt the most natural thing to do.

@imaginaryangle 5 ай бұрын

@@cangrejoxidao In this case, the choice of where to put index zero is more than a cosmetic preference, because the equalities explored in this video only work when indices are chosen this way.

@trucid2 4 ай бұрын

I think there's some fun to be had with Fibonacci and Lucas sequences in matrix form using phi and psi as the basis vectors.

@imaginaryangle 4 ай бұрын

There sure is! 😉

@austinisawesome2066 5 ай бұрын

So satisfying to see all of the phi and psi swaps and cancelations! And I love being able to watch the video and just understand all the math concepts right away. Great job explaining and showing visually everything!

@rugbybeef 5 ай бұрын

This is beautiful, which is both a thought and an expression of a feeling. Thank you. 🎉

@imaginaryangle 5 ай бұрын

I appreciate that 😊

@maix52 5 ай бұрын

This video is very interesting! I loved the "discover with me" vibe ! Also the fact that you have a voice that does't really fluctuate a lot is really enjoyable here. Please continue and I wish you a happy new year !

@imaginaryangle 5 ай бұрын

Thank you! Happy New Year to you too!

@bestwifi 5 ай бұрын

Beautiful explanation and animations. Massively underrated

@Jason4195 5 ай бұрын

I discovered your channel after SoME3 and I’m so glad I did! This is awesome!

@imaginaryangle 5 ай бұрын

Thank you! 😊

@robharwood3538 4 ай бұрын

Awesome! Many insights that have escaped me for years finally emerged by watching this video. Thanks!

@SandipChitale 5 ай бұрын

Brilliant. Don't just look ahead, look to the left, look to the right, look up, and look down and you will find more...

@imaginaryangle 5 ай бұрын

Thank you!

@SandipChitale 5 ай бұрын

@@imaginaryangle Truly out of the box or shall I say x-y plane thinking :)

@CJ-mk3nf 4 ай бұрын

Just found your channel and you seem to really have a knack for digging into exactly the questions raised by the topic that i find myself wondering, it makes these really satisfying. Add on to that your lovely visual and audio(al?) presentation style, and you've got a winning formula here. Tl;dr, love the content and keep it up!

@imaginaryangle 4 ай бұрын

Thank you! The videos are mostly inspired by questions I had that I couldn't find satisfying answers for. The audio is my own voice.

@SteveShahbazian 5 ай бұрын

I'm really enjoying this series! Awesome stuff and beautifully presented. Thank you and Happy New Year :)

@imaginaryangle 5 ай бұрын

I'm happy to hear that! Happy New Year to you too!

@shiftsky7130 5 ай бұрын

this is awesome

@wyboo2019 14 күн бұрын

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

@debblez 5 ай бұрын

didnt know fibonacci and lucas were that close

@johnadriandodge 3 ай бұрын

Thank you Mr. Imaginary Angle for sharing. This was a most fascinating and complex topic. Shalom

@imaginaryangle 3 ай бұрын

You are very welcome!

@The_Commandblock 5 ай бұрын

I discovered this relation in the infinite fraction x = 1/(1+x) (since x = 1/(1+x) you can reinsert the term into the equation giving you the fraction) When trying to calculate it i first got the golden ratio from the formula however when i showed the fraction to my mom she started adding the terms from "the start". The fractions always were a higher fibonacci number over the one before it. Its a really beautiful connection for me mainly because i found out about it myself

@imaginaryangle 5 ай бұрын

What I love about the Golden Ratio is how natural and friendly to exploration it is, so it creates a lot of these wonderful experiences. Thank you for sharing it!

@katten444 4 ай бұрын

This was magical! The animation is so good and intuitive I’m actually floored! Well done !

@imaginaryangle 4 ай бұрын

Thank you so much 😊

@lugyd1xdone195 5 ай бұрын

3:46 the numbers cant exactly be any, the sequence itself cant converge. For example -2 and 1 lead to -1, which leads to zero. Im also pretty sure that for some starting numbers the ratio converges to psi instead. At least thats what I heard. EDIT: I also dont understand what oscilation of what terms you mean in 9:47. This could be a mistake on my part, but Ive been replaying the segment for at least 10 times and I dont understand a bit of it. Why dont the graphs sometimes not touch the Fibonacci/Lucas numbers - for example the graph at around 10:00? I understood the one at the end where it was average that gave us the numbers, but most graphs were tough for me to make sense of. The equalities seemed precise and it didnt feel like the graphs were approximated so I dont get what happened there. It'd be nice if you better described what you're showing when it comes to graphs. Maybe some scale outlines and a note of what function it is would help. I dont think that would be at odds with the minimalist style you've got. Other than that I loved the various formulas. Keep up the good work!

@imaginaryangle 5 ай бұрын

Thanks! Try out the example with -2, 1, -1 on paper for a few more iterations, I think you'll be surprised ;) As for a combination of starting numbers that would converge to psi, that indeed does happen, but backwards (F_n/F_n-1 where n approaches negative infinity), and it does again for all such sequences. Going forward, it cannot happen, because the elements will consolidate to either all positive or all negative (for Real numbers), so they will grow (get further away from zero), enforcing a positive ratio > 1, eventually exactly phi. At 9:47, the oscillating graph is of f(x) = -psi^x/√5. The oscillation happens because psi is negative, so you could see it as (-1)^n * (phi^-1)^n. The (-1)^n term for integer powers of n goes: 1, -1, 1, -1,... and for non-integer powers it's a complex number.

@newwaveinfantry8362 4 ай бұрын

13:25 - No, it's because of the fact that you're dividing by sqrt(5) for ф but not psi. The if the difference was just the sign then they would converge very quickly as psi has absolute value

@imaginaryangle 4 ай бұрын

In Binet's Formula, the division with sqrt(5) happens to both phi and psi terms - their powers are substracted, and then this whole difference gets divided by the sqrt(5). The sign change between Binet's Formula and the Lucas closed-form formula has the effect that the difference between them is not only the factor of sqrt(5), but also that the psi fluctuations happen in the opposite order. If that was not the case, we'd have L_n = sqrt(5)*F_n for all n.

@assiddiq7360 5 ай бұрын

The thumbnail doesn't look quite attractive, but then I see the creator's name. Here I am

@jedglickstein 5 ай бұрын

Great video - great channel. Do similar relationships hold for other variants of the Fibonacci sequence (e.g., the Tribonacci sequence) and/or golden ratio (e.g., the silver ratio and its negative inverse)? I had never heard of these until watching the Mathologer video recently. Would love to see your take on it. Keep up the good work!

@imaginaryangle 5 ай бұрын

Thank you! That's an interesting suggestion, I haven't really looked into how these properties transfer onto other metallic means.

@jovetj 5 ай бұрын

This is really, really neat. 👍👍👍

@sertacatac0 4 ай бұрын

excellent video !

@imaginaryangle 4 ай бұрын

Thank you very much!

@danielgarciabustamante5315 Ай бұрын

Espectacular 💯🙌🏼

@imaginaryangle Ай бұрын

Gracias!

@JeffHanke 4 ай бұрын

Very cool video. I think it would be cool to show an animation flying along that final graph as the Lucas and Fibonacci graphs spiral around you.

@imaginaryangle 4 ай бұрын

Thank you! It's a cool idea, and I was thinking about it too, but I probably won't get around to it.

@Dreamut 5 ай бұрын

beautiful math

@weirdredstone42 5 ай бұрын

this is so cool! what do you use to render and animate your 3d graphs?

@imaginaryangle 5 ай бұрын

I use manim: www.manim.community/

@weirdredstone42 5 ай бұрын

@@imaginaryangle wow! i knew manim had lots of functionality, but 3d is something i haven't seen utilized as often.

@cangrejoxidao 5 ай бұрын

Wonderful insights, very well threaded together 😊. About the colors I love the dark background but I find the dark purple exponents not very easy to read at night.

@imaginaryangle 5 ай бұрын

Thank you! And also for the tip about colors, I'm still adjusting my palette to find the right combinations.

@AGamer_2010 5 ай бұрын

only know the lucas numbers because of the song "sing sing red indigo" by frums, for the rhythm game "a dance of fire and ice" dlc "neo cosmos", and i think i put "way too much" quotation marks

@realcygnus 5 ай бұрын

Nifty AF !

@user-zn4pw5nk2v 4 ай бұрын

I see complex rotation at the end around the average, where is my PIE. If you divided by the average you would have probably gotten a pretty spiral looking down from Y=1 towards XZ(or Z->X(RE)Y(IM) because usually Z is denoting the complex result (up direction) ).

@tot5299 4 ай бұрын

I would like to know the general form for converting powers of certain irrational numbers into linear combinations of sequences: x = a + b*sqrt(c) -> x^n = A(n) + B(n)*sqrt(c)

@imaginaryangle 4 ай бұрын

There is a generalization of simple recurrence relations with a linear combination of two previous elements, but it's a bit too much to explain in a comment. I'm making a video roughly related to the concept, though not directly, so I will try to put good resources for the general solution in the description. If you want to do some digging in the meantime, the term to search for is "second order homogeneous linear recurrence relation".

@Meow_YT 5 ай бұрын

It's always beautiful seeing the 2d graphs pop into the complex 3d view of what's going on... also maybe a little lonely that they'll be forever swirling around and never meeting.

@MagicGonads 4 ай бұрын

I've always heard of "the lucas numbers" as any sequence following the recurrence relation, that is to say that there are multiple "the lucas numbers", but "the fibonacci numbers" are the particular sequence we know. Is "the lucas numbers" shown here special? Can we generate a "the 'dual' lucas numbers" for any other "the lucas numbers" that has this same relationship across phi? Meaning: Say Fn is the 'dual' of Ln in this sense, what if we substitute Ln for any other lucas sequence, then reverse the formula to find the corresponding 'other' Fn = (2/sqrt5)(phi^n-Ln/2), are these guaranteed to be integers if Ln are (and ratios converge to phi)? If so then it would seem we only pick that particular Ln because Fn is so natural starting at 0 and 1 or 1 and 1 (depending on if you see the 'start' as n = 0 or n = 1).

@imaginaryangle 4 ай бұрын

The nomenclature is confusing here: there is a family of sequences called "Lucas Sequences", but there is one specific and particular Sequence (singular) of Lucas Numbers, which is the one presented in the video. This sequence is special, kind of in the same category as the Sequence of Fibonacci Numbers, in ways that are discussed in the video. Also worth noting, Lucas Sequences (plural) and Fibonacci Sequences (plural) do not refer to the same family and each contains its namesake numbers, -but not the other- *(

@MagicGonads 4 ай бұрын

@@imaginaryanglewell then should I change my note that I wrote in a document exploring different recursive forms of the fibonacci sequence in which I labelled the tail-recursive form a generator of the lucas numbers (meaning, every lucas sequence depending on the parameters given), when it should be generating the fibonacci sequences?

@MagicGonads 4 ай бұрын

btw said tail-recursive form is fib(0,a,b) = a fib(n,a,b) = fib(n-1,b,a+b) where 'the fibonacci numbers' are fib(n) = fib(n,0,1) writing it this way instead of the usual fib(0) = 0 fib(1) = 1 fib(n) = fib(n-1) + fib(n-2) makes it an O(n) time O(1) space algorithm instead of O(2^n) time O(2^n) space so what to call all sequences generated by all possible choices of a and b? (assume they are non-negative integers and a=b=0 is ignored if necessary)

@imaginaryangle 4 ай бұрын

I just realized I made a mistake in my previous response: both Lucas and Fibonacci numbers *are* members of both Lucas and Fibonacci sequence families. Sorry about that. Lucas Sequences generalize to a linear combination of two previous members with any (integer) factors allowed, while Fibonacci sequences specify adding two previous members as they are. So the naming you chose is not wrong. If you'd like to avoid ambiguity, I'd use "Recurrence Relations", as it's a clear and full generalization of a (usually linear) combination of any number of previous elements, and just specify the general expression.

@michaelparker6907 4 ай бұрын

So the cosmic filaments (plasma) observed, and found in the electric universe model.

@SomeoneCommenting 3 ай бұрын

How do you get the Z axis coordinate value 14:08 to be able to plot them in 3D?

@imaginaryangle 3 ай бұрын

The imaginary parts of F_n and L_n are mapped onto the z axis.

@reidflemingworldstoughestm1394 4 ай бұрын

Fibonacci numbers got no causal link to Φ. Using the same rule in the Fib. sequence starting with any two positive real numbers will approach Φ, just as you do starting with 1,1. The golden ratio is a result of the rules of the Fibonacci sequence, not of the numbers.

@imaginaryangle 4 ай бұрын

It's in the video 😉

@reidflemingworldstoughestm1394 4 ай бұрын

@@imaginaryangle Sure, tell me now.

@LeoStaley 4 ай бұрын

Nobody ever explained to me how we figured out that φ=(1/2)+(√5/2)

@imaginaryangle 4 ай бұрын

Do you understand it now, from the quadratic equation part of the video? Sorry, it wasn't clear to me if you are asking how or saying you now understand 😉

@stieli5816 5 ай бұрын

👍

@erikheddergott5514 4 ай бұрын

It is all about Root of 5

@RagibMahirAshhab-sn8qv 2 ай бұрын

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)

@imaginaryangle 2 ай бұрын

That's true. But that's one of three values you could assign to that expression, one of which is Real, while the remaining two are Complex. There are three values for Ψ^(1/3), and because that's a root of an odd degree of a negative number, one Real negative result will always exist, but it will not be alone. To get a single continuous curve through this multivaluedness, we decide ahead of time which solution will be picked when there are many, and here we are always picking the so-called "principal" solution, the one based on -1 being expanded to e^(i pi).