Golden Threeway - Fibonacci, Lucas and the Golden Ratio

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

[sizur]

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!

[austinisawesome]

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!

[trucid]

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.

[rugbybeef]

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

[maix]

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 !

[bestwifi]

Beautiful explanation and animations. Massively underrated

[newwaveinfantry]

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 <1.

[SteveShahbazian]

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

[Jason]

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

[robharwood]

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

[CJ-mknf]

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!

[katten]

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

[The_Commandblock]

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

[SandipChitale]

Brilliant.

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

[johnadriandodge]

Thank you Mr. Imaginary Angle for sharing.

This was a most fascinating and complex topic.

Shalom

[Billy-qots]

AHHHH lets go . video is exactly what i needed right now

[debblez]

didnt know fibonacci and lucas were that close

[wyboo]

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

[JeffHanke]

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.

[jedglickstein]

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!