The Fibonacci Sequence and the Golden Ratio

This video introduces the mysterious and mystical Fibonacci Sequence and explores its relationship to the Golden Ratio. While filmed with a fifth grade audience in mind, it should be accessible to kids as young as fourth grade . Hopefully it's interesting to everyone.
If you're interested in learning more about the golden ratio and other mathematical sorcery, I can't say enough about this book:

#khanacademytalentsearch, #phi, #goldenratio, #fibonacci, #math, #geometry

[tomwolf]

This is a perfect example on how the YouTube should be used to teach people. Great channel!

[MrKevin-ulqp]

Whoa! I just came back to this after a time away. Never thought so many people would view it. Thanks so much for your support. Maybe I should do another...

[sofiatheone]

Thanks for explaining The Golden Ratio in a simplistic way. I have known about it for some years now, but you have definitely enabled me to get a better understanding of how the patterns start to form, etc. 🙏🏽💕

[jacobblack]

I LOVE this.
Thank you for the explanation in PLAIN ENGLISH.

[seamus]

My favorite example is in a pine branch (pitch pine is best). It grows in two spirals in Fibonacci proportion. They run in opposite directions and everywhere they cross an event happens, a pine needle grows.

[beethovensg]

Credit to your effort, it is important and fantastic. The kids are forever more intuitive for this one act alone. Great for you!

[krishugmukhia]

If this teacher could have had taught me math back in school...

[josephtermeer]

Thank you very much for posting this. I am studying art and have come across this rule. I never learned this in the "advanced" art classes in high school. It was not even introduced in college. I must have been in the lower federally funded school district.

[tessellatiaartilery]

Great video. Explanation of the numbers sequence, how the ratio is 'derived' and showing visually with the graphic sections very clear. Nice activity with the calipers for seeing the ratio in nature outside too. Thanks for making and sharing this.

[BusinesssValues]

Thank you. I understood the adding of the numbers for the Fibonacci Sequence, but, I never knew the division to get Phi . I had heard of the Golden Ratio but didn't quite get how it tied to the Fibonacci Spiral. I had seen the callipers but had not seen such a fun experiment. Tis is excellent. Again, thank you.

[mattlast]

so cool that you engage your children that way

[professordeb]

The calipers are genius, Mr. Kevin! I may steal that! Very nice presentation.

[oliverwenath]

Thanks for explaining the number Phi, the golden ratio so nicely and understandable!

[solapowsj]

In fluid physics, there are 24 ones around a two, each power in an x-plane, forming a solid three with one set of fluid forces at 'c' (energy waves 🌊and fields), and the next having the graviton, fermi fermat, vortex, etc.... Just by intuition. Great presentation.

[scottielambert]

Exceptional post fella! Positive and thought provoking. I use this as a tool for trading but the eloquence and relatability of it is a bit astounding. Thanks for posting.

[batuhansonmez]

Also there is a relationship between Fibonacci Sequence and Euler’s Numbers.Please search on internet by this title “Quantum Perspective Model by Tahir Ölmez

[prosperity.]

How Amazing Are You!!! ♥️ Omg!! Saved!! Professors, Tutors, Textbooks, You Tube Vids... Left me clueless!!! Until You Saved the Day.. I can't thank you enough!!!! Thank you..

[marcuspi]

The dog was running in a Fibonacci spiral

[pantherplatform]

_Now_ my mind is blown. I've watched countless videos on this truth but none of them explained how the Fibonacci numbers were directly related to the golden ratio.

[CARLESIUS]

The difference equation that defines the Fibonacci sequence is:
Delta ^ 2 F + Delta F - F = 0
Its characteristic equation is:
r ^ 2 + r - 1 = 0
so the roots of this quadratic equation are
- (1 + sqrt (5)) / 2 and - (1 - sqrt (5)) / 2
The initial conditions of this difference equation are:
F (0) = 0 and Delta F (0) = 1 which leads to the algebreic expression for the nth term of this sequence:
F (k) = (((1 + sqrt (5)) / 2) ^ k - ((1-sqrt (5)) / 2) ^ k) / sqrt (5)
On the other hand, given a segment of length 1, divided into two segments of length x and (1-x), these 3 segments satisfy the golden ratio if the proportions are given:
x / 1 = (1-x) / x
This relationship also leads to the cadratic equation:
x ^ 2 + x - 1 = 0
So there is no mystery.