Complex Fibonacci Numbers?

Check out Ben Sparks's GeoGebra files.

My four-part series on Numberphile videos about Fibonacci Numbers (from 2014) starts here.

Here is me going on about the square root of five (Numberphile 2018).

This was the Fibonacci puzzle video from Matt Parker's Maths Puzzles.

Read a whole bunch about "Generalized Fibonacci Sequences and Binet-Fibonacci Curves".

This site has everything you'll ever need to know about Fibonacci Numbers.

Buttercup - The original buttercupchallenge

CORRECTIONS
This was a long video and in hindsight there are a few things I wish I had phrased better. Here they all are:
- I misspoke around 01:13 when I said "negative one, zero" as it is clearly "negative one, one, zero".
- At 07:53 I mean the negative values -5 to 0. I said it a weird way.
- My language at about the 1D input to 2D plot from 09:17 is a bit sloppy. The real values going into the Binet function are not the horizontal axis shown; the plot onscreen is solely the output.
- I say "axis" when I mean "plane" or even "complex plane". The big flat thing.
Let me know if you spot anything else!

Thanks again, as always, for Jane Street being my principal sponsor.

Thanks to my Patreon supports who do support these videos and make them possible. Here is a random subset:

Loren Thomas
Richard Dickins
Barry Salter
Susan Moury
Sarah Gerweck
Ulrich Kempken
Piotr
Gary Martin
Euler
Daniel DeJarnatt

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Filming and editing by Matt Parker
Music by Howard Carter (excluding Buttercup)
Design by Simon Wright and Adam Robinson

MATT PARKER: Stand-up Mathematician

[hexeddecimals]

Now I want to see a 3b1b style animation of the 2d inputs moving around to their 2d outputs

[nicolaom]

I’m wondering what are the properties of the loop that the two 1’s form... I don't know why, but it was the part that I found the coolest

[rashmi]

This is one of the awesome channel in you tube and I love it and learn from it.

Thank you so much sir.

[jacobyarinsky]

Still my favorite stand up maths video. That loop de loop is insane!!

[僕と契約しましょう]

Being a random person who doesn't understand maths really well but happened to stumble upon this video by chance I got anxious when he starts to get excited about more ways to mingle with the numbers...

Forgive me I'm scared, now that I learned that there's more that I don't know

Edit: I shouldn't look into the comments my head is exploding

[davidbledsoe]

I'm very curious about what Dr. Holly Krieger thinks about this. Complex numbers in a dynamical system... This is right up her alley! 😍

[McSlobo]

Old video, but you could use polar coordinates for improved visualization. Z-axis for absolute value and hue (red, yellow, green, cyan, blue, magenta, ...) for angle/phase .

[rom]

Ancient egyptians knew that PI - PHI^2 = Cubit = PI / 6. With that you can find that PHI = sqrt( 5 * PI / 6) and rewrite Binet's formula with PI as Fn = [ 1 / sqrt(5) * ((sqrt( 5 * PI / 6))^n - (sqrt( 5 * PI / 6) - 1)^n) ]

[Dodgerific]

I like the Fibonacci series where you start with 0, 0. Its easy to remember

[NyscanRohid]

φ : Let's see what's at the end of this infinite sum...
φ : π!?
π : Hey.
φ : What are you doing in complex space?
π : I work here. It's my job to be here at all times.

[punpcklbw]

The moments when his amazed face perfectly merges with himself are really trippy. Nice touch =P

[asailijhijr]

Missed opportunity: you could have had your amazed face trace the path of the graph shown on the screen at the time.

[robspiess]

Fun fact: Phi (1.618) is really close to the ratio between miles and kilometers (1.609) which means you can use adjacent Fibonacci numbers to quickly mentally convert back and forth between them.

For instance: 89 miles is nearly 144 km (it's actually 143.2), or 21 kilometers is roughly 13 miles (13.05).
You can even shift orders of magnitude to do longer distances! e.g., 210 miles is around 340 km (multiplying 21 and 34 by 10) which is close to the actual answer of 337.96 km.

[Rubrickety]

To explore this further would clearly require a large investment of time and effort. I suggest you apply for a Grant. Sanderson, ideally.

[BH-]

Well, I mean... the Fibonacci sequence was discovered thinking about the ideal procreation of rabbits, and it's pretty hard to have a negative rabbit mate with a positive rabbit

[dragoncurveenthusiast]

I have to say, I'm a tiny bit disappointed that his amazed face didn't follow the graph. It even pointed at his face!
6:45

[brandonfrancey]

I was waiting for the line, "And so I contacted Ben yet again and for some reason he blocked me and stopped responding to my e-mails."

[brianwestley]

The Fibonacci convention was huge this year -- it was as large as the previous two put together.

[haydenhoes]

this aint no sit down maths. we standin up now

[brianxx]

I've always preferred the 0, 1 start. With these numbers often found in nature, adding a moment of creation feels profound.