Complex Fibonacci Numbers?
Huge thanks to Jane Street! www.janestreet.com/join-jane-street/
Check out Ben Sparkss GeoGebra files.
Binet formula 2D complex output: www.geogebra.org/m/twvvzpga
3D imaginary output of Binet formula: www.geogebra.org/m/z6dy9cj5
3D plot of absolute output of Binet formula: www.geogebra.org/m/pb7hmxyd
My four-part series on Numberphile videos about Fibonacci Numbers (from 2014) starts here.
www.youtube.com/watch?v=D8ntDpBm6Ok
Here is me going on about the square root of five (Numberphile 2018).
www.youtube.com/watch?v=z1THaBtc5RE
This was the Fibonacci puzzle video from Matt Parkers Maths Puzzles.
www.youtube.com/watch?v=ILrqPpLpwpE
Read a whole bunch about «Generalized Fibonacci Sequences and Binet-Fibonacci Curves».
arxiv.org/pdf/1707.09151.pdf
The zero I found was at -9.14202391817 2.80064954276i and you can see the exact form here: www.wolframalpha.com/input/?i=(10*pi)/(pi + i*(log(sqrt(5) — 1) — log(1 + sqrt(5))))
Try it for yourself and put the Binet Formula (((1 sqrt(5))/2)^n — ((1-sqrt(5))/2)^n)/sqrt(5) in the Wolfram roots calculator: www.wolframalpha.com/widgets/view.jsp?id=b858339e64fa997454dd12f77cb1ece1
This site has everything youll ever need to know about Fibonacci Numbers.
www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html
Buttercup — The original buttercupchallenge
www.youtube.com/watch?v=l17l_KTAVJ0
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.
www.janestreet.com/
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
Support my channel and I can make more videos:
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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
Website: standupmaths.com/
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