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Author: Hal Snyder
Views : 231
Description: Celebrate #FibonacciDAy
Compute Environment: Ubuntu 18.04 (Deprecated)

Fun with Fibonacci numbers

Seen on Twitter: today is #FibonacciDay.

Month and day have same digits as first 4 Fibonacci numbers.

Contents:

  • BASH agrees, it's Fibonacci day.
  • Sagemath can tell you Fibonacci numbers (spoils the fun, sort of).
  • Fibonacci numbers aspire to the golden ratio.
  • Broccoli for mathematicians.
%sh # Bourne shell confirms it date +"%m-%d"
hello from init.sage 11-26

Sagemath

There is built-in support for Fibonacci numbers in Sagemath.

fl = list(fibonacci_sequence(1, 21)) fl
hello from init.sage [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765]
fr = [(fl[i+1]/fl[i]).n() for i in range(6)] fr fe = list(enumerate(fr)) fe
[1.00000000000000, 2.00000000000000, 1.50000000000000, 1.66666666666667, 1.60000000000000, 1.62500000000000] [(0, 1.00000000000000), (1, 2.00000000000000), (2, 1.50000000000000), (3, 1.66666666666667), (4, 1.60000000000000), (5, 1.62500000000000)]

the Golden Ratio

Sagemath has built in support for the golden ratio.

And the ratio of successive Fibonacci numbers approaches the Golden Ratio.

limnFn+1Fn=φ\lim_{n\to\infty}\frac{F_{n+1}}{F_n} = \varphi

# Here's a plot of the first few quotients. # Magenta line marks the golden ratio. p1 = plot(golden_ratio,(x, 0, 5), color="magenta") p2 = line(fe, marker='o', markersize=8) p1+p2

Romanesco

Romanesco Broccoli is a naturally occurring example of a Fibonacci fractal.

Image credit: Wikipedia on Romanesco broccoli