Phi-bonacci's Mysterious Series

 

I'm fascinated and intrigued by Leonardo Bonacci who was, oddly, called 'Fibonacci', possibly because he was filius (son of...) Bonacci, .  He introduced what we now call 'Arabic numerals' to Europe in 1202 in what was the forerunner of all mathematics texts in the Western World, the Liber Abaci.  In that book he introduced what today is known as 'the Fibonacci sequence':

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, &c.,

wherein each new term is the sum of the previous two, e.g.:

2 + 3 = 5.

And that about sums up what we know about Leonardo Bonacci.

What he didn't put in the book is also significant, mostly because he probably didn't realize it.  Understand that computing with Arabic numerals was brand-new in Europe, even for Bonacci.  The thing he missed was that the limit of the ratios of the Fibonacci sequence homes or 'asymptotes' to 1.6180339887..., e.g.:

832040 / 514229 = 1.6180339887...

"Big deal," I just heard someone mutter.  Well, yes, when you consider that the inverse is almost exactly identical:

514229 / 832040 = 0.6180339887... 514229 / 832040 = (832040 / 514229) - 1 Slightly weird.

Mathematicians find this number (1.618...) very useful and it lends itself to beautiful graphic explanations as well.  Since it's used so often, it has become a convention to use the Greek letter phi (Φ) to represent 1.618033... and its lower case form (φ) to represent 0.618033...  That means that

Φ - φ = 1 and Φ × φ = 1.

This number, Φ, is also called 'the golden ratio' and is found nearly everywhere you look, especially in architecture, botany, biology, and meteorology.  Φ is implicated in the spiral shapes of the chambered nautilus, sunflowers, and hurricanes.

YouTube has many great short videos that talk about Φ-bonacci, the Φ-bonacci sequence, and the Golden Ratio.