Da Vinci, Fibonacci, Assisi

I just finished reading Dan Brown's thriller, "The da Vinci Code". Nice book. Well crafted. Great thriller.

Essential to the plot are the "Da Vinci" numbers.......sorry, wrong historical Figure, I meant the "Fibonacci" numbers.

Those that go 1, 1, 2, 3, 5, 8, 13, 21....etc, each being the sum of the previous two. Remarkable? Not really.........unless you look closely at the world around you. These Fibonacci numbers are an integral part of the world we live in, as the book hinted at. Look at the spirals of the seeds in simple sunflowers, the way a tree branches, and you will find in the natural world, this number sequence is ubiquitous. If you take the ratio of successive terms, eg, 8/5, or 13/8, you get closer and closer to a "limiting number". This number is approximately 1.618. It is, of course, irrational.

Moreover, it is related to an entirely different branch of math.

Take a line and divide it into two parts "a" and "b" such that the ratio a/b is the same as b/(a+b). Simple algebra, upon substituting 1 for "a" will yield a simple quadratic equation. The roots to this equation are b = (1 +/- SQRT(5))/2, which gives us 1.618033988 and -0.618033988.

1.618. This is the exact same number from the Fibonacci sequence above. This did not go unnoticed by the Greeks, who called this number "The Golden Ratio", and gave it a letter like Pi. They called it Phi, another Greek letter. If you make a rectangle that is most pleasing to the eye, chances are, it will be close to a 3 by 5 rectangle, or an 8 by 13 rectangle. In other words, the "most pleasing" rectangle will be 1 by 1.618....

There's more.......

Phi is the only number in the cosmiverse to become its own reciprocal (that is, 1/phi) if 1 is subtracted.

(Phi - 1) equals (1/phi).

Before leaving sequences, check this one out -

1, 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, etc

It is a simply defined sequence, yet looks really erratic, especially when one gets to about 30 terms. I will reveal how it is generated in a future post. Meanwhile try and figure how it is generated. Hint - it is a relation to the Fibonacci sequence.


In closing, using cooking parlance, one might refer to both "Phi" and "Pi" as being as versatile as an egg.

Assisi.