The golden number

By Dustin Callegan on Wednesday, September 11, 2002 - 10:44 pm:

What is the meaning of the golden number?

By Julian Pulman on Thursday, September 12, 2002 - 12:50 am:

In a Fibonacci sequence, where terms are defined by the sum of its predecessor and its predecessor's predecessor (

f_{n} = f_{n - 1} + f_{n - 2}

f_{0} = 0

and

f_{1} = 1

) if you divide successive terms into each other you find that the higher the nth term, the closer you get to the golden number (we define as

ϕ = (1 + \sqrt{5}) / 2

). In fact,

f_{n} / f_{n + 1} \to ϕ

n

approaches infinity.

This "Golden Ratio" appears often in nature, owing to the fact that the difference between the golden ratio and its square is unity, and also that the difference bwteen the golden ratio and it's reciprocal is unity (giving its value as the continued fraction 1 + (1/[1+1/{1+1/...)

For more information, this link has a lot on the geometric implications.

Julian

By Gale Greenlee on Thursday, September 12, 2002 - 04:52 pm:

Dustin: Let me suggest looking under INSPIRATIONS and then "Pythagorean Golden Means", by Toni Beardon, on this very site. Gale