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Wednesday, April 23, 2008

Golden Ratio

The Fibonacci numbers are well known.

\\ \begin{matrix} n & f(n)\\ 0 & 0\\ 1 & 1\\ 2 & 1\\ 3 & 2\\ 4 & 3\\ 5 & 5\\ 6 & 8\\ 7 & 13 \end{matrix} \\ f(n)=\frac{1}{\sqrt{5}}(\frac{1+\sqrt{5}}{2})^n-\frac{1}{\sqrt{5}}(\frac{1-\sqrt{5}}{2})^n

But did you know that the function for the Fibonacci numbers is much more elegant if we explicitly use the Golden Ratio?

\\ f(n) = \frac{\phi^n-(1-\phi)^n}{\sqrt{5}}\\\\ \\ \phi = \frac{1+\sqrt{5}}{2}\\ \\ \phi * (\phi - 1) = 1\\ \\ \phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{\ddots}}}}

External Links
- Mathworld
Posted by at 7:46:00 PM
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1 comment:

  1. UnknownThursday, April 24, 2008 at 5:03:00 AM GMT+2

    I just posted a blog article about how the golden ratio and Fibonacci numbers came up in a statistical application at work today. A few minutes later I saw your post. It's amazing how Fibonacci numbers pop up unexpected.

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