Difference between revisions of "Golden ratio"
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[https://plus.google.com/u/0/+AndrewStacey/posts/Yvki1GcVywF] | [https://plus.google.com/u/0/+AndrewStacey/posts/Yvki1GcVywF] | ||
Revision as of 19:25, 5 September 2015
The golden ratio is $\varphi = \dfrac{1+\sqrt{5}}{2}.$
Properties
Theorem: The following formula holds: $$2\sin(i \log(\varphi))=i,$$ where $\sin$ denotes the sine function, $i$ denotes the imaginary number, $\log$ denotes the logarithm, and $\varphi$ denotes the golden ratio.
Proof: █
Theorem: The following formula holds: $$2\cos(i \log(1+\varphi))=3,$$ where $\cos$ denotes the cosine function, $i$ denotes the imaginary number, $\log$ denotes the logarithm, and $\varphi$ denotes the golden ratio.
Proof: █
Videos
The Golden Ratio & Fibonacci Numbers: Fact versus Fiction
