Golden ratio

The golden ratio $\varphi$ is the irrational algebraic number $$\varphi = \dfrac{1+\sqrt{5}}{2}.$$

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

References

[1]
[2]