We know that $i^i$ is real. But how to explain it geometrically maybe in terms of rotation. like we can explain geometrically multiplication of two complex numbers and so on. Can someone show me a little bit about geometric interpretation of $i^i$ and tell me if what I think below is true?
Note :(Some informations): It is equal to $e^{i \cdot \log(i)}$, but $\log$ is only well-defined up to adding integer multiples of $2 \pi i$. Thus in the correct setting $i^i$ is each of the numbers $e^{-\frac{(4n+1)\pi}{2}}$, $n$ an integer. Each of those are equally valid, thus finding a geometric interpretation would by a bit silly.
Thank you for any help
Your note is precisely the point. There is a geometric meaning for addition and multiplication and real powers, but imaginary exponents don't have any because they must be defined in terms of $\exp$ and $\ln$ or something equivalent. And there is no geometric interpretation of $\exp$.