If you have $xy = i $ and $x^2 + y^2 = 1$ then you get the solutions that have the golden ratio in them.
Wolfram calculation: https://www.wolframalpha.com/input/?i=xy%3Di%2C+x%5E2%2By%5E2%3D1
What is exactly happening here? Why is that $-1$ in $ -1+\sqrt{5} $ instead of $+1$. Is there some spiral going on? Hopefully someone can show me what is happening here, I'm quite curious. I also don't really know how to visualise where $xy=i$ and $x^2 + y^2 =1 $ intersect.
It probably has to do with self-similarity, since that is basically the gist of the golden ratio.
The polynomial you get from those equations is $x^4 -x^2 -1 = y$, which is basically really similar to the golden ratio polynomial.
Also if you swap the constants: $x^2 + y^2 = i$ and $xy=1$, you seem to rotate the solutions wolfram calculation
Since $e$ is the hyperbolic constant and $\pi $ the circle constant, can you relate them all nicely in one equation?
Summing my questions and making them concrete:
- Why exactly is the golden ratio showing up here geometrically?
- How can you visualise what a 'complex' hyperbola and a 'real' circle is. They seem their 'conjugates' almost.
- Is there a way to better understand what is going on here? It is probably quite trivial but it seems very interesting.
$xy$ is basically a square that you squish and stretch while keeping the area constant. Maybe you are somehow constructing a golden square, but I don't clearly see it.
Thanks in advance!
EDIT: I still don't quite understand it, but I found this that maybe is related: Ideal triangle