@johncarlosbaez
Interesting!
I have some unfinished work, which I left lying for a couple of years, that is related.
The idea was to relate equations like the Klein Quartic (xy^3+yz^3 + zx^3) to tilings of the complex plane. The equation (u+v+w=0) can be interpreted as a triangle, and the tiling would correspond to different ways of mapping (xy^3 -> u, yz^3 -> v, zx^3 -> w).
As I mentioned, I didn’t finish this project. But I found one tantalising link to Ramanujan’s taxi cab number 1729!
The triangle pattern in the picture does not have reflection symmetry in the circles, I was still figuring out how to do that.
I have a lot of other projects, but maybe I could continue the work….