Geometry in theoretical physics: alternatives having been tried ?

[lalbatros]

String theory opened the barrier of 4 dimensions up to 11.
This is a step outside common sense that apparently opened new horizons.

I would be interrested to know if larger steps have already been tried and if papers are available on these attempts.

For example, I have the feeling that challenging the triangular inequality would be the most audacious.
Did some physicist try to imagine the consequences of such hypothesis?

By the way, I would find it much more interresting to break such an axiom, than adding up new dimensions ...

 

[Crosson]

In mathematics, a metric space is a space in which the triangle equality holds (which is the only thing needed in order to sensibly define distance). Many people believe that on some "quantum foam" level, spacetime is not a metric space.