Say we have equilateral triangles coming together at a common vertex with each angle is $ 2 \pi/n= 2 \pi/6$. As the hexagon side tends to zero we have a flat plane defined with hexagonally packed tiling and the Gauss curvature of the plane
$$K_6=0$$
Now in a similar manner $K_7, K_8$ repeated 3d tiling would form surfaces of negative curvature as $ \text{side of triangle}\to 0$, a defined polyhedral solid angle forms.
How to parameterize the surface in order to visualize it?
Thanks for all pointers, how to use GB theorem and so on..