Let's take a graph (with vertices and edges), which is in some sense 2D. In the simplest case consider a flat square lattice (or triangular, or hexagonal). But as well it can be a graph on a sphere (e.g. icosahedron) or a tessellation, e.g. uniform tilling in hyperbolic plane.
Is there a way to assign its Gaussian curvature? (Especially if it is constant.)
For a simple case, inspired by a game HyperRogue (a screenshot is below):
- each vertex (tile) has 6 or 7 edges (neighbouring tiles),
- length of each edge is 1 (it takes one turn to get to every neighbouring site).
It that case, how does suffice to know ratio of heptagons to hexagons?
The standard way to assign a curvature would be to build a piecewise linear surface, where each face of the graph is a regular $n$-gon. Then the curvature would be sitting on the vertices, and the curvature at a vertex $v$ would be $K(v)= 1 - \frac{\alpha(v)}{2\pi}$, where $\alpha(v)$ is the sum of the angles of the faces at $v$.