0
$\begingroup$

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?

enter image description here

1 Answers 1