Here's a math puzzle: I have a two-dimensional sphere, the unit sphere. I want to form a geodesic triangulation such that there are at least N vertices, and the edge length connecting vertex \(i\) with vertex \(j\) is denoted \(\ell_{ij}=|e_{ij}|\) but specifically satisfies \(|\ell_{ij}-\ell|<\varepsilon\) where \(\varepsilon\) is a given tolerance (we can assume \(0<\varepsilon\ll\ell\)).

Puzzle: is there a relation or equation expressing the number of vertices \(N\) with the maximum deviation from equilateral triangles \(\varepsilon\)?

I'm curious about the cases where \(5810\leq N<10^{4}\) and \(\varepsilon\lesssim 10^{-4}\), if this helps any.

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