On 10/06/2020 23:06, Dan Asimov wrote:

Since the Gauss-Bonnet theorem states that for a compact surface M,

Integral K dA = 2π X(M)

(K = Gaussian curvature, X(M) = Euler characteristic of M),

we could in the case of constant negative curvature K = -1 (say), divide both sides by K. Then using the face that X(M) = V - E + F, we get

area(M) = 2π (-V + E - F).

So for a hyperbolic surface, that is, for a surface whose Euler characteristic is negative (the connected sum of ≥ 2 tori or of ≥ 3 projective planes), we could interpret the V, E, and F as negative (V' = -V, E' = -E, F' = -F) to get

area(M) = 2π (V' - E' + F')

if we like.

Aha, yes. That's the kind of thing I meant by

Something something Gauss-Bonnet something

in what I originally wrote. Except that you actually know what you're talking about and remembered the details of what Gauss-Bonnet says, whereas I just had a vague idea that it said the right sort of thing to make something like this work :-). Thanks! -- g