Galois orbits in the moduli space of all triangles

Every a in the torus A = R³/2Z³ determines a unique spherical, Euclidean or hyperbolic triangle $T(a)$ with angles (π aᵢ). In this paper we study the Galois orbits Gal(a) of torsion points a ∈ A, focusing on the ramification density ρ(a) = | ∈ Gal(a) \: : \: T(b) \;is spherical\||Gal(a)| · We show that the closure R̄ of the set of values of ρ(a) is a countable subset of $[0, 1]$, with 0 and 1 as isolated points. The spectral gaps at 0 and 1 lead to general finiteness statements for the classical triangle groups Δ(p,q,r) ⊂ SL₂(R). For example, we obtain a conceptual proof, based on equidistribution, that the set of arithmetic triangle groups is finite.