  Let $\Delta=\Delta(a,b,c)$ be a hyperbolic triangle group, a Fuchsian group
obtained from reflections in the sides of a triangle with angles
$\pi/a,\pi/b,\pi/c$ drawn on the hyperbolic plane. We define the arithmetic
dimension of $\Delta$ to be the number of split real places of the quaternion
algebra generated by $\Delta$ over its (totally real) invariant trace field.
Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic
dimension $1$, corresponding to the arithmetic triangle groups. We show more
generally that the number of triples with fixed arithmetic dimension is finite,
and we present an efficient algorithm to completely enumerate the list of
triples of bounded arithmetic dimension.
