  We consider triangulations of closed surfaces in which every vertex is
incident to exactly $d$ edges.
  These triangulations can be identified with subgroups of the triangle group
$\langle a,b,c\mid a^2,b^2,c^2,(ab)^3,(ac)^2,(bc)^d\rangle$ that intersect
$\langle a,b\rangle$, $\langle a,c\rangle$, and $\langle b,c\rangle$ trivially.
  The term geodesic duality refers to an external symmetry introduced by Wilson
in 1979. Our main result is the characterisation of all subgroups corresponding
to geodesic self--dual regular triangulations, together with a complete
enumeration for $d < 10$.
