  We address the question of determining the eigenvalues $\lambda\_n$ (listed
in nondecreasing order, with multiplicities) for which Courant's nodal domain
theorem is sharp i.e., for which there exists an associated eigenfunction with
$n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to
Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat
equilateral torus are the first and second, and that the only Courant-sharp
Dirichlet eigenvalues of the equilateral triangle are the first, second, and
fourth eigenvalues. In the last section we sketch similar results for the
right-angled isosceles triangle and for the hemiequilateral triangle.
