  We find the Courant-sharp Neumann eigenvalues of the Laplacian on some
2-rep-tile domains. In $\R^{2}$ the domains we consider are the isosceles right
triangle and the rectangle with edge ratio $\sqrt{2}$ (also known as the A4
paper). In $\R^{n}$ the domains are boxes which generalize the mentioned planar
rectangle. The symmetries of those domains reveal a special structure of their
eigenfunctions, which we call folding\textbackslash{}unfolding. This structure
affects the nodal set of the eigenfunctions, which in turn allows to derive
necessary conditions for Courant-sharpness. In addition, the eigenvalues of
these domains are arranged as a lattice which allows for a comparison between
the nodal count and the spectral position. The Courant-sharpness of most
eigenvalues is ruled out using those methods. In addition, this analysis allows
to estimate the nodal deficiency - the difference between the spectral position
and the nodal count.
