Propeller domains

A propeller domain consists of seven congruent reflected triangles: one central triangle and three arms of two triangles each. There are different coloured gluing patterns whose Dirichlet Laplacian and Neumann Laplacian spectra agree by the transplantation theorem. Varying the three side lengths of a scalene triangle in a suitable open region gives three parameters; the central triangle is identifiable from the three reflex corners, allowing a direct nonisometry proof.