  We calculate the quantum states of regular polygons made of 1D quantum wires
treating each polygon vertex as a scatterer. The vertex scattering matrix is
analytically obtained from the model of a circular bend of a given angle of a
2D nanowire. In the single mode limit the spectrum is classified in doublets of
vanishing circulation, twofold split by the small vertex reflection, and
singlets with circulation degeneracy. Simple analytic expressions of the energy
eigenvalues are given. It is shown how each polygon is characterized by a
specific spectrum.
