Fig. 3: Frequency doublets of eigenmodes in the bent ring.

a Eigenmode frequencies as a function of mode number, starting from the lowest-frequency mode. Except for the zero-frequency mode (the ferromagnetic resonance, FMR), all modes appear as nearly degenerate pairs−frequency doublets. A similar pattern is observed in the Möbius strip. b Spatial profiles of an example doublet: mode #5 and mode #6. Both modes have the same wavelength and identical numbers of nodes and antinodes, but differ in the spatial positioning of their nodes and antinodes, which are shifted relative to each other along the ring. Color indicates the oscillation amplitude, with red corresponding to antinodes (maximum amplitude) and blue to nodes (minimum amplitude).