It has been proposed that Floquet topological states can be used to generate period-doubled oscillations that break discrete time-translation symmetry. This behavior originates from topological $\pi$ modes—boundary states pinned at half the driving frequency. Although such oscillations resemble those in time crystals, the underlying mechanisms are different: the period-doubled Floquet edge states arise from a single-particle Hamiltonian and are protected by the space-time symmetries of the lattice. Recently, period-quadrupled oscillations have been realized in acoustic and photonic systems, governed by $\pi /2$ modes—boundary states pinned at one quarter of the driving frequency. These studies establish the sequence $2T$ to $4T$, where $T$ is the driving period, and further generalizations within the same framework lead to $8T, 16T$, and other powers of two, rather than to $3T$ or $6T$. In this work, we realize another set of edge states—$2\pi /3$ and $\pi /3$ modes, with quasienergies pinned at $\pm 2\pi /3T$ and $\pm \pi /3T$. Using designed acoustic waveguides, we observe $3T$- and $6T$-periodic responses from an initially localized boundary excitation, providing evidence for the existence of these states.

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• Acoustic metamaterials
• Edge states
• Symmetry protected topological states
• Topological materials