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Owing to smooth surface transitions and open-cell connectivity, minimal surfaces can be conveniently designed to form singly, doubly, triply periodic, and quasi-periodic structures, that provide new opportunities for novel metamaterial design of potential engineering relevance. Prior studies have demonstrated band gaps and engineered mechanical properties in minimal surface structures. However, their potential for the exploitation of topological phenomena through elastic waves remains largely unexplored. In this work, we design periodic and quasi-periodic minimal surface metamaterials and explore their topological properties. We start with the construction of 1D/2D periodic minimal surfaces with dimerized-like parametrizations. The designs facilitate the opening of band gaps upon breaking the inversion symmetry of the unit cells through a band inversion process that defines distinct topological phases. Simulations demonstrate the existence of 0D localized modes in 1D interfaced structures, and robust waveguiding along a valley-type 1D interface separating distinct 2D domains. These designs are fabricated with additive manufacturing technologies and tested with laser vibrometry, confirming the presence of the predicted topological states. We then investigate 1D quasi-periodic minimal surfaces through a quasi-periodic modulation of the dimerization parameter. Such structures support topological gaps forming a fractal spectrum that resemble the Hofstadter butterfly. The existence of non-trivial gaps and localized modes in quasiperiodic systems extend the avenues for wave localization and transport exploring higher dimensional topologies. With the growth of additive manufacturing techniques, the presented framework of minimal surfaces provides remarkable design freedom to explore a variety of symmetries in 1D, 2D, and 3D domains, enabling a variety of other wave physics and topological phenomena to be explored.
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