Higher-order topological insulators have been widely explored in classical wave systems, characterized by topological states localized at hinges or corners. Conventional approaches to realizing higher-order corner states necessitate bulk materials, which typically result in inefficient use of the bulk. Here we introduce topological vertex states in three-dimensional hollow pyramid structures, achieved by tailoring and folding planar phononic crystals. These vertex states are highly localized at the vertices of hollow pyramid structures formed by the connection of capsule-shaped acoustic cavities. The frequencies of these vertex states can be precisely tuned by deforming the pyramid geometry or integrating external tubes. Interestingly, the pyramids support anomalous acoustic wave partitions of hinge states that do not follow the original planar valley-selective transport rule, stemming from the emergence of projected two-dimensional topologically nontrivial valley configurations of distorted lattices. Notably, the partition ratio can be effectively tuned by the vertex states. Our approach provides an efficient alternative for wave localization and manipulation, reducing bulk material usage while enabling the design of lightweight and compact detectors for advanced sensing applications.

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• Acoustic metamaterials
• Kagome lattice
• Phononic crystals
• Topological materials