In a Japanese woven basket, triangles and hexagons combine into a pattern that physicists call a kagome lattice. This pattern turns out to be more than just pretty: electrons in such a material behave unusually because the lattice splits into three groups of nodes, three sublattices. A wave traveling through the crystal is delayed differently at each of them, and this affiliation determines whether the material will be a superconductor or develop charge density waves. But seeing exactly how the wave function is distributed across the sublattices remained elusive for a long time: electronic methods are blind to phase, and real crystals are too complex.

Physicists 3D-printed a kagome lattice with a lattice constant of 28 mm. Each node consists of three resonators, one for each sublattice, surrounding a central cavity. A speaker with a 1 mm aperture was placed at a node of a chosen sublattice and emitted a sinusoidal signal with a linear frequency sweep from 100 to 12000 Hz over 30 seconds. A microphone on a robotic arm sequentially scanned all nodes, recording the complex response. Measurements were repeated for each of the three sublattices, and then real space was transformed into momentum space using a two-dimensional discrete Fourier transform. This yielded the dispersion relations and the intensity distribution across the Brillouin zone.

The experimental band structure beautifully reproduces the characteristic features of the kagome lattice: two van Hove singularities at 5.5 kHz and 8.0 kHz, a Dirac cone around 6.7 kHz, and a flat band at 10.9 kHz. Fitting to a tight-binding model gave a hopping parameter t = 1.4 kHz and an on-site energy ε₀ = 8.05 kHz. At low frequencies the dispersion is linear, meaning sound perceives the lattice as a continuous medium with an effective speed of about 256 m/s. Near the Brillouin zone boundaries, the spectrum matches the tight-binding model.

Mapping the sublattice character of the flat band revealed a cyclic change in composition: A → AB → B → BC → C → CA → A along the Γ–K and Γ–M directions. In real space, at the flat band frequency, the wave is localized on two neighboring rings, and adjacent resonators oscillate in antiphase, that is, with a shift of π. This directly confirms the destructive interference mechanism: waves from neighboring nodes cancel each other, and the flat band arises precisely from such cancellation. For the van Hove singularities, it was found that the lower one (5.5 kHz) has a mixed type with pairs of sublattices, while the upper one (8.0 kHz) is pure, dominated by a single sublattice. Fourier filtering at selected M points showed that in real space the amplitude is strictly confined to the expected sublattices: for p-type, only sublattice B lights up; for m-type, A and C, forming chains at an angle of 150° with interference fringes.

Such a sonic snapshot of an invisible pattern provides the first direct evidence of the sublattice texture of the kagome band structure, including phase information inaccessible to electronic methods. Unlike neighboring metamaterial studies that focused on topological edge states or the skin effect, this work measured the sublattice polarization of the wave function across the entire Brillouin zone. It confirms theoretical predictions about the role of sublattices in correlation effects and provides a benchmark for interpreting data from complex electronic materials. It is shown that the anisotropic distribution of spectral weight on the flat band can serve as an indicator of its kagome origin, distinguishing it from trivially localized d- or f-orbitals. The technique is also applicable to designing acoustic and photonic devices with tailored localization and interference properties.