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In this work, we investigate the symmetry of the phonon landscape of twistedkagome lattices across their duality boundary. The study is inspired by recentwork by Fruchart et al. [Nature, 577, 2020], who specialized the notion ofduality to the mechanistic problem of kagome lattices and linked it to theexistence of duality transformations between configurations that aresymmetrically located across a critical transition point in configurationspace. Our first goal is to elucidate how the existence of matching phononspectra between dual configurations manifest in terms of observable wavefieldcharacteristics. To this end, we explore the possibility to aggregate dualkagome configurations into bi-domain lattices that are geometricallyheterogeneous but retain a mechanically homogeneous response. Our secondobjective is to extend the analysis to structural lattices of beams, which arerepresentative of realistic cellular metamaterials. We show that, in this case,the symmetry of the phonon landscape across the duality boundary is broken,implying that the conditions for dual behavior do not merely depend on thegeometry, but also on the dominant mechanisms of the cell, suggesting adichotomy between geometric and functional duality.

Symmetry of the phonon landscape across the duality boundary of twisted kagomes lattices