  In this work, we investigate the symmetry of the phonon landscape of twisted
kagome lattices across their duality boundary. The study is inspired by recent
work by Fruchart et al. [Nature, 577, 2020], who specialized the notion of
duality to the mechanistic problem of kagome lattices and linked it to the
existence of duality transformations between configurations that are
symmetrically located across a critical transition point in configuration
space. Our first goal is to elucidate how the existence of matching phonon
spectra between dual configurations manifest in terms of observable wavefield
characteristics. To this end, we explore the possibility to aggregate dual
kagome configurations into bi-domain lattices that are geometrically
heterogeneous but retain a mechanically homogeneous response. Our second
objective is to extend the analysis to structural lattices of beams, which are
representative 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 the
geometry, but also on the dominant mechanisms of the cell, suggesting a
dichotomy between geometric and functional duality.
