Nonlocal twist sequences in floppy kagome chains

The twisted kagome family comprises a spectrum of configurations that can be realized through the sweep of a single configurational degree of freedom known as a twist angle. Recently, it was shown that certain pairs of configurations along this sweep were linked by duality transformations and displayed matching phonon spectra. In this work, we introduce an intercell-connection system that spreads the lattice in the dimension orthogonal to the tessellation plane. The resulting three-dimensional character of the lattice allows us to sweep the entirety of the twist-angle spectrum, including all the compact configurations featuring overlapping triangles that, in a strictly two-dimensional space, are forbidden. Duality provides precious guidance for interpreting the availability of floppy mechanisms arising in the compact configurations through the one-to-one correspondence with their expanded counterparts. Our focus is on the compact configuration corresponding to a null twist angle, where the lattice degenerates to a chain. From the perspective of the chain, several of the local connections between neighboring lattice cells play the role of nonlocal long-range interactions between cells of the chain. We demonstrate experimentally some peculiar behavior that results from such nonlocality, including a selective activation of floppy sequences that is informed by the direction of loading.