  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 twist angle. Recently, it has been shown that certain pairs of
configurations along this sweep are linked by duality transformations and
display matching phonon spectra. In this work, we introduce an inter-cell
connection system that spreads the lattice in the dimension orthogonal to the
tessellation plane. The resulting 3D 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 2D space,
are forbidden. Duality provides precious guidance in 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
non-local long-range interactions between cells of the chain. We demonstrate
experimentally some peculiar behavior that results from such non-locality,
including a selective activation of floppy sequences that is informed by the
direction of loading.
