Table 1. General classification of lattices with uniform soft twistings.

The spatially varying floppy modes are expressed in terms of the wave number in the y direction when a plane wave of wave number k propagates in the x direction (see discussions after equation (4)). The bulk phonon spectra show example phonon frequency contour plots (darker colour for lower frequency) as a function of k_x, k_y. The example lattices are shown as rigid polygons (triangles or parallelograms) connected by free hinges at their corners⁴⁵, and they can be directly mapped into strut-hinge frames by replacing the triangles by three connected struts on their edges and the parallelograms by five connected struts with four on edges and one on the diagonal to make them rigid. Thus the structures consisting of triangles (deformed kagome lattices as defined the text) have 〈z〉=4=2d and the structures consisting of parallelograms (deformed checkerboard lattice) have 〈z〉=5>2d.