KirigamiProgramme.zip from Folding a tessellated uniform-thick plate into compact stacks

Many large deployable rigid arrays for aerospace and civil applications, e.g. solar panels of satellites, reflectarray antennas and foldable bridges, are preferably constructed from identical panels of uniform thickness. Achieving compact packing for easy transportation remains a significant challenge. This paper develops a mathematical framework to determine when a monohedral tessellated surface of uniform thickness can be rigidly folded into two compact stacks. We first divide the planar surface into a tessellation and define the cuts and folds along a Hamiltonian Circuit (HC) planned within the tessellation, producing a loop of panels. We then establish the necessary and sufficient conditions for the panels to be rigidly folded into two stacks. The first condition ensures that the panels can be folded into two equal stacks by making an imaginary cut along one edge and verifying whether the two ends of the resulting chain rejoin after an equal number of folds. The second condition requires the absence of entanglements or twists during folding when the loop is not cut open. The framework is validated through analytical examples and physical prototypes. This work advances the theoretical foundations of kirigami-inspired deployable structures and provides practical design guidelines for compact foldable planar structures.