  Kirigami, the art of paper cutting, has become the subject of study in
mechanical metamaterials in recent years. The basic building blocks of any
kirigami structures are repetitive deployable patterns, the design of which has
to date largely relied on inspirations from art, nature, and intuition,
embedded in a choice of the underlying pattern symmetry. Here we complement
these approaches by clarifying some of the connections between kirigami
patterns and symmetry in terms of the wallpaper groups, the set of seventeen
plane symmetry groups that fully characterize the space of periodic tilings of
the plane. We start by showing that deployable kirigami patterns in any of the
seventeen wallpaper groups can be constructed, and then design
symmetry-preserving cut patterns that can be effectively applied for achieving
arbitrary size change throughout deployment. We further prove that different
symmetry changes can be achieved by controlling the shape and connectivity of
the tiles. Finally, we connect these results using the underlying lattice
structures of the kirigami patterns. All together, our work provides a
systematic approach for creating deployable structures with any prescribed size
and symmetry properties, thereby paving the way for the design of a wide range
of metamaterials by harnessing kirigami.
