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 groupscan 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 .
Kirigami, the art of paper cutting, has become a paradigm for mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns that derive inspiration from geometric art forms and simple planar tilings . Here we complement these approaches by directly linking kirigami patterns to the symmetry associated with the set of seventeen repeating patterns that fully characterize the space of periodic tilings of the plane. We start by showing how to construct deployable kirigami patterns using any of the wallpaper groups , and then design symmetry-preserving cut patterns to achieve arbitrary size changes via deployment. We further prove that different symmetry changes can be achieved by controlling the shape and connectivity of the tiles and connect these results to the underlying kirigami-based lattice structures . All together, our work provides a systematic approach for creating a broad range of kirigami-based deployable structures with any prescribed size and symmetry properties .
R
the subject of study in
a paradigm for
R
, 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
that derive inspiration from geometric art forms and simple planar tilings
R
clarifying some of the connections between kirigami patterns and symmetry in terms of the wallpaper groups, the
directly linking kirigami patterns to the symmetry associated with the
R
plane symmetry groups
repeating patterns
R
that
how to construct
R
in
using
R
seventeen wallpaper groupscan be constructed
wallpaper groups
R
that can be effectively applied for achieving arbitrary size change throughout
to achieve arbitrary size changes via
R
. Finally, we
and
R
using the underlying lattice structures of the kirigami patterns
to the underlying kirigami-based lattice structures
A
a broad range of kirigami-based
D
, thereby paving the way for the design of a wide range of metamaterials by harnessing kirigami