  The ability to change a surface's profile allows biological systems to
effectively manipulate and blend into their surroundings. Current surface
morphing techniques rely either on having a small number of fixed states or on
directly driving the entire system. We discovered a subset of scale-independent
auxetic metamaterials have a state trajectory with a star-graph structure. At
the central node, small nudges can move the material between trajectories,
allowing us to locally shift Poisson's ratio, causing the material to take on
different shapes under loading. While the number of possible shapes grows
exponentially with the size of the material, the probability of finding one at
random is vanishingly small. By actively guiding the material through the node
points, we produce a reprogrammable surface that does not require inputs to
maintain shape and can display arbitrary 2D information and take on complex 3D
shapes. Our work opens new opportunities in micro devices, tactile displays,
manufacturing, and robotic systems.
