Metamaterials are macroscopic systems composed of many unit cells made of conventional materials, such as metal, plastic or rubber. The unit cells and their stacking are designed in such a way that the macroscopic system will exhibit the desired properties, usually ones not exhibited by the original conventional material, including negative response, cloaking, topological insulation and programmability. In existing realizations, the metamaterial consists of a periodic stacking of unit cells, which limits their potential, as at large scales such metamaterials behave homogenously. 

Here we introduce a combinatorial strategy to create a vast number of distinct, three-dimensional mechanical metamaterials, each with a unique spatially heterogeneous texture and response. These materials consist of aperiodic stackings of anisotropic unit cells, and their functionality rests on the unit cell design and more importantly on the stacking order. 

Our experimental collaborators created such “metacubes” by 3D printing, and demonstrated that when compressed, the information embedded in the stacking order translates to a well-defined spatial texture at their surface. Conversely, when brought into contact with a rigidly textured surface, their apparent stiffness depends non-trivially on the mismatch between these patterns. 

Compatible structures where the stacking order allows the unit cells to deform without frustration are determined by a non-trivial combinatorial 3D tiling or equivalently, a spin-ice problem, and we show that the number of distinct compatible configurations already exceeds Avogadro’s number for metacubes consisting 9×9×9 unit cells. Our combinatorial design strategy can easily be extended to other unit cells, dimensions, and lattices.