Engineering materials properties in codimension > 0

When thin nanomaterials spontaneously deform into nonflat geometries (e.g., nanorods into nanohelices, thin sheets into ruffled forms), their properties may change by orders of magnitude. We discuss this phenomenon in terms of a formal mathematical concept: codimension c = D - d, the difference between the dimensionality of space D, and that of the object d. We use several independent examples such as the edge stress of graphene nanoribbons, the elastic moduli of nanowires, and the thermal expansion of a modified bead-chain model to demonstrate how this framework can be used to generically understand some nanomaterial properties and how these properties can be engineered by using mechanical constraints to manipulate the codimension of the corresponding structure.