Inspired by Nature’s ability to make complex structures, self-assembly has emerged as a powerful technique for synthesizing nanoscale materials. While living systems are brimming with examples of functional nanomaterials that have one or more self-limited dimensions, such as virus capsids, cytoskeletal filaments, or complex molecular machines, most examples of synthetic self-assembly tend to produce periodic lattice structures with macroscopically uncontrolled dimensions. In this talk, I will describe one route to engineering the self-limiting assembly of synthetic materials based on self-closure, in which the subunits assemble with a preferred curvature that drives the growing assembly to close upon itself at a specific dimension. In particular, I will explore the trade-offs between geometric specificity and interaction specificity using a specific class of self-closing structure--helical tubules. We make triangular subunits using DNA origami that have specific, valence-limited interactions and designed binding angles, and study their assembly into tubules that have a self-limited width and pitch that can be arbitrarily large compared to the size of an individual subunit. In contrast to crystallization, we find that the self-closing assembly of tubules is very susceptible to polymorphism, and that a simple colloidal mixture can produce an ensemble of structures with different widths, helicity, and chirality. By effectively balancing geometric specificity and interaction specificity, I will show how this type of polymorphism can be tamed by increasing the number of subunit species to yield a single target geometry with 100% yield using a minimal number of distinct components. I will conclude by showing how these ideas can be generalized to other complex, self-limiting geometries, like helicoids, toroids, and periodic porous frameworks using a kirigami-like design approach. Together, our findings highlight the essential roles that dynamics, geometry, and assembly complexity play in the programmable assembly of self-closing structures.