The control of structure on the nanoscale relies on intermolecular interactions whose specificity and geometry can be treated on a predictive basis. With this criterion in mind, DNA is an extremely favorable construction medium: The sticky-ended association of DNA molecules occurs with high specificity, and it results in the formation of double helical DNA, whose structure is well known. The use of stable branched DNA molecules permits one to make stick-figures. We have used this strategy to construct a covalently closed DNA molecule whose helix axes have the connectivity of a cube. The molecule has twelve double helical edges; each edge is two helical turns in length, resulting in a hexacatenane, each of whose strands corresponds to a face of the object. The cube has been fabricated in solution, which is inefficient. We have developed a solid-support-based synthetic methodology that is much more effective. The key features of the technique are control over the formation of each edge of the object, and the topological closure of each intermediate. Each edge results from the restriction of two hairpins, which are then ligated together. The isolation of individual objects on the surface of the support permits one to use both symmetric and asymmetric sites in the formation of edges that close polygons. We have used solid-support-based methodology to construct a molecule whose helix axes have the connectivity of a truncated octahedron. This figure has 14 faces, of which six are ideally square and eight are hexagonal; this Archimedean polyhedron contains 24 vertices and 36 edges. Control of topology is strong in this system, but control of 3-D structure remains elusive. Topological control is enhanced by the use of topological protection techniques. Our key aim is the formation of prespecified 2-D and 3-D periodic structures with defined topologies. Applications envisioned include nanomanipulators and scaffolding for molecular electronic devices.