A method for designing a structure includes selecting an initial cage, defining a secondary cage by positioning a plurality of tiles, reduced tiles, or larger patches obtained or derived from a selected one of the uniform Archimedean tilings over the faces of the initial cage, resizing edges of the secondary cage such the cage is equilateral, and planarizing the cage faces. In an embodiment the patch comprising a network of edges and vertices from a uniform tiling decorates the faces of a polyhedron to define a non-polyhedral cage that is transformed by planarizing the faces. In an embodiment the secondary cage comprises tiles derived from an Archimedean tiling that decorate faces of the initial cage comprising a polyhedron. In an embodiment the secondary cages resemble a nanotube.