Archimedean cages, polyhedra, and nanotube structures and methods

A method of defining a convex equilateral cage comprising:

• selecting an initial cage comprising a plurality of edges extending from vertices, wherein the edges define a plurality of faces;
• defining a secondary cage comprising edges and vertices from a plurality of tiles, reduced tiles, or larger patches obtained or derived from a selected one of the uniform Archimedean tilings, wherein the plurality of tiles, reduced tiles, or patches are positioned over the plurality of faces of the initial cage;
• resizing at least some of the edges of the secondary cage such that all of the edges of the secondary cage have the same length, the resized cage defining a plurality of polygons, wherein at least some of the plurality of polygons are not planar;
• solving for a set of interior angles for the plurality of polygons that transforms the plurality of polygons to be planar or solving for a set of vertex coordinates for the plurality of polygons that transforms the plurality of polygons to be planar;
• and
• constructing a convex equilateral cage structure comprising members defining edges having the same length and oriented according to the set of interior angles or the set of vertex coordinates.