A vented honeycomb structure with a plurality of honeycomb cells arranged in a hierarchical order and having a plurality of truss walls, each truss wall including a plurality of members. The vented honeycomb structure is fabricated by joining a plurality of sheets of trusses using any one of an expansion, a corrugation, and a slotting process. Fabrication can also occur by deposition, casting, additive, extrusion, or aligning and joining methods. The honeycomb cells, truss walls, truss wall openings, and truss wall members can be functionally graded.