text: Cellular composites extend stretch-dominated lattices to the ultralight regime (below ten milligrams per cubic centimeter). Performance depends positively on the framework rigidity of the lattice, node connectivity, slenderness of strut members and the scaling of the density cost of mechanical connections.
  Conventional fiber composites make truss cores and structural frames, with bonded assembly of substructures or continuous fiber winding. Examples of such truss cores have been reported with continuous two-dimensional (2D) geometric symmetry and nearly ideal but highly anisotropic specific modulus scaling.
  Three-dimensional open-cell lattice materials occur in natural and engineered systems, spanning many length scales. Their mechanical properties scale with relative density according to the geometry. They display either stretch-dominated or transverse beam bending-dominated microstructural behavior, based on periodic mechanical models. For Young’s modulus E, ideal stretch-dominated scaling with density ρ follows a proportional law E∝ρ, while common stochastic foams follow a quadratic law E∝ρ2 otherwise associated with transverse beam bending-dominated behavior. At ultralight densities a further reduced cubic scaling law E∝ρ3 is common, such as with aerogels and aerogel composites.
  The dependence of scaling on geometry is seen in periodic lattice-based materials that have nearly ideal E∝ρ scaling, with high node connectedness relative to stochastic foams. These structures have previously been implemented only in relatively dense engineered materials. For the ultralight regime the E∝ρ2 scaling seen in denser stochastic cellular materials applies to electroplated tubular nickel micro-lattices, as well as carbon-based open-cell stochastic foams, including carbon microtube aerographite and graphene cork.