Abstract: Regular tetrahedral lattices (RTL's) can be used as building blocks in continuous one, two, and three-dimensional structures of superior lightness, strength, and rigidity. Such lattice structure can be thought of as consisting of at least 3 imaginary, edgeless cubic cells, wherein each cell contains a pair of RTL's and each RTL pair is formed entirely from six right-angle crossing elements of equal arm length and joined at eight vertices. Each of the eight vertices corresponds to a corner of one imaginary cubic cell and each of said cubic cells, together with each of said RTL pairs, shares an interface with at least one other cubic cell and RTL pair of said lattice structure. These structures can be formed into straight-line chains, make ninety-degree turns and three-dimensional right-angle crosses, thereby lending themselves to conventional construction with simple fabrication methods and rapid repairs in place.