The Spatial Lattice Design from a Tetrapod-Shaped Element

Two discoveries demonstrate that hexagon-type spatial rigid bar structures built of tetrapod-shaped superelements are the best for solving of optimal lattice and optimal two dimensional multiscale tessellation problems. Unexplored, however, remain the topological transformations and multiscale tessellation possibilities of regular spatial lattices space in analogy to the 2-dimensional transformations considered so far. Therefore this article concentrates specifically on derivation of hexagon-type spatial lattices and the multiscale tessellation compatibility possibilities of the spatial lattice cells obtained. In order to determine the shapes of the regular spatial hexagonal lattices, two choices are available. The first one is to apply topological transformations to spheres, placed at regular points in space. The second one is to search for all unique combinations of the superelements. However, both of the approaches ask for proof that all possible lattice configurations have been found. A combined approach is to use the 2-dimensional hexagonal lattice as a starting point for expanding to three dimensions. This yields two unique spatial hexagon-type cell analogues and, consequently, two lattice variants. Considering multiscale tessellation possibility of the lattices, an examination of multiscaled lattice cell vertex coincidence reveals that repeating patterns do exist for the second lattice variant.