term "aperiodic hierarchical tiling" is a convenient shorthand, meaning something along the lines of "a set of tiles admitting only non-periodic tilings with a hierarchical structure". Each of these sets of tiles, in any tiling they admit, forces a particular hierarchical structure. (In many later examples, this structure can be described as a substitution tiling system; this is described below). No tiling admitted by such a set of tiles can be periodic, simply because no single translation can leave the entire hierarchical structure invariant. Consider Robinson's 1971 tiles: Any tiling by these tiles can only exhibit a hierarchy of square lattices: the centre of any orange square is also a corner of a larger orange square, ad infinitum. Any translation must be smaller than some size of square, and so cannot leave any such tiling invariant. Robinson proves these tiles must form this structure inductively; in effect, the tiles must form blocks which themselves fit together as larger versions of the original tiles, and so on. This idea — of finding sets of tiles that can only admit hierarchical structures — has been used in the construction of most known aperiodic sets of tiles to date. Substitutions Substitution tiling systems provide a rich source of aperiodic tilings. A set of tiles that forces a substitution structure to emerge is said to enforce the substitution structure. For example, the chair tiles shown below admit a substitution, and a portion of a substitution tiling is shown at right below. These substitution tilings are necessarily non-periodic, in precisely the same manner as described above, but the chair tile itself is not aperiodic—it is easy to find periodic tilings by unmarked chair tiles. However, the tiles shown below force the chair substitution structure to emerge, and so are themselves aperiodic. The Penrose tiles, and shortly thereafter Amman's several different sets of tiles, were the first example based on explicitly forcing a substitution tiling structure to emerge. Joshua Socolar, Roger Penrose, Ludwig Danzer, and Chaim Goodman-Strauss have found several subsequent sets. Shahar Mozes gave the first general construction, showing that every product of one-dimensional substitution systems can be enforced by matching rules. Charles Radin found rules enforcing the Conway-pinwheel substitution tiling system. In 1998, Goodman-Strauss showed that local matching rules can be found to force any substitution tiling structure, subject to some mild conditions. Cut-and-project method Non-periodic tilings can also be obtained by projection of