Covering units of aperiodic tilings

Penrose tilings have two prototiles but no example exists of a single aperiodic prototile in R2 . However, Conway showed Penrose tilings can be covered by copies of a single decagonal cluster of tiles. We extend this result to give covering units for other aperiodic tilings in various dimensions.; If copies of a cluster of tiles C can cover tiling T , it is a weak covering unit. If C covers T so that every subfacet star is covered by a single copy of C , it is a strong covering unit.; Fibonacci tilings are examined and covering units are identified, such as the smallest strong and weak covering units for every Fibonacci tiling. We also show that an inflation tiling satisfying certain criteria has a family of strong covering units which can be generated from a single strong covering unit.; Other one-dimensional tilings are constructed by projection of a subset of Z2 onto a line E . We show all such tilings have strong covering units. Longer strong covering units are constructed from the appropriate unions of smaller strong covering units. The lengths of these longer strong covering units are shown to exhibit a special relationship with a continued fraction representation for the slope of E⊥ .; Two-dimensional tilings constructed by the projection of the point lattice Z3 onto a plane are also examined. The one-dimensional results are utilized to create strong covering units of the zones of the tiling.; We then define a cluster of tiles U to be a strong semi-covering unit of T if overlapping copies of T can be used to partially cover T so that T is uniquely determined by the subset of covered tiles. We combine two zone strong covering units to create a strong semi-covering unit.; Tilings of (n - 1)-dimensional spaces E are then created by projecting the point lattice Zn onto E . The lower dimensional results are generalized, including the identification of a strong semi-covering unit W which is formed from the union of n - 1 subfacet zone strong covering units.