apply to pairs of nonadjacent tiles. Alternatively, an undecorated tile with no matching rules may be constructed, but the tile is not connected. The construction can be extended to a three-dimensional, connected tile with no matching rules, but this tile allows tilings that are periodic in one direction, and so it is only weakly aperiodic. Moreover, the tile is not simply connected. The existence of a strongly aperiodic tile set for the Euclidean plane consisting of one connected tile without matching rules is an unsolved problem. See also Binary tiling, a weakly aperiodic tiling of the hyperbolic plane with a single tile References Aperiodic tilings Unsolved problems in geometry