The effect of tile rotation on tile set size

Modification of the underlying state machine of Wang tiles, or the rules about the way the tiles go together, affects the size of the tile set needed for a given self-assembly. Specifically, allowing Wang tiles to rotate, so that they can attach in new ways affects the number of tiles needed in tile sets that self-assemble into limited size squares [1]. Rotation introduces complexity that must be controlled by some other mechanism. The mechanism proposed here is polarity of edge matching. In certain examples, the described change cuts the number of tiles needed by half or more. Simulations have been conducted with tilings formed with both squares and hexagons.