 (the term "cyclic list" indicates that the last letter may pair with the first one) The proof involves Truchet tiles.The hexagonal grid permits up to six different rotations, which are notated here as N (no change), R1 (60° clockwise), R2 (120° clockwise), U (180°), L2 (120° counter-clockwise), L1 (60° counter-clockwise).A further extension of Langton's ants is to consider multiple states of the Turing machine – as if the ant itself has a color that can change.