cp's OEIS Frontend

The ant lives on a centrally symmetric Penrose rhomb tiling with a "Sun" patch (S configuration, cf. A242935) at the center and starts on one of the thick rhombs of that patch, looking towards one of the outward edges of that tile. On a white rhomb, turn to the next edge of that cell in clockwise direction, flip the color of the rhomb, then move forward one unit. On a black rhomb, turn to the next edge of that cell in counterclockwise direction, flip the color of the rhomb, then move forward one unit.

In contrast to the corresponding sequences for Langton's ant on periodic tilings, like the square tiling (A255938) or a hexagonal tiling (A269757), this sequence is most likely not unique. A Penrose tiling lacks translational symmetry, meaning any two finite regions in the tiling that are identical are surrounded by different patches of tiles when examining a large enough region of the surrounding tiles. Therefore I suspect that, unless the trajectory of the ant is bounded to stay inside a finite region of the tiling, the trajectories of any two ants placed at different starting points on the tiling will diverge at some point.

The two ants meet seven times; in that case, the color of the current square is flipped only once. Eventually, both ants build a recurrent highway pattern. - Rémy Sigrist, Jul 28 2019

The ant begins on the inner corner of a subtile.

On a white tile, turn 90 degrees right, flip the color of the tile, then move forward until reaching a new tile, moving as far as possible within the tile.

On a black tile, turn 90 degrees left, then continue as above.

The chair tiling used for this automaton is, like all aperiodic hierarchical tilings, not unique (see for example Goodman-Strauss, p. 490). See "Remarks, 2019" in links for clarification which tiling the ant lives on.

The ant starts on a completely white grid.

Order of operations: turn 90 degrees, change cell color, move forward one unit.

The rules governing the movement of the ant and the change of cell colors are shown in the following table:

Cell color at | Direction of | Color to which the

start of iteration | 90-degree turn | cell is changed

-------------------+----------------+-------------------

white | right | black

black | left | blue

blue | up | yellow

yellow | down | white

The first differences of this sequence are ultimately periodic with period 28 beginning at generation 93475, see A308563. - Charlie Neder, Jun 10 2019

The ant starts on a completely white grid.

Order of operations: turn 90 degrees, change cell color, move forward one unit.

The rules governing the movement of the ant and the change of cell colors are shown in the following table:

Cell color at | Direction of | Color the cell is

start of iteration | 90-degree turn | changed to

--------------------------------------------------------

white | right | black

black | left | blue

blue | up | yellow

yellow | down | white

For the rules applying to this ant, see A325953.

The sequence is finite, with 49928 being the last term. The ant never reaches the origin again after that, since it starts building a highway pattern at iteration 93475.

On a white square, turn 90 degrees right, change the color to black, then move forward one unit.

On a black square, turn 90 degrees left, change the color to gray, then move forward one unit.

On a gray square, turn 180 degrees, change the color to white, then move forward one unit.

The complete configuration, including the position and direction of the ant and the color of all cells, has period 276. - Rémy Sigrist, Jul 19 2019

On a white square, turn 90 degrees right, change the color to black, then move forward one unit.

On a black square, turn 90 degrees left, change the color to gray, then move forward one unit.

On a gray square, turn 180 degrees, change the color to white, then move forward one unit.