A308590 - OEIS

%I #14 Jul 23 2019 08:45:37

%S 0,1,2,3,4,5,4,5,6,7,8,9,8,7,6,7,6,7,8,9,8,9,10,11,10,9,8,7,8,7,8,9,

%T 10,9,10,11,12,11,12,13,14,13,14,15,16,15,16,17,18,17,16,15,14,13,12,

%U 11,12,11,12,13,14,13,14,13,14,15,14,13,12,13,12,13,14

%N Langton's ant on a Penrose rhomb tiling: number of black cells after n moves of the ant.

%C 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.

%C 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.

%H Felix Fröhlich, <a href="/A308590/a308590.pdf">Illustration of iterations 0-72 of the ant</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Langton%27s_ant">Langton's ant</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Penrose_tiling">Penrose tiling</a>

%e See illustration in links.

%Y Cf. A255938, A269757, A325953, A325954, A325955.

%K nonn

%O 0,3

%A _Felix Fröhlich_, Jun 09 2019