Cellular automaton on quasiperiodic tiling

Any tiling can be used to define a cellular automaton. The tiles (squares, triangles, rhombs and other polygons) are simply the cells. Each tile has all other tiles with a common edge in its von Neumann neighborhood. I use the dualization method for the tiling. Then the lines of the generating grid give the interaction between the cells.

Here I show results for the stampfli tiling. All starts at the center of perfect 12-fold rotational symmetry. The state of its 12 rhombs is put equal to one and all others equal to zero. To get the new state of a cell I calculate the sum of all cells in its neighborhood. Then the rest of a division by three gives the new state. State number zero is shown as grey, one as black and three as white. I limit the cellular automaton to a finite circular region around the center. Thus I get perfect 12-fold rotational symmetry.

The results do not show the self-similar structure of the tiling. This is a bit disappointing but expected. Drawing the exact triangle, square and rhombic shapes gives rather pixellated results. Thus I am smoothing the shapes and get rose windows. Here are same examples:

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