## ornaments of eight-fold rotational symmetry

The square lattice of cells has eight directions to nearest neighbors. Four directions are vertical or horizontal and the other four go diagonally. Thus I tried to create ornaments of eight-fold rotational symmetry with my modified cellular automaton and using these eight directions. This is not so easy, because the diagonal directions are different to the horizontal and vertical ones. We can correct this problem, but eventually the basic square lattice breaks down the eight-fold rotational symmetry.

The tips growing diagonally grow faster at each step because the diagonal of the squares is longer than the side, see the figure at the left. Thus we need more growth steps for the tips growing horizontally or vertically to get equal lengths for all branches. See the figure at the right. The added steps are shown in brown. Their number is [sqrt(2)-1] times the number of the growth steps in diagonal direction.

If the side cells propagate in all eight directions we get a square, see the figure at the left. Its sides are horizontal or vertical. At each growth step, their lengths increase by two cells. If the cells propagate only horizontally or vertically we get squares too, see the figure at right. But now its sides have diagonal directions and their length increases only by one at each growth step. Using both we can get octagons. The octagon has equal sides if the number of steps in only horizontal and vertical direction is equal to sqrt(2) times the number of growth steps in all eight directions, see the figure at the left. Again, yellow shows the growth steps in all directions and the other steps are shown in brown.

My results show that overall the eight-fold rotational symmetry persists for a surprisingly long time. We find small deviations in details.

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