## The basic Amman-Beenker tiling

With the details presented in the earlier posts we can easily get large parts of this tiling. I have tried to choose colors, which are not too contrasting and too hard on the eyes:

Basic Amman-Beenker tiling of 8-fold rotational symmetry.

We see that small patterns are repeated throughout in a rather systematic but not periodic way. Thus this tiling is “quasiperiodic”. Note particularly the stars made of eight rhombs with a common corner. They are always surrounded by eight squares and eight rhombs. Further out the patterns vary. But larger parts of the tiling repeat too – we have just to search further away.

The same distances appear throughout between the stars of rhombs. They lie at the corners of larger rhombs and squares. Really, the centers of these stars are the points of the same tiling, but of larger lengths:

The centers (red dots) of the stars of the tiling are points of a same tiling with larger lengths. Red lines show the edges of the new tiles. Squares of the new tiling darken the basic tiling and rhombs bleach it out.

For the large rhombs and squares we see always the same pattern of smaller squares and rhombs. The pattern inside the square is not symmetric and has four different orientations. We easily get that the edges of the large tiling are larger by a factor of (3+2*sqrt(2)) than the edges of the basic tiling.

Obviously, the stars of the larger tiling are again corners of the tiles of an even larger tiling, and so on. All these tilings have the same shape and thus the Amman-Beenker tiling is self-similar at all scales in the same way as some fractals, such as the Sierpinsky triangle. Thus we can make a quasiperiodic tiling with eight-fold rotational symmetry with only squares of different sizes.

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