Decomposition of a rhomb.

Decomposition of a square.

To find iterative methods is an amusing pastime. While shopping with wife and daughter I found a nice decomposition of rhombs, squares and octagons. Their sides are then reduced by the factor 2+sqrt(2) = 3.14, which is not related to the corresponding factor 1+sqrt(2) of the Ammann-Beenker tiling. Thus, this is a new tiling and not a mere redecoration of the squares and rhombs of the Ammann-Beenker tiling. For simplicity I use only one single color for each tile.

Decomposition of an octagon.

The new tiling has rings of octagons connected by corner points:

A new quasiperiodic tiling with eight-fold rotational symmetry containing octagons.

Because of self-similarity and arising from the iterative method most rings of eight octagons are again connected to form hyper-rings, and so on.

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