I am addicted to iterative methods. They are easy to program and yield surprising results. I like to work out new iterative schemes, which have often their particular beauty. But the great suspense arises when running them first time on the computer. Like the seed of same exotic plant one never nows what will come out of it.
Iterative methods for quasiperiodic tilings in general use several regular polygons, rhombs and other polygons as tiles. They are then dissected into smaller polygons. Some of the smaller polygons are half inside and half outside of the large polygon. For the Amman-Beenker tiling the squares are thus divided into triangles, see “An efficient iterative method for the Ammann-Beenker tiling“. There the length of the sides of the polygons is reduced by a factor of 1+sqrt(2)=2.41 for each iteration step. Note that this factor and all its powers are irrational numbers. This is typical for quasiperiodic tilings.
For quasiperiodic tilings with eight-fold rotational symmetry there arise other factors too. In “A tiling of octagons, squares and rhombs” I have shown a tiling with a factor of 2+sqrt(2)=3.41 and in “Iteration of rhombs: Filling the gap” the factor is sqrt(2+sqrt(2))=sqrt(3.41)=1.84. I expect that it is more difficult to find iteration schemes with a smaller factor. Thus I wanted to make up more tilings with the same factor as the Ammann-Beenker tiling.
An octagon can be dissected into eight octagons which share their sides and have their centers at the corners of the large octagon. Their sides are smaller than the large octagons by this factor of 1+sqrt(2), see the figure. In the center there is a large gap with the shape of two squares making an eight-pointed star. This star-like shape can be dissected into octagons if we allow octagons to overlap. After two iterations, an octagon looks like this:
The resulting tiling looks like this: