Doubling the tessellation of octagons and squares

Regular octagons and squares make up a well-known semiregular tessellation that is often used to decorate floors

Semiregular tessellation of octagons and squares.

Its dual grid is essentially a square grid with diagonal lines added.

Dual grid of the tessellation.

Four lines cross at the corner points giving the octagons of the tesselation.The crossing of the diagonals corresponds to the squares.

We now put two such grids together, rotated at an angle of 45 degrees with respect to each other. Using the dualization method we get a quasiperiodic tiling with 8-fold rotational symmetry.

Quasiperiodic tiling obtained from doubling the semiregular tessellation of octagons and squares.

Note the distinct rings of eight octagons connected by a common corner. This is different to the earlier tiling presented in “A tiling of octagons, squares and rhombs” where the octagons shared a common side. How do these rings of octagons arise here ? Let us look at the dual grid of the tiling.

Dual grid of the tiling made of two dual grids of the periodic tessellation. The center of a ring of octagons is indicated in yellow. The crossings of lines at the red points give the octagons.

Surrounded by the yellow line we find intersections of several lines. They result in two squares and four rhombs making up a dissected octagon at the center of the ring of octagons. Further out, perpendicular crossings correspond to squares of the tiling. Then there are four lines crossing in one point (indicated in red) making the octagons. Clearly, a ring of eight octagons arises if the squares here shown with black sides form an eight pointed star.

Note that this is the same condition as for stars of eight rhombs in the Ammann-Beenker tiling. Thus here the centers of the rings of octagons are the corner points of an Ammann-Beenker tiling. In the picture of the tiling above you can draw rhombs and squares that have their corners at the centers of the rings of octagons. Although this tiling is closely related to the Ammann-Beenker tiling it has a rather different appearance.

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